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href="../2122/21-22-introducing-functions-.html">2.1 & 2.2 Introducing Functions </a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2127/23-24-key-features-of-functions-and-graphing-.html">2.3 & 2.4 Key Features of Functions And Graphing </a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2207/25-linear-models.html">2.5 Linear models</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2256/25-quadratic-models.html">2.5 Quadratic Models</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2257/25-exponential-models.html">2.5 Exponential Models</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2186/25-trigonometric-models-.html">2.5 Trigonometric Models </a></label></li></ul></li><li class=""><label style="padding-left: 0px"><a class="expander" href="#" style="font-size: .9em"><i class="fa fa-fw fa-caret-right"></i></a><a href="../1070/geometry-and-trigonometry.html">Geometry and Trigonometry</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2398/31-volume-and-surface-area.html">3.1 Volume and surface area</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2260/32-right-angled-trigonometry.html">3.2 Right angled Trigonometry</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2261/32-non-right-angled-trigonometry.html">3.2 Non-right angled trigonometry</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2196/33-applications-of-trigonometry.html">3.3 Applications of trigonometry</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../1862/34-circles.html">3.4 Circles</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2120/35-36-perpendicular-bisectors-voronoi-diagrams-.html"> 3.5 & 3.6 Perpendicular Bisectors & Voronoi Diagrams </a></label></li></ul></li><li class=""><label style="padding-left: 0px"><a class="expander" href="#" style="font-size: .9em"><i class="fa fa-fw fa-caret-right"></i></a><a href="../1065/statistics-and-probability.html">Statistics and Probability</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2188/41-statistical-concepts.html">4.1 Statistical concepts</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2099/42-43-cumulative-frequency-box-plots-.html">4.2 & 4.3 Cumulative Frequency & Box Plots </a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2189/43-central-tendency-and-dispersion.html">4.3 Central tendency and Dispersion</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2266/44-bivariate-data-linear-correlation.html">4.4 Bivariate data & Linear Correlation</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2258/45-46-probability.html">4.5 & 4.6 Probability</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2190/47-discrete-random-variables.html">4.7 Discrete random variables</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2403/48-binomial-distribution.html">4.8 Binomial Distribution</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2191/49-normal-distribution.html">4.9 Normal Distribution</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2407/410-spearmans-rank-correlation-coefficient.html">4.10 Spearman's Rank Correlation Coefficient</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2259/411-chi-squared-independence-tests.html">4.11 Chi Squared Independence tests</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2413/411-chi-squared-goodness-of-fit-and-the-t-test.html">4.11 Chi Squared Goodness of Fit and the T-Test</a></label></li></ul></li><li class=""><label style="padding-left: 0px"><a class="expander" href="#" style="font-size: .9em"><i class="fa fa-fw fa-caret-right"></i></a><a href="../1078/calculus.html">Calculus</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2400/51-53-introduction-to-calculus.html">5.1 & 5.3 Introduction to Calculus</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2401/52-54-tangents-normals-and-the-second-derivative.html">5.2 & 5.4 Tangents, normals and the second derivative</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2402/56-57-stationary-points-and-optimisation.html">5.6 & 5.7 Stationary Points and Optimisation</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2197/55-58-integration-and-trapezoidal-rule.html">5.5 & 5.8 Integration and Trapezoidal rule</a></label></li></ul></li><li class=""><label style="padding-left: 0px"><a class="expander" href="#" style="font-size: .9em"><i class="fa fa-fw fa-caret-right"></i></a><a href="../2800/pre-ib-preparation-.html">Pre-IB Preparation </a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2812/pre-ib-weekly-quiz-1.html">Pre-IB Weekly Quiz 1</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2814/pre-ib-weekly-quiz-2.html">Pre IB Weekly Quiz 2</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2815/pre-ib-weekly-quiz-3.html">Pre-IB Weekly Quiz 3</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2819/pre-ib-weekly-quiz-4.html">Pre-IB Weekly Quiz 4</a></label></li></ul></li><li class=""><label style="padding-left: 0px"><a class="expander" href="#" style="font-size: .9em"><i class="fa fa-fw fa-caret-right"></i></a><a href="../1082/exams-assessment.html">Exams & Assessment</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../1083/ia-technology.html">IA & technology</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../1085/exam-advice.html">Exam Advice</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../1084/sixes-sevens.html">Sixes & sevens</a></label></li></ul></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2928/for-teachers.html">For Teachers</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Practice Questions <a href="#" class="mark-page-favorite pull-right" data-pid="1094" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i> Home</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Practice Questions</span></li> </ol> <article id="main-article"> <p>This page is a place where all the quizzes from the different chapters are written in one place. Please don't forget to go back to all of the individual pages for help in between if and when you need it! Remember, that whilst practising questions is an excellent way to revise, you will then have to do something about what you weren't able to do the first time, by going back to your resources for revision.</p> <hr> <div class="panel panel-darkorange panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Number and Algebra</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-darkorange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exponents</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-567-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find</p><p><span class="math-tex">\({ 5 }^{ -2 }\times { 5 }^{ 5 }\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span class="math-tex">\({ 25 }^{ -10 }\)</span></label> </p><p><label class="radio"> <input type="radio"> <span class="math-tex">\({ 25 }^{ 3 }\)</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span class="math-tex">\({ 5 }^{ 3 }\)</span></label> </p><p><label class="radio"> <input type="radio"> <span class="math-tex">\({ 5 }^{ -10 }\)</span></label> </p></div><div class="q-explanation"><p>When the bases are the same, multiply the exponents only.</p><p><span class="math-tex">\({ =5 }^{ (-2+5) }\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Simplify</p><p><span class="math-tex">\({ { (2 }^{ 3 }) }^{ 4 }\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\( { 2 }^{ 12 }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\( { 8 }^{ 4 }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\( { 16 }^{ 12 }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\( { 2 }^{ 7 }\)</span></span></label> </p></div><div class="q-explanation"><p>This is equivalent to <span class="math-tex">\({ { 2 }^{ 3 } } \times { { 2 }^{ 3 } } \times{ { 2 }^{ 3 } } \times { { 2 }^{ 3 } } .\)</span> As the bases are the same we can now simply add together the exponents.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Simplify</p><p><span class="math-tex">\(3{ y }^{ -2 } \)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac { 3 }{ y^2 } \)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 1 }{ 9y^2 } \)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-6x\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac { 1 }{ 3y^2 } \)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(3{ y }^{ -2 }=3{ \quad \times \quad y }^{ -2 }\)</span></p><p><span class="math-tex">\({ y }^{ -2 }=\frac { 1 }{ y^2 } \)</span></p><p>Therefore</p><p><span class="math-tex">\({ 3y }^{ -2 }=3\times \frac { 1 }{ y^2 } \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\({ a }^{ 3 }\times { a }^{ 4 }={ a }^{ n }\)</span>, what is <span class="math-tex">\(n\)</span>?</p></div><div class="q-answer"><p>n = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\({ a }^{ m }\times { a }^{ n }={ a }^{ m+n }\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\({ ({ 2p }^{ 3 }) }^{ 2 }=a{ p }^{ n }\)</span>, what are the values of <span class="math-tex">\(a\)</span> and <span class="math-tex">\(n\)</span>?</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p>n = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\({ ({ 2p }^{ 3 }) }^{ 2 }={ 2 }^{ 2 }\times { p }^{ 3\times 2 }=4{ p }^{ 6 }\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\({ 5 }^{ -1 } =n\)</span>, what is the value of <span class="math-tex">\(n\)</span>? (<em>Without a calculator</em>)</p></div><div class="q-answer"><p>n = <input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span> </p></div><div class="q-explanation"><span class="math-tex">\({ 5 }^{ -1 }=\frac { 1 }{ { 5 } } =0.2\)</span></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\({ ({ 3 }^{ 0 }) }^{ -3 }=\frac { m }{ n } \)</span>, what are the values of <span class="math-tex">\(m\)</span> and <span class="math-tex">\(n\)</span>? (<em>Without a calculator</em>)</p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> </p><p>n = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> </p></div><div class="q-explanation"><span class="math-tex">\({ ({ 3 }^{ 0 }) }^{ -3 }={ 1 }^{ -3 }=\frac { 1 }{ { 1 }^{ 3 } } =\frac { 1 }{ 1 } \)</span></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\({ (\frac { { 9x }^{ 5 } }{ { 4y }^{ 7 } } ) }^{ -3 }=\frac { a{ y }^{ b } }{ m{ x }^{ n } } \)</span>, what are the values of <span class="math-tex">\(a\)</span>, <span class="math-tex">\(b\)</span>, <span class="math-tex">\(m\)</span> and <span class="math-tex">\(n\)</span>?</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="64"> <span class="review"></span> </p><p>b = <input type="text" style="height: auto;" data-c="21"> <span class="review"></span> </p><p>m = <input type="text" style="height: auto;" data-c="729"> <span class="review"></span> </p><p>n = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span> </p></div><div class="q-explanation"><span class="math-tex">\({ (\frac { { 9x }^{ 5 } }{ { 4y }^{ 7 } } ) }^{ -3 }=\frac { 1 }{ { (\frac { { 9x }^{ 5 } }{ { 4y }^{ 7 } } ) }^{ 3 } } =1\times { (\frac { { 4y }^{ 7 } }{ { 9x }^{ 5 } } ) }^{ 3 }=\frac { { ({ 4y }^{ 7 }) }^{ 3 } }{ { ({ 9x }^{ 5 }) }^{ 3 } } =\frac { 64{ y }^{ 21 } }{ 729{ x }^{ 15 } } \)</span></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\({ (\frac { { 4a }^{ 3 }\times 2{ a } }{ { 12{ a }^{ 4 }\times a } } ) }^{ -3 }=\frac { { ma }^{ n } }{ x{ a }^{ y } } \)</span>, what are the values of <span class="math-tex">\(m\)</span>, <span class="math-tex">\(n\)</span>, <span class="math-tex">\(x\)</span> and <span class="math-tex">\(y\)</span>?</p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="1728"> <span class="review"></span> </p><p>n = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span> </p><p>x = <input type="text" style="height: auto;" data-c="512"> <span class="review"></span> </p><p>y = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span> </p></div><div class="q-explanation"><span class="math-tex">\({ (\frac { { 4a }^{ 3 }\times 2{ a } }{ { 12{ a }^{ 4 }\times a } } ) }^{ -3 }=\frac { 1 }{ { (\frac { { 4a }^{ 3 }\times 2{ a } }{ { 12{ a }^{ 4 }\times a } } ) }^{ 3 } } =1\times { (\frac { 12{ a }^{ 4 }\times a }{ { { 4a }^{ 3 }\times 2{ a } } } ) }^{ 3 }={ (\frac { { 12a }^{ 5 } }{ { 8{ a }^{ 4 } } } ) }^{ 3 }=\frac { { { ({ 12a }^{ 5 }) }^{ 3 } } }{ { { (8a }^{ 4 }) }^{ 3 } } =\frac { 1728{ a }^{ 15 } }{ 512{ a }^{ 12 } } \)</span></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\frac { 5 }{ { a }^{ 4 } } \times { 3a }^{ 8 }=m{ a }^{ n }\)</span>, what are the values of <span class="math-tex">\(m\)</span> and <span class="math-tex">\(n\)</span>? </p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span> </p><p>n = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p></div><div class="q-explanation"><span class="math-tex">\(\frac { 5 }{ { a }^{ 4 } } \times { 3a }^{ 8 }=5{ a }^{ -4 }\times { 3a }^{ 8 }=15{ a }^{ -4+8 }=15{ a }^{ 4 }\)</span></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-darkorange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Amortisation and Annuities</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-520-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Katie wants to save to put a deposit of $50000 down on a house in five years. She saves every year in an annuity that gives 3% interest. Calculate how much she needs to save each year (to 3sf)</p></div><div class="q-answer"><p>$ <input type="text" style="height: auto;" data-c="9420"> <span class="review"></span> </p></div><div class="q-explanation"><p><img alt="" height="389" src="../../annuity-1.png" width="589"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Joe is saving to buy a new pizza oven, which costs $400. He would like to buy it six months time. He saves every month in an annuity that gives 2% interest. How much does he need to save each month? (to 2dp)</p></div><div class="q-answer"><p>$ <input type="text" style="height: auto;" data-c="63.41"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" height="386" src="../../annuity-2.png" width="582"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Adrian is saving to buy a record player, which costs $200. He would like to buy it a year's time, and saves every month in an annuity that gives 4% interest. How much does he need to save each month? (to 2dp)</p></div><div class="q-answer"><p>$ <input type="text" style="height: auto;" data-c="13.31"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" height="342" src="../../annuity-3.png" width="511"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Diane is saving to buy an expensive painting, which is valued at $1,000,000. She would like to buy it in 5 years time and saves each year in an annuity that gives 7% interest. Calculate how much she needs to save each year (to 2 dp)</p></div><div class="q-answer"><p>$ <input type="text" style="height: auto;" data-c="173890.69"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" height="400" src="../../annuity-4.png" width="634"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-darkorange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Accuracy and Estimation</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-515-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 14.7854 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>15</span></label> </p><p><label class="radio"> <input type="radio"> <span>14.785</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>14.8</span></label> </p><p><label class="radio"> <input type="radio"> <span>14.7</span></label> </p></div><div class="q-explanation"><p>14.8 - the third non zero digit is 7, which is rounded up to 8 because of the 8 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 0.003046 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"><input type="radio">0.003</label></p><p><label class="radio"><input type="radio"><span>0</span></label> </p><p><label class="radio"> <input type="radio">0.00304</label></p><p><label class="radio"> <input class="c" type="radio"> 0.00305</label></p></div><div class="q-explanation"><p>0.00305 - the third non zero digit is 4, which is rounded up to 7 because of the 6 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 12348 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"><input type="radio">12350</label></p><p><label class="radio"> <input type="radio">12400</label></p><p><label class="radio"><input type="radio">123</label></p><p><label class="radio"> <input class="c" type="radio"> 12300</label></p></div><div class="q-explanation"><p>12300 - the third non zero digit is 3, which stays the same because of the 4 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 6504905 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"> <input type="radio">6510000</label></p><p><label class="radio"><input type="radio">650 </label></p><p><label class="radio"><input type="radio">6505000</label></p><p><label class="radio"> <input class="c" type="radio"> 6500000</label></p></div><div class="q-explanation"><p>6500000 - the third non zero digit is 0, which stays the same because of the 4 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 452756 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"><input type="radio">453 </label></p><p><label class="radio"><input type="radio">450000</label></p><p><label class="radio"> <input class="c" type="radio"> 453000</label></p><p><label class="radio"> <input type="radio">452000</label></p></div><div class="q-explanation"><p>453000 - the third non zero digit is 2, which becomes 3 because of the 7 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>A rectangular field measures 23m by 63m, correct to the nearest metre. What is the smallest possible area of the field in square metres? (Give the exact answer)</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>1414.76</span></label> </p><p><label class="radio"> <input type="radio"> <span>1417.50</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>1406.25</span></label> </p><p><label class="radio"> <input type="radio"> <span>1449</span></label> </p></div><div class="q-explanation"><p>The lower limt of both measurements are 22.5 and 62.5.</p><p>22.5 x 62.5 = 1406.25</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>A rectangular field measures 15m by 21m, correct to the nearest metre. What is the upper limit for the area of the field in square metres? (Round your answer to 3sf))</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>297</span></label></p><p><label class="radio"> <input class="c" type="radio"> <span>333</span></label></p><p><label class="radio"> <input type="radio"> <span>332</span></label></p><p><label class="radio"> <input type="radio"> <span>315</span></label></p></div><div class="q-explanation"><p>The upper limt of both measurements are 15.5 and 21.5.</p><p>15.5 x 21.5 = 333.25, which is 333 when rounded to 3sf</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>A bucket is used to lower the level of water in a swimming pool - 100 full buckets were removed. The bucket is advertised with a capcity of 3 litres, The actual capacity of the bucket is 3.2 litres. What is the percentage error if the marketed capacity is used to calculate the volume of water that is removed? </p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>6.25%</span></label> </p><p><label class="radio"> <input type="radio"> <span>6.30%</span></label> </p><p><label class="radio"> <input type="radio"> <span>6.67%</span></label> </p><p><label class="radio"> <input type="radio"> <span>6.5%</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(Volume\quad using\quad marketing\quad capacity\quad =\quad 300l\\ Actual\quad volume\quad =\quad 320l\\ Percentage\quad error\quad =\quad \frac { 320-300 }{ 320 } \times 100=6.25\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Hannah estimates that the sum of the first 20 terms of the following sequence, <span class="math-tex">\({ U }_{ n }=12+8(n-1)\)</span>, is 2000. What is her percentage error</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>12%</span></label> </p><p><label class="radio"> <input type="radio"> <span>13.64%</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>13.6%</span></label> </p><p><label class="radio"> <input type="radio"> <span>12.0%</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\({ U }_{ n }=12+8(n-1)\\ n=20,\quad { U }_{ 1 }=12,\quad d=8\\ S_{ 20 }=\frac { 20 }{ 2 } (24+8(20-1))=1760\\ Error\quad =\quad \frac { 2000-1760 }{ 1760 } \times 100=13.6\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Nick converted 300 Zambia Kwechas (ZMW) to Euros (EUR) and the bank were offering an exchange rate of 1 ZMW = 0.049 EUR. If he converted 300 and got 14 EUR, how much commission was he charged?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>6.7%</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>4.76%</span></label> </p><p><label class="radio"> <input type="radio"> <span>4.75%</span></label> </p><p><label class="radio"> <input type="radio"> <span>2.4%</span></label> </p></div><div class="q-explanation"><p>The conversion rate is</p><p><span class="math-tex">\(1\quad ZMW\quad =\quad 0.049\quad EUR\)</span></p><p>so converting 300 ZMW gives:</p><p><span class="math-tex">\(300\quad ZMW\quad =\quad 14.7\quad EUR\)</span></p><p>Buzz got 14 so he paid 0.7 EUR in commission, making the percentage difference:</p><p><span class="math-tex">\(\frac{0.7}{14.7}\cdot100=4.76\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-darkorange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Standard Form</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-456-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Convert the speed of light, <span class="math-tex">\(300000000\space ms^{-1}\)</span>, into standard form. </p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(3\times10^{7}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(3\times10^{8}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(0.3\times10^{9}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(30\times10^{7}\)</span></label></p></div><div class="q-explanation"><p>The first number has to be a number between <span class="math-tex">\(1\)</span> and <span class="math-tex">\(10\)</span>(<span class="math-tex">\(10\)</span> not included). If we insert a decimal point after the digit <span class="math-tex">\(3\)</span>, then count how many times we must move that point to get to the end of the number, that provides the power of <span class="math-tex">\(10\)</span> required.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Write 50200 in the form <span class="math-tex">\(a\times 10^{k}, where\space 1\leq a<10\space and\space k \in \mathbb{Z}\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(5.02\times10^{4}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(5.02\times10^{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(5\times10^{4}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(5.2\times10^{4}\)</span></span></label> </p></div><div class="q-explanation"><p>The first number has to be a number between <span class="math-tex">\(1\)</span> and <span class="math-tex">\(10\)</span>(<span class="math-tex">\(10\)</span> not included). If we insert a decimal point after the digit <span class="math-tex">\(5\)</span>, then count how many times we must move that point to get to the end of the number, that provides the power of <span class="math-tex">\(10\)</span> required.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The population of Beijing is approximately <span class="math-tex">\(21.5\)</span> million people. Write this in standard form.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(21.5\times10^{6}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(2.15\times10^{6}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(21.5\times10^{7}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(2.15\times10^{7}\)</span></span></label> </p></div><div class="q-explanation"><p>Since <span class="math-tex">\(1\)</span> million is <span class="math-tex">\(10^6\)</span>, we start by writing the number as <span class="math-tex">\(21.5\times10^{6}\)</span>. The first number has to be a number between <span class="math-tex">\(1\)</span> and <span class="math-tex">\(10\)</span>(<span class="math-tex">\(10\)</span> not included), we must divide it by <span class="math-tex">\(10\)</span>. To balance out that division so as not to change the number, we must multiply <span class="math-tex">\(10^6\)</span> by <span class="math-tex">\(10\)</span>. This gives the answer shown.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Convert <span class="math-tex">\(1.2\times10^{3}\)</span> into normal form.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="1200"> <span class="review"></span> </p></div><div class="q-explanation"><p>Start by writing out the first number, <span class="math-tex">\(1.2\)</span>, and since the power is <span class="math-tex">\(3\)</span>, move the decimal point to the right <span class="math-tex">\(3\)</span> places. Then fill in the gaps with zeros.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Convert <span class="math-tex">\(5.5\times10^{-4}\)</span> into normal form.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.00055"> <span class="review"></span> </p></div><div class="q-explanation"><p>Start by writing out the first number, <span class="math-tex">\(5.5\)</span>, and since the power is minus <span class="math-tex">\(4\)</span>, move the decimal point to the left <span class="math-tex">\(4\)</span> places. Then fill in the gaps with zeros.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>The average width of a red blood cell is <span class="math-tex">\(0.006\space mm\)</span>. Write this in the form <span class="math-tex">\(a\times 10^{k}, where\space 1\leq a<10\space and\space k \in \mathbb{Z}\)</span>.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(0.6\times10^4\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(6\times10^{-3}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(6\times10^4\)</span></label></p></div><div class="q-explanation"><p>The first number has to be a number between <span class="math-tex">\(1\)</span> and <span class="math-tex">\(10\)</span>(<span class="math-tex">\(10\)</span> not included). If we insert a decimal point after the digit <span class="math-tex">\(6\)</span>, then count how many times we must move that point to get to the decimal, that provides the power of <span class="math-tex">\(10\)</span> required.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>How many zeros are there beteen the decimal point and the first significant figure of <span class="math-tex">\(4.63\times10^{-12}\)</span>?</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="11"> <span class="review"></span> </p></div><div class="q-explanation"><p>The number given is <span class="math-tex">\(0.00000000000463\)</span>. There are <span class="math-tex">\(11\)</span> zeros between the decimal point and the first significant figure. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Water flows through a tube at a rate of <span class="math-tex">\(12\space ms^{-1}\)</span>. How long will it take to fill a pool with volume <span class="math-tex">\(1.7\times10^{3}\space m^{3}\)</span>? Leave your answer to the nearest second. </p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="142"> <span class="review"></span> </p></div><div class="q-explanation"><p>Division: <span class="math-tex">\((1.7\times10^{3})\div12=141.667 s\)</span>. The answer is shown to the nearest second. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>It takes light approximately 8 minutes to reach the Earth from the Sun. If light travels at <span class="math-tex">\(3\times10^{8}\space ms^{-1}\)</span>, calculate the distance between the Earth and the Sun. Give your answer in km, written in standard form.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(1.8\times10^{10}\space km\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(1.44\times10^{8}\space km\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(1.44\times10^{11}\space km\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2.4\times10^{6}\space km\)</span></label></p></div><div class="q-explanation"><p>Since <span class="math-tex">\(8\)</span> minutes is <span class="math-tex">\(420\)</span> seconds, multiply <span class="math-tex">\(3\times10^{8}\times420=1.44\times10^{11}\space m\)</span>. To change metres into kilometres, we divide by <span class="math-tex">\(1000\)</span>. However, dividing by <span class="math-tex">\(10^{3}\)</span> is equivalent to subtracting <span class="math-tex">\(3\)</span> from the power, leaving <span class="math-tex">\(1.44\times10^{8}\space km\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Write <span class="math-tex">\(600.7\times10^{-6}\)</span> the form <span class="math-tex">\(a\times 10^{k}, where\space 1\leq a<10\space and\space k \in \mathbb{Z}\)</span>.</p></div><div class="q-answer"><p><span class="math-tex">\(a=\)</span> <input type="text" style="height: auto;" data-c="6.007"> <span class="review"></span> </p><p><span class="math-tex">\(k=\)</span> <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span> </p></div><div class="q-explanation"><p>Since the first number has to be a number between <span class="math-tex">\(1\)</span> and <span class="math-tex">\(10\)</span>(<span class="math-tex">\(10\)</span> not included), we must divide it by <span class="math-tex">\(100\)</span> or <span class="math-tex">\(10^{2}\)</span>. To balance out that division so as not to change the number, we must multiply <span class="math-tex">\(10^{-6}\)</span> by <span class="math-tex">\(10^2\)</span>. This gives the answer shown.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-darkorange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Arithmetic Sequences</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-314-1094"><p></p><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following sequences could be arithmetic</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> 7, 11, 15, 19.......</label></p><p><label class="checkbox"><input type="checkbox"> 3, 6, 10, 15.....</label></p><p><label class="checkbox"><input type="checkbox"> 1, 1, 2, 3, 5......</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 3.5, 3.75, 4, 4.25......</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 175, 350, 525, 700.......</label></p><p><label class="checkbox"><input type="checkbox"> 1, 2, 4, 8.....</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 65, 57.5, 50, 42.5, 35.......</label></p><p><label class="checkbox"><input type="checkbox"> 3, 9, 15, 20.........</label></p></div><div class="q-explanation"><p>Arithmetic sequences must have a constant difference between the terms. Four of the answers do not. The others could be arithmetic sequences because they do.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the common difference in this sequence? 2, 1.8, 1.6, 1.4 (no spaces in your answer)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-0.2"> <span class="review"></span> </p></div><div class="q-explanation"><p>Sequence goes down by 0.2 each time so the difference is -0.2</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The general term of an arithmetic sequence is given by U<sub>n</sub> = 3 + 4(n - 1), what is the 10th term of this sequence?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="39"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitute n = 9 into the nth term and get U<sub>10</sub> = 3 + 4 X 9, so U<sub>10</sub> = 39</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following sequence, Un = -2 + 0.5(n - 1), work out the 8th and 15th terms and add them together.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="6.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitute n = 8 and 15 in to the general term. The 8th term is 1.5, 15th term is 5, together they sum to 6.5</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>An arithmetic sequence begins 5, 14, 23, 32..... Which of the following is the nth term of that sequence</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> 5 + 8(n - 1)</label></p><p><label class="checkbox"><input type="checkbox"> 9 + 5(n - 1)</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 5 + 9(n - 1)</label></p><p><label class="checkbox"><input type="checkbox"> 9n - 5</label></p></div><div class="q-explanation"><p>Using the formula for the nth term, U<sub>n</sub> = U<sub>1</sub> + d(n - 1), substitue U<sub>1 </sub>= 5 and d = 9</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the arithmetic sequence that begins 15, 31, 47.... Calculate the sum of the the first ten terms of this sequence.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="870"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitue U<sub>1</sub> = 15 and d = 16 in to the sum formula, or work out that U<sub>10</sub> = 159 and use the other formula</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Calculate the sum of the first 20 terms of this sequence Un = 10 - 4(n - 1)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-560"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitue U<sub>1 </sub>= 10 and d = -4 in to the sum formula, or work out that U<sub>20</sub> = -66 and use the other formula</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>An arithmetic sequence has U<sub>4</sub> = 18 and U<sub>7</sub> = 30, what is the first term of the sequence?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> </p></div><div class="q-explanation"><p>Work out that the common difference ias added 3 times between the 4th and 7th term and this is +12 all together so d = 4. Subtract the value of d 3 times to get back to the first term.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>An arithmetic sequence has U<sub>2</sub> = 18 and U<sub>9</sub> = 53, what is the 10th term of the sequence?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="58"> <span class="review"></span> </p></div><div class="q-explanation"><p>Work out that the common difference ias added 7 times between the 2nd and 9th term and this is +35 all together so d = 5. Add the value of d once to get from the 9th to 10th terms.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which is greater, </p><p>a) The sum of the sequence U<sub>n</sub> = 150 + 25(n - 1) to the first 20 terms or</p><p>b) The sum of the sequence U<sub>n</sub> = 150 + 20(n - 1) to the first 25 terms </p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> a) </label></p><p><label class="checkbox"><input class="c" type="checkbox"> b)</label></p></div><div class="q-explanation"><p>The first sum is 7750 and the second is 9750</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-darkorange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Geometric Sequences</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-315-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following sequences could be geometric?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> 2, 4, 8, 16, 31</label></p><p><label class="checkbox"><input type="checkbox"> 3, 6, 9, 12</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 5, 2.5, 1.25, 0.625</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 2, -6, 18, -36</label></p><p><label class="checkbox"><input type="checkbox"> 3, 6, 9, 15</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 5, 10, 20, 40</label></p></div><div class="q-explanation"><p>Onnly the sequences where each term is found by multiplying the previous term by the same common ratio. Example, 5, 10, 20 works because each term is the previous term multiplied by 2.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the common ratio of this geometric sequence?</p><p>7, 21, 63, 189</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p>7 x 3 = 21, 21 x 3 = 63, 63 x 3 = 189</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the next term of this geometric sequence?</p><p>4, 6, 9.....</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="13.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>Each term is 1.5 times bigger than the previous term</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the 6<sup>th</sup> term of the geometric sequencethat begins 3, 15, 75 ?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="9375"> <span class="review"></span> </p></div><div class="q-explanation"><p>3 x 5<sup>5</sup> = 9375</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the 10th term of the geometric sequence Un = 2.5 x (-2)<sup>(n-1)</sup><span id="cke_bm_1086C" style="display: none;"> </span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-1280"> <span class="review"></span> </p></div><div class="q-explanation"><p>2.5 x (-2)<sup>9</sup></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following could be the nth term of the geometric sequence that begins...... 2, 8, 32, 128?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> 4 x 2<sup>(n-1) </sup></label></p><p><label class="checkbox"><input class="c" type="checkbox"> 2 x 4<sup>(n-1)</sup> </label></p><p><label class="checkbox"><input type="checkbox"> 4<sup>(n-1)</sup></label></p><p><label class="checkbox"><input type="checkbox"> 2 x 4<sup>n</sup></label></p></div><div class="q-explanation"><p>The general expression for the nth term of a geometric sequence is U<sub>n</sub> = U<sub>1</sub> x r<sup>(n-1)</sup>, where U<sub>1</sub> is the first term, r is the common ratio and n is the postion/number of terms. We can see that the common ratio is 4 and the first term is 2.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following sequences could be a part of the geometric sequence with the general expression Un = 3 x 2<sup>(n-1)</sup>?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> 6, 12, 24</label></p><p><label class="checkbox"><input type="checkbox"> 1.5, 3, 6</label></p><p><label class="checkbox"><input type="checkbox"> 30, 60, 120</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 192, 384, 768</label></p></div><div class="q-explanation"><p>6, 12, 24 are the first 3 terms. If you divide the terms 192, 384, 768 by 3 (the first term) you get 64, 128, 256, which are successive powers of 2 and so each of these terms are 3 x a power of 2 and so they must be part of the sequence.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the geometric sequence that begins 4.5, 9, 18....</p><p>What is the sum of the first 8 terms of this sequence?<span id="cke_bm_1086C" style="display: none;"> </span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="1147.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitute n = 8, U1 = 4.5 and r = 2 in to the sum formula.</p><p><span class="math-tex">\({ S }_{ n }={ u }_{ 1 }\left( \frac { { r }^{ n }-1 }{ r-1 } \right) \\ in\quad this\quad case,\quad { u }_{ 1 }=4.5\quad and\quad r=2\\ SO,\\ { S }_{ 8 }=4.5\left( \frac { { 2 }^{ 8 }-1 }{ 2-1 } \right) \\ { S }_{ 8 }=4.5\left( \frac { 255 }{ 1 } \right) \\ { S }_{ 8 }=1147.5\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the sum of the first 20 terms of this geometric sequence, U<sub>n</sub> = 15 x 1.2<sup>(n - 1)</sup>. Give you answer correct to 3 sf.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="2800"> <span class="review"></span> </p></div><div class="q-explanation"><p>Substitute n = 20, U<sub>1</sub> = 15 and r = 1.2 in to the sum formula. Then round carefully.</p><p><span class="math-tex">\({ S }_{ n }={ u }_{ 1 }\left( \frac { { r }^{ n }-1 }{ r-1 } \right) \\ in\quad this\quad case,\quad { u }_{ 1 }=15\quad and\quad r=1.2\\ SO,\\ { S }_{ 20 }=15\left( \frac { { 1.2 }^{ 20 }-1 }{ 1.2-1 } \right) \\ { S }_{ 20 }=2800.319994\\ { S }_{ 20 }=2800\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>A geometric sequence has U3 = 9 and U6 = -243, what is its common ratio?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\({ u }_{ 3 }\times { r }\times { r }\times { r }={ u }_{ 6 }\\ { u }_{ 3 }\times { r }^{ 3 }={ u }_{ 6 }\\ 9\times { r }^{ 3 }=-243\\ { r }^{ 3 }=\frac { -243 }{ 9 } =-27\\ r=\sqrt [ 3 ]{ -27 } =-3\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> <p><a class="expander" href="#"><span class="fa fa-plus"></span></a></p> </div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-orange"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Functions</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Trigonometric Models</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-575-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>On a unit circle, <span class="math-tex">\(\sin\left(30\right)\)</span> represents...</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> The length of a point on the circumference at 30 degrees from (0, 0). </label></p><p><label class="radio"><input class="c" type="radio"> The height of a point on the circumference at 30 degrees from (0,0).</label></p></div><div class="q-explanation"><p>See answer given. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>The period of the function, <span class="math-tex">\(f\left(x\right)=\cos\left(x\right)\)</span>, is...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>180 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>270 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>90 degrees</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>360 degrees</span></label> </p></div><div class="q-explanation"><p><img alt="" src="../../images/period.png" style="width: 400px; height: 166px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The period of the function, <span class="math-tex">\(f\left(x\right)=\cos\left(2x\right)\)</span>, is...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>90 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>360 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>270 degrees</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>180 degrees</span></label> </p></div><div class="q-explanation"><p>Utilising the formula, <span class="math-tex">\(Period=\frac{360}{b}\)</span>;</p><p><span class="math-tex">\(Period=\frac{360}{2}=180 ^{\circ}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>The period of the function, <span class="math-tex">\(f\left(x\right)=2\cos\left(x\right)\)</span>, is...</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>90 degrees</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>360 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>270 degrees</span></label> </p><p><label class="radio"> <input type="radio"> <span>180 degrees</span></label> </p></div><div class="q-explanation"><p>In this function the amplitude has doubled but the period has not changed from <span class="math-tex">\(f\left(x\right)=\cos\left(x\right)\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function is given by <span class="math-tex">\(f\left(x\right)=4\cos\left(kx\right)\)</span>. If the period of the function is 120 degrees, find the value of <span class="math-tex">\(k\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>2</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>1</span></label> </p><p><label class="radio"> <input type="radio"> <span>4</span></label> </p><p><label class="radio"> <input type="radio"> <span>3</span></label> </p></div><div class="q-explanation"><p>Utilising the formula, <span class="math-tex">\(b=\frac{360}{Period}\)</span>;</p><p><span class="math-tex">\(b=\frac{360}{120}=3\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function, <span class="math-tex">\(g\)</span>, is given by <span class="math-tex">\(g\left(x\right)=3\sin\left(4x\right)+2\)</span>.</p><p>Identify the amplitude, period and principal axis of <span class="math-tex">\(g.\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>Amplitude: <span class="math-tex">\(3\)</span></span></label>, Period: <span class="math-tex">\(90\)</span> degrees; Principal Axis: <span class="math-tex">\(y=2\)</span> </p><p><label class="radio"> <input type="radio"> <span>Amplitude: <span class="math-tex">\(3\)</span></span></label>, Period: <span class="math-tex">\(4\)</span>, Principal Axis: <span class="math-tex">\(y=2\)</span><label class="radio"></label><br> </p></div><div class="q-explanation"><p>The point of this question is to show that the <span class="math-tex">\(4\)</span> inside the brackets is not itself the period of the wave, but it does of course affect the period. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function, <span class="math-tex">\(g\)</span>, is given by <span class="math-tex">\(g\left(x\right)=\sin\left(10x\right)-1\)</span>.</p><p>Identify the amplitude, period and principal axis of <span class="math-tex">\(g.\)</span></p></div><div class="q-answer"><p>Amplitude: <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> </p><p>Period: <input type="text" style="height: auto;" data-c="36"> <span class="review"></span> </p><p>Principal Axis: <input type="text" style="height: auto;" data-c="y=-1"> <span class="review"></span> </p></div><div class="q-explanation"><p>See answers given.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function, <span class="math-tex">\(g\)</span>, is given by <span class="math-tex">\(g\left(x\right)=5\sin(\frac{x}{2})\)</span>.</p><p>Identify the amplitude, period and principal axis of <span class="math-tex">\(g.\)</span></p></div><div class="q-answer"><p>Amplitude: <input type="text" style="height: auto;" data-c="5"> <span class="review"></span> </p><p>Period: <input type="text" style="height: auto;" data-c="720"> <span class="review"></span> </p><p>Principal Axis: <input type="text" style="height: auto;" data-c="y=0 or x-axis"> <span class="review"></span> </p></div><div class="q-explanation"><p>In this case for the period, rewrite <span class="math-tex">\(\frac{x}{2}\)</span> as <span class="math-tex">\(\frac{1}{2}x\)</span> and utilise the formula, <span class="math-tex">\(Period=\frac{360}{b}\)</span>;</p><p><span class="math-tex">\(Period=\frac{360}{0.5}=720 ^{\circ}\)</span>. Other answers are given and more easily understood. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>The equation of the curve below is given as, <span class="math-tex">\(f\left(x\right)=a\cos\left(bx\right)+c\)</span>. </p><p>Find the values of <span class="math-tex">\(a\)</span>, <span class="math-tex">\(b\)</span> and <span class="math-tex">\(c\)</span>.</p><p><img alt="" src="../../images/period2.png" style="width: 400px; height: 215px;"></p></div><div class="q-answer"><p>a: <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> </p><p>b: <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p><p>c: <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> </p></div><div class="q-explanation"><p>You can use the equations given in video 4, or find these values more intuitively.</p><p>The principal axis (or centre line) or the curve is the most logical to identify first. This corresponds to the value <span class="math-tex">\(c\)</span> which is <span class="math-tex">\(-1\)</span>.</p><p>The amplitude is the height between the principal axis and the maximum (or minimum points. This corresponds to the value <span class="math-tex">\(a\)</span> which is <span class="math-tex">\(2\)</span>. </p><p>The period of the wave is <span class="math-tex">\(90\)</span> degrees. Intuitively this means that <span class="math-tex">\(4\)</span> waves fit into a <span class="math-tex">\(360\)</span> degree domain which means that the value <span class="math-tex">\(b\)</span> is <span class="math-tex">\(4\)</span>. You might also find this using <span class="math-tex">\(b=\frac{360}{Period}\)</span>;</p><p><span class="math-tex">\(b=\frac{360}{90}=4\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The equation of the curve below is given as, <span class="math-tex">\(f\left(x\right)=a\sin\left(bx\right)+c\)</span>. </p><p>Find the values of <span class="math-tex">\(a\)</span>, <span class="math-tex">\(b\)</span> and <span class="math-tex">\(c\)</span>.</p><p><img alt="" src="../../images/period3.png" style="width: 400px; height: 196px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=1.5\sin\left(\frac{x}{3}\right)+1\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(f\left(x\right)=1.5\sin\left(\frac{x}{3}\right)\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=1.5\sin\left(\frac{2x}{3}\right)+1\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=1.5\sin\left(\frac{2x}{3}\right)\)</span></span></label> </p></div><div class="q-explanation"><p>Utilising the formulae in this case: </p><p><span class="math-tex">\(a=\frac{\max-\min}{2}=\frac{1.5--1.5}{2}=\frac{3}{2}=1.5\)</span></p><p>To find <span class="math-tex">\(b\)</span>, first identify the period of the wave. Half of the wave is shown with the co-ordinates given, so half the period is <span class="math-tex">\(810-270=540\)</span>. The period is therefore <span class="math-tex">\(540\cdot2=1080\)</span>. Hence, <span class="math-tex">\(b=\frac{360}{1080}=\frac{1}{3}\)</span>.</p><p><span class="math-tex">\(c=\frac{\max+\min}{2}=\frac{1.5+-1.5}{2}=\frac{0}{2}=0\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Features of Functions and Graphing</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-554-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>What can you call the points shown on the diagram below?</p><p><img alt="" src="../../picture1(9).png" style="width: 632px; height: 490px;"></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Zeros</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Roots</label><label class="radio"></label></p><p><label class="checkbox"><input class="c" type="checkbox"> x-intercepts</label></p><p><label class="checkbox"><input type="checkbox"> Vertices</label></p></div><div class="q-explanation"><p>The only incorrect response to this question is "vertices". </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>A function, <span class="math-tex">\(f\)</span>, is defined by <span class="math-tex">\(f\left(x\right)=x^{2}+6x+3\)</span></p><p>Find the x-intercepts of the graphs of <span class="math-tex">\(f\)</span></p></div><div class="q-answer"><p>Answers: <input type="text" style="height: auto;" data-c="-5.45"> <span class="review"></span> , <input type="text" style="height: auto;" data-c="-0.551"> <span class="review"></span> </p></div><div class="q-explanation"><p>Notice that the answers have not been given as co-ordinates. This is strictly correct, but not necessary in an exam scenario. Answers should also be given to 3 s.f. where appropriate. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of these function contain a vertical asymptote? (there is more than one correct answer)</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=\frac{1}{x}\)</span></span></label> </p><p><label class="checkbox"><input class="c" type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=\frac{6x+2}{x-3}\)</span></span></label> </p><div class="ck-mark" id="right-mark" style="display:none;"> </div><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=3^{x}-2\)</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=x^{3}+3x^{2}-2x\)</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(f\left(x\right)=\sin\left(x\right)\)</span></span></label> </p><div class="ck-mark" id="wrong-mark" style="display:none;"> </div></div><div class="q-explanation"><p>Only the first two rational functions contain vertical asymptotes (they also contain horizontal asymptotes). The exponential function, <span class="math-tex">\(f\left(x\right)=3^{x}-2\)</span>, contains a horizontal asymptote but no vertical asymptote. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Tick the answers below which are most commonly required when you are asked to "sketch a graph."</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span>Key features of the graph, such as x-intercepts and y-intercepts, should be clearly labelled. </span></label> </p><p><label class="checkbox"><input class="c" type="checkbox"> <span>Axes must be labelled.</span></label> </p><div class="ck-mark" id="right-mark" style="display:none;"> </div><p><label class="checkbox"><input type="checkbox"> <span>You must you graph paper.</span></label> </p><p><label class="checkbox"><input type="checkbox"> <span>You must label every number on each axis. </span></label> </p><p><label class="checkbox"><input type="checkbox"> <span>You must use a pencil.</span></label> </p></div><div class="q-explanation"><p>A sketch requires you to be relatively accurate, but it does not require the same level of detail as when you are asked to "draw a function."</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Tick the answers below which are most commonly required when you are asked to "draw a graph."</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> You must use a pencil and a ruler.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> You must adhere to any scales given for the axes. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> You must use graph paper.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> The curve must pass directly through any points plotted. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> You must plot individual points on the graph if you are given the data to do so. </label></p></div><div class="q-explanation"><p>Drawing a graph requires precision, hence why all answers above are correct. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>The formuala, <span class="math-tex">\(A=\pi{r}^{2}+\frac{100}{r}\)</span>, is given for the surface area, <span class="math-tex">\(A\)</span>, of a cylinder with radius, <span class="math-tex">\(r\)</span>.</p><p>Find the local minimum point which minimises the surface area. </p></div><div class="q-answer"><p>Minimum Surface Area: <input type="text" style="height: auto;" data-c="59.6"> <span class="review"></span> <span class="math-tex">\({cm}^2\)</span></p></div><div class="q-explanation"><p>Graph the function in your GDC and identify the local minimum. On the GDC you will be given (2.52, 59.6), but you can discard 2.52 given that this is the value of the radius, <span class="math-tex">\(r\)</span>, which results in a minimum surface area, and that is not needed for the question.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Below is a sketch of the function, <span class="math-tex">\(f\left(x\right)=\frac{1}{3}x^{3}+\frac{3}{4}x^{2}-x-1\)</span></p><p>Identify what is missing from the skecth below:</p><p><img alt="" src="../../picture2(5).png" style="width: 686px; height: 480px;"></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Vertex points have not been labelled.</label></p><p><label class="checkbox"><input type="checkbox"> The sketch should include numbers along both axes.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> The x-intercepts and y-intercept has not been labelled. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> The axes have not been labelled. </label></p></div><div class="q-explanation"><p>See answers given.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function, <span class="math-tex">\(f\left(x\right)=3^{x}+\frac{1}{x+2}\)</span> </p><p>Sketch the grap of <span class="math-tex">\(y=f(x)\)</span> for <span class="math-tex">\({-2}\leq{x}\leq{2}\)</span> and <span class="math-tex">\({-2}\leq{y}\leq{10}\)</span>.</p></div><div class="q-answer"><p>Type "done" into the textbook to check your answer: <input type="text" style="height: auto;" data-c="done"> <span class="review"></span> </p></div><div class="q-explanation"><p><img alt="" src="../../picture1(10).png" style="width: 701px; height: 448px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the functions <span class="math-tex">\(f\left(x\right)=x+2\)</span> and <span class="math-tex">\(g\left(x\right)=3^{x}-3\)</span></p><p>Solve <span class="math-tex">\(f\left(x\right)=g\left(x\right)\)</span>.</p></div><div class="q-answer"><p>Answers: <input type="text" style="height: auto;" data-c="-5.00"> <span class="review"></span> , <input type="text" style="height: auto;" data-c="1.74"> <span class="review"></span> </p></div><div class="q-explanation"><p>When you graph the two functions, there are two points of intersection, (-5.00, -3.00) and (1.74, 3.74).</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Finish the sentence - </p><p>A straight line can intersect a quadratic function in:</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> 2 points</label></p><p><label class="checkbox"><input class="c" type="checkbox"> No points </label></p><p><label class="checkbox"><input class="c" type="checkbox"> 1 point - if it is a tangent to the curve. </label></p></div><div class="q-explanation"><p>See answers.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Introducing Functions</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-470-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Is this a function or a relation?</p><p><img alt="" src="../../graph-1.png" style="width: 560px; height: 444px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> One-to-one function</label></p><p><label class="radio"><input type="radio"> Many-to-one function</label></p><p><label class="radio"><input class="c" type="radio"> Relation</label></p></div><div class="q-explanation"><p>Since it fails the vertical line test - i.e. there is a section in which a vertical line passes through more than one point on the curve - it must be a relation.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Identify which functions are linear.</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span><span class="math-tex">\(f(x)=6x+5\)</span></span></label> </p><p><label class="checkbox"><input class="c" type="checkbox"> <span><span class="math-tex">\(g(x)=10-2x\)</span></span></label> </p><div class="ck-mark" id="right-mark" style="display:none;"> </div><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(w(x)=3x+7+x^2 \)</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(q(x)=\frac{1}{3x-2}\)</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span><span class="math-tex">\(v(x)=2^{x+5}\)</span></span></label> </p><div class="ck-mark" id="wrong-mark" style="display:none;"> </div></div><div class="q-explanation"><p>Linear functions - which are straight lines on a co-ordinate grid - have an <span class="math-tex">\(x\)</span> term with a maximum power of <span class="math-tex">\(1\)</span>, which we of course do not write in mathematics. <span class="math-tex">\(w(x)\)</span> is a quadratics function, <span class="math-tex">\(q(x)\)</span> is a rational function and <span class="math-tex">\(v(x)\)</span> is an exponential function.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="math-tex">\(f(x)=\frac{1}{3}x^2+\frac{2}{5}x-2\)</span></p><p>Find <span class="math-tex">\(f(2)\)</span>, giving your answer to 3 significant figures. </p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.133"> <span class="review"></span> </p></div><div class="q-explanation"><p>If you substitute <span class="math-tex">\(x=2\)</span> into the functin you obtain 0.133.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A toy sales person sells <span class="math-tex">\(t\)</span> toys per month. </p><p>Her monthly profit in US dollars can be modelled by: <span class="math-tex">\(N(t)=t^3+2t-100\)</span></p><p>Find the value of <span class="math-tex">\(N\)</span> if no toys are sold.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-100"> <span class="review"></span></p></div><div class="q-explanation"><p>If no toys are sold, <span class="math-tex">\(t=0\)</span>, and hence <span class="math-tex">\(N(0)=-100\)</span> dollars. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="math-tex">\(f(x)=\frac{3+x}{x^2}\)</span></p><p>A point <span class="math-tex">\((x, 14)\)</span> lies on this function. Find the value of <span class="math-tex">\(x \)</span></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.25"> <span class="review"></span> </p></div><div class="q-explanation"><p>Start by substituting <span class="math-tex">\(14\)</span> into the equation: <span class="math-tex">\(14=\frac{3+x}{x^2}\)</span></p><p>Using your GDC, graph each side of the equation and find the point of interestion of these graphs.</p><p>Since we are only interested in the <span class="math-tex">\(x\space co-ordinate\)</span> of that intersection point, you find <span class="math-tex">\(x=0.25\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>If you know that <span class="math-tex">\(f(10)=25\)</span>, find <span class="math-tex">\(f^{-1}(25)\)</span></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="10"> <span class="review"></span> </p></div><div class="q-explanation"><p>If <span class="math-tex">\(f\)</span> takes an input of <span class="math-tex">\(10\)</span> and produces an output of <span class="math-tex">\(25\)</span>, then the inverse of <span class="math-tex">\(f\)</span>, <span class="math-tex">\(f^{-1}\)</span>, does the opposite. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the domain of the function, <span class="math-tex">\(f(x)=\sqrt{x}\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(x \geq 0\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(x > 0\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(x \geq 1\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(x \leq 0\)</span></span></label> </p></div><div class="q-explanation"><p>If you input the graph of <span class="math-tex">\(f(x)=\sqrt{x}\)</span> into your GDC, you will notice that it does not exist for any negative value of <span class="math-tex">\(x \)</span>, because we cannot take the square root of a negative number and obtain a real answer. The domain is therefore, <span class="math-tex">\(x \geq 0\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the range of the function <span class="math-tex">\(f(x)=x^2\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(f(x)>0\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(f(x) <0\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(f(x) \geq 0\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(f(x) \geq1\)</span></label></p></div><div class="q-explanation"><p>If you completed the previous question, then you will already be aware the the domain of a function, is the same the range of its inverse. Therefore, the range of <span class="math-tex">\(f(x)=x^2 \)</span> is <span class="math-tex">\(f(x) \geq0\)</span>. This can also been seen through inputting the graoh into your GDC, or noting that when you square any input number, the output will always be positive or zero. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>What can you say about the domain and range of any diagonal straight line (linear function).</p></div><div class="q-answer"><p>Domain: <input type="text" style="height: auto;" data-c="all real numbers"> <span class="review"></span> </p><p>Range: <input type="text" style="height: auto;" data-c="all real numbers"> <span class="review"></span> </p><p>Choose options below to write into the text input boxes - note that it may be the same option. </p><p>a. all real numbers</p><p>b. x / f(x) must be greater than zero</p><p>c. x / f(x) must be less than zero</p></div><div class="q-explanation"><p>The domain and range of diagonal straight lines is all real numbers. This is not the case for horizontal or vertical lines. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>To graphically obtain an inverse function, you reflect it on the co-ordinate grid in the diagonal line, .....................................</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>y = x</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = 0</span></label> </p><p><label class="radio"> <input type="radio"> <span>y = -x</span></label> </p><p><label class="radio"> <input type="radio"> <span>x = 0</span></label> </p></div><div class="q-explanation"><p>To obtain the inverse function graphically, reflect it in the diagonal line, y = x. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Sinusoidal Functions</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-360-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Match the functions shown in the diagram below with the letters</p><p style="text-align: center;"> </p></div><div class="q-answer"><p>Drag the correct letter in the box <span class="q-text-draggable draggable" draggable="true">A</span> <span class="q-text-draggable draggable" draggable="true">B</span> <span class="q-text-draggable draggable" draggable="true">C</span> <span class="q-text-draggable draggable" draggable="true">D</span> </p><p>sinx is graph <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>sin2x is graph <input type="text" style="height: auto;" data-c="D"> <span class="review"></span></p><p>2sinx is graph <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p><p>sinx+2 is graph <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Match the functions shown in the diagram below with the letters</p><p style="text-align: center;"> </p></div><div class="q-answer"><p>Drag the correct letter in the box <span class="q-text-draggable draggable" draggable="true">A</span> <span class="q-text-draggable draggable" draggable="true">B</span> <span class="q-text-draggable draggable" draggable="true">C</span> <span class="q-text-draggable draggable" draggable="true">D</span> </p><p>cos3x is graph <input type="text" style="height: auto;" data-c="C"> <span class="review"></span></p><p>0.5cosx is graph <input type="text" style="height: auto;" data-c="D"> <span class="review"></span></p><p>2cosx is graph <input type="text" style="height: auto;" data-c="B"> <span class="review"></span></p><p>cosx is graph <input type="text" style="height: auto;" data-c="A"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following sinusoidal function <span class="math-tex">\(f(x)=asinbx+c\)</span> as shown in the diagram below. Determine the values of a, b and c</p><p style="text-align: center;"> </p></div><div class="q-answer"><p>a= <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> b= <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> c= <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following sinusoidal function <span class="math-tex">\(f(x)=asinbx+c\)</span> as shown in the diagram below. Determine the values of a, b and c</p><p style="text-align: center;"> </p></div><div class="q-answer"><p>a= <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> b= <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> c= <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following sinusoidal function <span class="math-tex">\(f(x)=asinbx+c\)</span> as shown in the diagram below. Determine the values of a, b and c</p><p style="text-align: center;"> </p></div><div class="q-answer"><p>a= <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> b=<input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span> c= <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following sinusoidal function <span class="math-tex">\(f(x)=acosbx+c\)</span> as shown in the diagram below. Determine the values of a, b and c</p><p style="text-align: center;"> </p></div><div class="q-answer"><p>a= <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> b=<input type="text" style="height: auto;" data-c="1"> <span class="review"></span> c= <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>Watch out! For cos functions, the y-intercept is c+a</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following sinusoidal function <span class="math-tex">\(f(x)=acosbx+c\)</span> as shown in the diagram below. Determine the values of a, b and c</p><p style="text-align: center;"> </p></div><div class="q-answer"><p>a= <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span> b=<input type="text" style="height: auto;" data-c="2"> <span class="review"></span> c= <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>Watch out! For cos functions, the y-intercept is c+a</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following sinusoidal function <span class="math-tex">\(f(x)=acosbx+c\)</span> as shown in the diagram below. Determine the values of a, b and c</p><p style="text-align: center;"> </p></div><div class="q-answer"><p>a= <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> b=<input type="text" style="height: auto;" data-c="3"> <span class="review"></span> c= <input type="text" style="height: auto;" data-c="+2"> <span class="review"></span></p></div><div class="q-explanation"><p>Watch out! For cos functions, the y-intercept is c+a</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Quadratic Models</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-317-1094"><p>For the first seven questions of the is queuiz you need to refer to the quadratic function y = x<sup>2</sup> -2x - 15 and its graph, drawn below,</p><p style="text-align: center;"><img alt="" src="../../files/MM/QM/Quiz1.jpg" style="width: 500px; height: 375px;"></p><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the exact value of y, when x = 2.5? (do not include spaces in your answer)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-13.75"> <span class="review"></span> </p></div><div class="q-explanation"><p>Type the function in to your GDC, use the table menu, set the table to go up in steps of 0.5 and read the answer from the table. OR, use the the 'value' or 'y solve' tool on your GDC.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>......</p><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the exact value of x, when y = 128? (do not include spaces in your answer)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="13"> <span class="review"></span> </p></div><div class="q-explanation"><p>Type the function in to your GDC, use the table menu, set the table appropriately and read the answer from the table. OR, use the 'x solve' tool on your GDC.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is the correct factorisation of the given quadratic?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> y = (x + 3)(x - 5)</label></p><p><label class="checkbox"><input type="checkbox"> y = (x - 3)(x +5)</label></p><p><label class="checkbox"><input type="checkbox"> y = (x + 3)(x + 5)</label></p><p><label class="checkbox"><input type="checkbox"> y = (x - 3)(x - 5)</label></p></div><div class="q-explanation"><p>Expand the brackets to be sure. SInce the coefficient of x is -2, this must come from +3, -5. SInce the value of 'c' is - 15, this corresponds.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use your GDC to find the 'zeros' or 'roots' of this quadratic equation. What are the coordinates of the positive root? Answer in the form (x,y) with no spaces.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="(5,0)"> <span class="review"></span> </p></div><div class="q-explanation"><p>USe the 'Calc' or 'gsolve' menu on your calculator and choose 'zeros' or 'roots'.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>What are the coordinates of the y-intercept of the function? Answer in the form (x,y) with no spaces.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="(0,-15)"> <span class="review"></span> </p></div><div class="q-explanation"><p>The y - intercept occurs when x = 0. This leaves y = -15.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the equation of the axis of symmetry for this function? (no spaces)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="x=1"> <span class="review"></span> </p></div><div class="q-explanation"><p>The axis of symmetry is given by x = -b/2a, where y = ax<sup>2</sup> + bx + c</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use your GDC, or otherwise, to work out the coordinates of the vertex of the function. (no spaces)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="(1,-16)"> <span class="review"></span> </p></div><div class="q-explanation"><p>Ues the 'calc' or 'gsolve' menu on your calculator and choose 'min'. Alternatively, sonce you know the axis of symmetry is x = 1, work out the value of y when x = 1.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following functions is shown in the diagram below?</p><p style="text-align: center;"><img alt="" src="../../files/MM/QM/Quiz2.jpg" style="width: 400px; height: 408px;"></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> 2x<sup>2</sup> + 3x - 5</label></p><p><label class="checkbox"><input type="checkbox"> x<sup>2</sup> + 3x - 5</label></p><p><label class="checkbox"><input type="checkbox"> 2x<sup>2</sup> - 3x + 5</label></p><p><label class="checkbox"><input type="checkbox"> x<sup>2</sup> - 3x + 5</label></p></div><div class="q-explanation"><p>We can see that the function intersects th y axis at (0,-5) and so c = -5. This narrows it down to 2 functions. You can see the the function has zeros at x = -2.5 and 1. These values of x should give y = 0 when substituted. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>The profit (P) in dollars made by a given enterprise is given by <img align="middle" alt="negative 1 over 10 x squared plus 4 x minus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»10«/mn»«/mfrac»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»-«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage.php?formula=c8ca3b60e9becb56817f98ae6f457139.png"> where x is the number units sold. What was the cost of their intial investment? How many units need to be sold before they break even. (this means they have covered their costs - answer should be a whole number)</p></div><div class="q-answer"><p>Cost of initial investment <input type="text" style="height: auto;" data-c="15"> <span class="review"></span> dollars</p><p>Number of units sold to break even <input type="text" style="height: auto;" data-c="5"> <span class="review"></span> units</p></div><div class="q-explanation"><p>The initial investment is the money they spent before anything was sold. This is the y value when x = 0 , in other words, the y-intercept. The number of units for breakeven is the value of the first zero. Up until that point, the profit is negative. 5 is the fisrt whole number x - value that yields a profit of greater than or equal to zero.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The profit (P) in dollars made by a given enterprise is given by <img align="middle" alt="negative 1 over 10 x squared plus 4 x minus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»10«/mn»«/mfrac»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»-«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage.php?formula=c8ca3b60e9becb56817f98ae6f457139.png"> where x is the number units sold. What is the maximum profit that can be made with this model? How many units need to be sold for this maximum profit?</p></div><div class="q-answer"><p>Maximum profit <input type="text" style="height: auto;" data-c="25"> <span class="review"></span> </p><p>Number of units <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> </p></div><div class="q-explanation"><p>The maximum profit is the y coordinate of the maximum point on the curve. The number of units is the corresponding x coordinate. Use your GDC to find these values.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exponential Models</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-318-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the exponetial function <img align="middle" alt="f left parenthesis x right parenthesis space equals space 2 cross times 3 to the power of x plus 1" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»+«/mo»«mn»1«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-1.php?formula=e090344d6e297a2d9342ed90edda7ecd.png">,</p><p>Fill in the corresponding values of the function for the given values of x.</p></div><div class="q-answer"><p>x = 0, so f(x) = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>x = 5, so f(x) = <input type="text" style="height: auto;" data-c="487"> <span class="review"></span> </p></div><div class="q-explanation"><p>Either substitue the value of x in to the function and calculate directly OR type the function in to your GDC and read your answers from the table. Note a<sup>0</sup>=1.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the exponetial function <img align="middle" alt="f left parenthesis x right parenthesis space equals space minus 1.5 cross times 2 to the power of x minus 2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»1«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#215;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»-«/mo»«mn»2«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-2.php?formula=8e0171ae1701d6cce7191db314c7b5c3.png">,</p><p>Fill in the corresponding values of the function for the given values of x. (give exact answers)</p></div><div class="q-answer"><p>x = 1 so f(x) = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span> </p><p>x = -2, so f(x) = <input type="text" style="height: auto;" data-c="-2.375"> <span class="review"></span> </p></div><div class="q-explanation"><p>Either substitue the value of x in to the function and calculate directly OR type the function in to your GDC and read your answers from the table. Note a<sup>0</sup>=1.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Write down the y-coordinate of the y-intercpt and the equation of the asymptote for the following exponential function.</p><p><img align="middle" alt="y equals 2 cross times 1.5 to the power of x plus 3.2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»5«/mn»«mi»x«/mi»«/msup»«mo»+«/mo»«mn»3«/mn»«mo».«/mo»«mn»2«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-3.php?formula=cb44752b08b1fd503b75d26e98c61d09.png"></p></div><div class="q-answer"><p>The y-intercept is at (0, <input type="text" style="height: auto;" data-c="5.2"> <span class="review"></span> )</p><p>The asymptote has the equation y = <input type="text" style="height: auto;" data-c="3.2"> <span class="review"></span> </p></div><div class="q-explanation"><p>For the y - intercept, substitute x = 0. So, 1.5<sup>0 </sup>= 1 therefore y = 2 + 3.2. SInce <img align="middle" alt="2 cross times 1.5 to the power of x" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»5«/mn»«mi»x«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-4.php?formula=473c9e0ceb1889f8e7f2a150ad76216e.png"> can never be zero, y can never be 3.2</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Write down the y-coordinate of the y-intercpt and the equation of the asymptote for the following exponential function.</p><p><img align="middle" alt="y equals 1.2 cross times 3 to the power of negative x end exponent minus 18" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»1«/mn»«mo».«/mo»«mn»2«/mn»«mo»§#215;«/mo»«msup»«mn»3«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«mo»-«/mo»«mn»18«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-5.php?formula=3185aa72b48bb6d09bb8e89c27aca522.png"></p></div><div class="q-answer"><p>The y-intercept is at (0, <input type="text" style="height: auto;" data-c="-16.8"> <span class="review"></span> )</p><p>The asymptote has the equation y = <input type="text" style="height: auto;" data-c="-18"> <span class="review"></span> </p></div><div class="q-explanation"><p>For the y - intercept, substitute x = 0. So, 3<sup>0 </sup>= 1 therefore y = 1.2 -18. SInce <img align="middle" alt="1.2 cross times 3 to the power of negative x end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo».«/mo»«mn»2«/mn»«mo»§#215;«/mo»«msup»«mn»3«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-6.php?formula=5f0a44c4b3d8538cb3b8bfb5c258b43a.png"> can never be zero, y can never be -18</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the graphs on the axes below. Write the corresponding letter next to each function. (use upper case)</p><p style="text-align: center;"><img alt="" src="../../files/MM/EM/q1.jpg" style="width: 500px; height: 359px;"></p></div><div class="q-answer"><p>y = 2<sup>x </sup>, <input type="text" style="height: auto;" data-c="D"> <span class="review"></span> </p><p>y = 2<sup>x</sup>+1, <input type="text" style="height: auto;" data-c="A"> <span class="review"></span> </p><p>y = -2<sup>x</sup>, <input type="text" style="height: auto;" data-c="B"> <span class="review"></span> </p><p>y = 2<sup>-x</sup>, <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> </p></div><div class="q-explanation"><p>You can simply put these in your calculators. BUT you should recognises that A is the same as D but moved 1 up the y - axis. C has -x in the exponent to produce that reflection and B is -2<sup>x</sup> , the negative of 2<sup>x</sup> .</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>An experiment is done to monitor how a population of bacteria grows over time. At the start, there are 200 bacteria and they triple every hour. If the function we would use to model this is in the form <img align="middle" alt="f left parenthesis x right parenthesis equals k cross times a to the power of x" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»k«/mi»«mo»§#215;«/mo»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-7.php?formula=acbeb6269524b8098787e5a441d244e1.png"> , what are the values of k and a?</p></div><div class="q-answer"><p>k = <input type="text" style="height: auto;" data-c="200"> <span class="review"></span> </p><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p>200 must be the starting value, when x = 0. Next, 3 is the multiplier because the population triples for every x-increase by one.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The value of a car depreciates by 20% each year after purchase. It was purchased for $4000. If the function we would use to model this is in the form <img align="middle" alt="f left parenthesis x right parenthesis equals k cross times a to the power of x" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»k«/mi»«mo»§#215;«/mo»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-7.php?formula=acbeb6269524b8098787e5a441d244e1.png"> , what are the values of k and a?</p></div><div class="q-answer"><p>k = <input type="text" style="height: auto;" data-c="4000"> <span class="review"></span> </p><p>a = <input type="text" style="height: auto;" data-c="0.8"> <span class="review"></span> </p></div><div class="q-explanation"><p>4000 must be the starting value, when x = 0. Next, 0.8 is the multiplier because the value becomes 80% of its previous value with each passing year.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the model <img align="middle" alt="T equals 59 cross times 1.8 to the power of negative x end exponent plus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»=«/mo»«mn»59«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»8«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«mo»+«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-8.php?formula=84419022efb953fd85a8bfe7ffdeb02c.png">, which measures temperature (T) as time (x, in minutes) passes. </p><p>What is the initial temperature at the start of the experiment? </p></div><div class="q-answer"><p>T = <input type="text" style="height: auto;" data-c="74"> <span class="review"></span> </p></div><div class="q-explanation"><p>This is when x = 0.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the model <img align="middle" alt="T equals 59 cross times 1.8 to the power of negative x end exponent plus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»=«/mo»«mn»59«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»8«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«mo»+«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-8.php?formula=84419022efb953fd85a8bfe7ffdeb02c.png">, which measures temperature (T) as time (x, in minutes) passes. </p><p>What is the temperature after 4 minutes? (3sf)</p></div><div class="q-answer"><p>T = <input type="text" style="height: auto;" data-c="20.6"> <span class="review"></span> </p></div><div class="q-explanation"><p>This is when x = 4. Read this from the table on your GDC.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the model <img align="middle" alt="T equals 59 cross times 1.8 to the power of negative x end exponent plus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»=«/mo»«mn»59«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»8«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«mo»+«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-8.php?formula=84419022efb953fd85a8bfe7ffdeb02c.png">, which measures temperature (T) as time (x, in minutes) passes. </p><p>After how many minutes does the temperature fall below 18 degrees? (nearest minute)</p></div><div class="q-answer"><p>After <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> minutes.</p></div><div class="q-explanation"><p>Read this from the table on your GDC.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Statistics and Probability</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-lightgreen"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Goodness of Fit and T-tests</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-645-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>A Chi-sqaured goodness of fit test has a p-value of 0.0231. If the test is conducted at a 5% significance level, what is the conclusion of the test. </p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Since <span class="math-tex">\(0.0231<0.05\)</span></label>, we reject the null hypothesis. </p><p><label class="radio"><input type="radio"> Since <span class="math-tex">\(0.0231<0.05\)</span></label>, we accept the null hypothesis. </p></div><div class="q-explanation"><p>If the p-value is less than or equal to the significance level, we reject the null hypothesis. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>A t-test is has a p-value of 0.516. If the test is conducted at the 5% significance level, what is the conclusion of the test. </p></div><div class="q-answer"><p><label class="radio"><input type="radio"> Since <span class="math-tex">\(0.516>0.05\)</span></label>, we reject the null hypothesis.</p><p><label class="radio"><input class="c" type="radio"> Since <span class="math-tex">\(0.516>0.05\)</span></label>, we accept the null hypothesis. </p></div><div class="q-explanation"><p>If the p-value is greater than the significance level, we accept the null hypothesis. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The formula for the degrees of freedom of a Chi-Squared goodness of fit test is: </p></div><div class="q-answer"><p><label class="radio"><input type="radio"> df = (r - 1)(c - 1), where r is the number of rows and c is the number of columns in a table. </label></p><p><label class="radio"><input class="c" type="radio"> df = n - 1, where n is the number of categories. </label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>When conducting a 2-sample t-test on your GDC, you start by inputting the data for each sample into your lists. The calculator will then ask you whether the data is Pooled. Which answer do you enter?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Yes</label></p><p><label class="radio"><input type="radio"> No</label></p><p><label class="radio"><input type="radio"> It depends on whether it is a one-tailed test or two-tailed test. </label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>When conducting a 2 sample t-test, what assumptions do you make in performing a test?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> The data sets follow a normal distribution. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> The variance of each data set is approximately equal. </label></p><p><label class="checkbox"><input type="checkbox"> The mean of each data set is approximately equal.</label></p></div><div class="q-explanation"><p>See answers above. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>A school decided to pilot a reward based system if students attended over 95% of the academic year. Any students in year 10 who attended over this percentage would be allowed to go on a reward trip. At the end of the year, the school decided to compare the year 10 attendance results with that of the previous year 10. It was assumed that students in the current year would be motivated by the reward trip and therefore attend school more regularly.</p><p>The analysis involved a t-test. What type of test would this involve?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> A one-tailed test, i.e. <span class="math-tex">\(\mu_1>\mu_2\)</span></label></p><p><label class="radio"><input type="radio"> A two-tailed test, i.e. <span class="math-tex">\(\mu_1 \neq \mu_2\)</span></label></p></div><div class="q-explanation"><p>This is a one-tailed test because we are hypothesising that the mean of the current group of year 10 students is going to be greater than the mean of the previous group of year 10 students. i.e. we have been given information in the question which suggests that one data set will have a higher mean than the other. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>A school decided to pilot a reward based system if students attended over 95% of the academic year. Any students in year 10 who attended over this percentage would be allowed to go on a reward trip. At the end of the year, the school decided to compare the current year 10 attendance results with that of the previous year 10. It was assumed that students in the current year would be motivated by the reward trip and therefore attend school more regularly. A sample was taken of eacy year group and the results are shown below: </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Current Year 10</td><td>90%</td><td>96%</td><td>96%</td><td>96%</td><td>92%</td><td>100%</td><td>100%</td><td>99%</td><td>97%</td><td>99%</td></tr><tr><td>Previous Year 10</td><td>90%</td><td>95%</td><td>89%</td><td>90%</td><td>96%</td><td>97%</td><td>100%</td><td>90%</td><td>93%</td><td>92%</td></tr></tbody></table><p>Conduct a t-test and identify the p-value for this data. Give your answer to 3 significant figures. </p></div><div class="q-answer"><p>p-value: <input type="text" style="height: auto;" data-c="0.0250"> <span class="review"></span> </p></div><div class="q-explanation"><p>Insert each data set into a list, then perform a 2 sample t-test, with <span class="math-tex">\(\mu_1>\mu_2\)</span>. Ensure you enter "Yes" on Pooled data. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>For a Chi-squared goodness of fit test, the critical value is given as 14.1. The <span class="math-tex">\(χ_2
\)</span> statistic iscalculated to be 10.5. Identify the conclusion for the test. </p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Since 10.5 < 14.1, we accept <span class="math-tex">\(H_0\)</span></label></p><p><label class="radio"><input type="radio"> Since 10.5 < 14.1, we reject <span class="math-tex">\(H_0\)</span></label></p></div><div class="q-explanation"><p>See answer above. </p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>For football stickers, the manufacturer claims the following distribution for the proportions of sticker type:</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Percentage</td></tr><tr><td>Very Common</td><td>50%</td></tr><tr><td>Common</td><td>35%</td></tr><tr><td>Rare</td><td>10%</td></tr><tr><td>Very Rare</td><td>5%</td></tr></tbody></table><p>Sophie is skeptical of the manufacturers claim and tests the result on 200 stickers based on the sticker website information. Her results are shown below: </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Frequency </td></tr><tr><td>Very Common</td><td>110</td></tr><tr><td>Common</td><td>79</td></tr><tr><td>Rare</td><td>7</td></tr><tr><td>Very Rare</td><td>4</td></tr></tbody></table><p>Find the expected frequency of how rare each category of sticker is. </p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Expected Frequency</td></tr><tr><td>Very Common</td><td> <input type="text" style="height: auto;" data-c="100"> <span class="review"></span> </td></tr><tr><td>Common</td><td> <input type="text" style="height: auto;" data-c="70"> <span class="review"></span> </td></tr><tr><td>Rare</td><td> <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> </td></tr><tr><td>Very Rare</td><td> <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> </td></tr></tbody></table></div><div class="q-explanation"><p>To find the expected frequencies, multiply the total number of trials by the expected proportions. i.e. </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Expected Frequency</td></tr><tr><td>Very Common</td><td><span class="math-tex">\(200\times0.5\)</span></td></tr><tr><td>Common</td><td><span class="math-tex">\(200\times0.35\)</span></td></tr><tr><td>Rare</td><td><span class="math-tex">\(200\times0.1\)</span></td></tr><tr><td>Very Rare</td><td><span class="math-tex">\(200\times0.05\)</span></td></tr></tbody></table></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>For football stickers, the manufacturer claims the following distribution for the proportions of sticker type:</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Percentage</td></tr><tr><td>Very Common</td><td>50%</td></tr><tr><td>Common</td><td>35%</td></tr><tr><td>Rare</td><td>10%</td></tr><tr><td>Very Rare</td><td>5%</td></tr></tbody></table><p>Sophie is skeptical of the manufacturers claim and tests the result on 200 stickers based on the sticker website information. Her results are shown below: </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rarity of Sticker</td><td>Frequency </td></tr><tr><td>Very Common</td><td>110</td></tr><tr><td>Common</td><td>79</td></tr><tr><td>Rare</td><td>7</td></tr><tr><td>Very Rare</td><td>4</td></tr></tbody></table><p>Conduct a Chi-Squared goodness of fit test to determine whether the manufacturers claims are correct. Enter the p-value in the space below, given to 3 significant figures, and determine the conclusion of the test at a 1% significance level.</p></div><div class="q-answer"><p>Degrees of Freedom: <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>p-value: <input type="text" style="height: auto;" data-c="0.00264"> <span class="review"></span> </p></div><div class="q-explanation"><p>Since 0.00264 < 0.01, we reject the null hypothesis which means that the manufacturers claim about the proportions of sticker types is incorrect.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-lightgreen"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Spearman's Rank</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-635-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Estimate Spearman's rank correlation coefficient, <span class="math-tex">\(r_{s}\)</span>, for the following scatter graph. </p><p><img alt="" src="../../images/dans-images/picture1(1).png" style="width: 300px; height: 231px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 0.725</label></p><p><label class="radio"><input type="radio"> 0.322</label></p><p><label class="radio"><input class="c" type="radio"> 0.964</label></p><p><label class="radio"><input type="radio"> 1</label></p></div><div class="q-explanation"><p>The data is strictly increasing, with the exeption of the final point, and given that the final point is still relatively close to the previous point, <span class="math-tex">\(r_{s}\)</span>, is high. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Estimate Spearman's rank correlation coefficient, <span class="math-tex">\(r_{s}\)</span>, for the following scatter graph. </p><p><img alt="" src="../../images/dans-images/picture2.png" style="width: 300px; height: 232px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> -1</label></p><p><label class="radio"><input type="radio"> 0</label></p><p><label class="radio"><input type="radio"> 0.999</label></p><p><label class="radio"><input class="c" type="radio"> 1</label></p></div><div class="q-explanation"><p>Given the data is strictly increasing (monotonic), <span class="math-tex">\(r_{s}\)</span> equals one. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Estimate Spearman's rank correlation coefficient, <span class="math-tex">\(r_{s}\)</span>, for the following scatter graph. </p><p><img alt="" src="../../images/dans-images/picture3.png" style="width: 300px; height: 225px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input class="c" type="radio"> -1</label></p><p><label class="radio"><input type="radio"> 0</label></p><p><label class="radio"><input type="radio"> -0.855</label></p></div><div class="q-explanation"><p>Given the data is strictly increasing (monotonic), <span class="math-tex">\(r_{s}\)</span> equals minus one. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Two judges score a diving competition with five competitors. After ranking the scores, the <span class="math-tex">\(r_{s}\)</span> value is found to be 0.952. Comment on <span class="math-tex">\(r_{s}\)</span>in this context.</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> There is a strong, positive correlation between the ranks. The judges strongly agree with each other. </label></p><p><label class="radio"><input type="radio"> There is no significant correlation between the ranks. The judges do not agree with each other. </label></p><p><label class="radio"><input type="radio"> There is a moderate, positive correlation between the ranks. The judges partially agree with each other. </label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>The amount of time studying for a Biology test is compared with the assessment scores. The <span class="math-tex">\(r_{s}\)</span> value for the data was 0.650. Interpret this value.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> There is a strong, positive correlation between the ranks of study time and test scores. Study time is strongly associated with test scores. </label></p><p><label class="radio"><input type="radio"> There is a weak, positive correlation between the ranks of study time and test scores. Study time is weakly associated with test scores. </label></p><p><label class="radio"><input class="c" type="radio"> There is a moderate, positive correlation between the ranks of study time and test scores. Study time is moderately associated with test scores. </label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Before calulating Spearman's Rank Correlation Coeffiient, we create a ranked table. Create a ranked table for the data below. Rank from smallest to largest.</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>x</td><td>2</td><td>2</td><td>5</td><td>3</td><td>8</td></tr><tr><td>y</td><td>3</td><td>2</td><td>3</td><td>4</td><td>7</td></tr></tbody></table></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" height="64" width="742"><tbody><tr><td>Rank, x</td><td> <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></td></tr><tr><td>Rank, y</td><td> <input type="text" style="height: auto;" data-c="2.5"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="2.5"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"><p>Given that the first position and second position are equal for the variable x, we average these positions to get 1.5 in each case. This is similarly the case for the second and third position for variable y, giving 2.5 for these positions. Note that it is equally valid to rank from largest to smallest, as long as you do this for bothh variables.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The temperature of coffee is taken over time. The results are given in the table below.</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Time (t), mins</td><td>0</td><td>10</td><td>20</td><td>30</td><td>40</td></tr><tr><td>Temperature, T, celsius</td><td>68</td><td>45</td><td>30</td><td>25</td><td>25</td></tr></tbody></table><p>Find <span class="math-tex">\(r_{s}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input class="c" type="radio"> - 0.975</label></p><p><label class="radio"><input type="radio"> -1</label></p><p><label class="radio"><input type="radio"> 0.975</label></p></div><div class="q-explanation"><p>The ranked table should be:</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Rank Time</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>Rank Temperature</td><td>5</td><td>4</td><td>3</td><td>1.5</td><td>1.5</td></tr></tbody></table><p>After inserting this data into your GDC, you find <span class="math-tex">\(r_{s}\)</span> to be - 0.975.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>The table below shows a class sample of study time compared with scores on a test.</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Study Time, t</td><td>1</td><td>1</td><td>5</td><td>3.5</td><td>1</td><td>4</td></tr><tr><td>Score, S</td><td>68</td><td>60</td><td>92</td><td>70</td><td>64</td><td>80</td></tr></tbody></table><p>Fill in the gaps in the ranked table below.</p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Study Time, t</td><td> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></td><td> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></td><td>6</td><td>4</td><td> <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></td><td>5</td></tr><tr><td>Score, S</td><td>3</td><td>1</td><td>6</td><td>4</td><td>2</td><td>5</td></tr></tbody></table><p> </p></div><div class="q-explanation"><p>Given that the first, second and third position for time spent studying are the same, we will find the average of these positions by adding them and dividing by three. In this case, that gives a rank of 2 for all three empty boxes.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>From the list below, pick the advantages of using Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient.</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> Spearman's rank provides a more accurate correlation than Pearson's. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> Spearman's rank can be used on data which is not linear. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> Spearman's rank can be useful for data which is less easy to quanitfy. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> Spearman's rank is less effected by outliers. </label></p></div><div class="q-explanation"><p>See answers above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>A set of data is strictly increasing. If the final data point is brought below the previous data point, but so that it is still above all other data points. How will this effect <span class="math-tex">\(r_{s}\)</span>?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> It will decrease to zero. </label></p><p><label class="radio"><input class="c" type="radio"> It will decrease only slightly from 1. </label></p><p><label class="radio"><input type="radio"> It will decrease to 0.5.</label></p></div><div class="q-explanation"><p>See answer above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-lightgreen"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Statistical Concepts</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-534-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of these are examples of discrete data?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span>The number of students in a class</span></label> </p><p><label class="checkbox"><input class="c" type="checkbox"> <span>Results of rolling a dice</span></label> </p><div class="ck-mark" id="right-mark" style="display:none;"> </div><p><label class="checkbox"><input type="checkbox"> <span>A cat's weight</span></label> </p><p><label class="checkbox"><input type="checkbox"> <span>The length of a leaf</span></label> </p><p><label class="checkbox"><input type="checkbox"> <span>Time taken to run a race</span></label> </p><div class="ck-mark" id="wrong-mark" style="display:none;"> </div></div><div class="q-explanation"><p>Discrete data is data that can be counted, continuous data is data that can be measured</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of these are examples of continuous data?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="checkbox"><span>The length of a leaf</span></span></label> </p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="checkbox"> <span>A cat's weight</span></span></label> </p><div class="ck-mark" id="right-mark" style="display:none;"> </div><p><label class="checkbox"><input type="checkbox"> <span class="checkbox"><span>Results of rolling a dice</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> <span class="checkbox"><span>The number of students in a class</span></span></label> </p><p><label class="checkbox"><input type="checkbox"> Colours in a rainbow</label></p><div class="ck-mark" id="wrong-mark" style="display:none;"> </div></div><div class="q-explanation"><p>Discrete data is data that can be counted, continuous data is data that can be measured</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which type of data is data that can be measured, and can take an infinite number of values?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="continuous"> <span class="review"></span> data</p></div><div class="q-explanation"><p>continuous data is data that can be measured, whereas discrete data is data that is counted</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which type of data is data that can be counted, and could only take a finite number of values?</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="discrete"> <span class="review"></span> data</p></div><div class="q-explanation"><p>continuous data is data that can be measured, whereas discrete data is data that is counted</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>An <input type="text" style="height: auto;" data-c="outlier"> <span class="review"></span> is a value that lies outside other values.</p></div><div class="q-answer"></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Below is a table of students' test results. One of these results is an outlier.</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Craig</td><td>45%</td></tr><tr><td>Laura</td><td>36%</td></tr><tr><td>Luisa</td><td>56%</td></tr><tr><td>Josh</td><td>98%</td></tr><tr><td>Maia</td><td>40%</td></tr><tr><td>Becca</td><td>42%</td></tr></tbody></table><p>a. Whose test result is the outlier?</p><p>b. What is the mean if you remove the outlier?</p></div><div class="q-answer"><p>a. <input type="text" style="height: auto;" data-c="Josh"> <span class="review"></span></p><p>b. <input type="text" style="height: auto;" data-c="43.8"> <span class="review"></span> %</p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>An outlier is a piece of data that is distant from the other data in a data set. It can occur because of inaccurate data collection, but also just through variation in the data that is collected. As statisticians, we can choose to either include or exlude outliers, depending on whether we think they contribute to our analysis. This decision can change how we interpret a data set, as it impacts how we calculate central tendency, especially the <span style="background-color: rgb(240, 240, 240);"><input type="text" style="height: auto;" data-c="mean"> <span class="review"></span> </span></p></div><div class="q-answer"></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Match the word to the correct description:</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>The whole group we are interested in</td><td> <input type="text" style="height: auto;" data-c="Population"> <span class="review"></span></td></tr><tr><td>A collection of data from the whole population</td><td> <input type="text" style="height: auto;" data-c="Census"> <span class="review"></span></td></tr><tr><td>A collection of data from part of the population</td><td> <input type="text" style="height: auto;" data-c="Sample"> <span class="review"></span></td></tr></tbody></table><p style="text-align: center;"> </p></div><div class="q-answer"><span class="q-text-draggable draggable" draggable="true">Census</span> <span class="q-text-draggable draggable" draggable="true">Population</span> <span class="q-text-draggable draggable" draggable="true">Sample</span></div><div class="q-explanation"><p>Use this optional field for teacher explanations.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Match the word to the correct description of different types of sampling</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>Randomly chosen data, but in the ratio of group size</td><td> <input type="text" style="height: auto;" data-c="Stratified"> <span class="review"></span></td></tr><tr><td>Data chosen from a random starting point, and then at even intervals</td><td> <input type="text" style="height: auto;" data-c="Systematic"> <span class="review"></span></td></tr><tr><td>Data picked randomly from a list</td><td> <input type="text" style="height: auto;" data-c="Simple"> <span class="review"></span></td></tr><tr><td>A sample created from individuals that fit certain required characteristics</td><td> <input type="text" style="height: auto;" data-c="Quota"> <span class="review"></span></td></tr><tr><td>A sample chosen by ease of collection</td><td> <input type="text" style="height: auto;" data-c="Convenience"> <span class="review"></span></td></tr></tbody></table></div><div class="q-answer"><span class="q-text-draggable draggable" draggable="true">Systematic</span> <span class="q-text-draggable draggable" draggable="true">Quota</span> <span class="q-text-draggable draggable" draggable="true">Simple</span><span class="q-text-draggable draggable" draggable="true">Convenience</span> <span class="q-text-draggable draggable" draggable="true">Stratified</span></div><div class="q-explanation"><p>Use this optional field for teacher explanations.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>What type of sampling are you using if you survey every fifth person who enters the room?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>Systematic</span></label> </p><p><label class="radio"> <input type="radio"> <span>Stratified</span></label> </p><p><label class="radio"> <input type="radio"> <span>Quota</span></label> </p><p><label class="radio"> <input type="radio"> <span>Simple</span></label> </p></div><div class="q-explanation"><p>Use this field for hidden/teacher's explanations</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-lightgreen"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Discrete Random Variables</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-521-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following table shows the probability distribution of a discrete random variable X.</p><table border="0" cellpadding="0" cellspacing="0" height="45" width="277"><tbody><tr><td>X</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>P(X=x)</td><td>0.2</td><td>0.3</td><td> x </td><td>0.4</td></tr></tbody></table><p>Find the value of x.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.1"> <span class="review"></span> </p></div><div class="q-explanation"><p>The sum of probabilities in a probability distribution must be one (i.e. 100%). Therefore, 0.2+0.3+x+0.1=1 and so x=0.1.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following table shows the probability distribution of a discrete random variable X.</p><table border="0" cellpadding="0" cellspacing="0" height="45" width="277"><tbody><tr><td>X</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>P(X=x)</td><td>0.3</td><td>0.4</td><td>0.2</td><td>0.1</td></tr></tbody></table><p>Find E(X).</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="2.1"> <span class="review"></span></p></div><div class="q-explanation"><p>To find the expected value, E(X): <span class="math-tex">\(1\times0.3+2\times0.4+3\times0.2+4\times0.1=2.1\)</span></p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following are examples of discrete random variables?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> The lengths of leaves on a tree. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> Marks on a test out of 10.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> The score on a die.</label></p><p><label class="checkbox"><input type="checkbox"> The heights of students in a school.</label></p></div><div class="q-explanation"><p>Discrete random variables can only take a countable number of values. The marks on a test and the score on a die are both countable and so they are discrete random variables. In contrast, lengths of leaves and heights of students are measured, and they can take an infinite number of values. They are not discrete random variables.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>X is a discrete random variable. Find the value of k in the following probability distribution.</p><table border="0" cellpadding="0" cellspacing="0" height="45" width="233"><tbody><tr><td>X</td><td>1</td><td>2</td><td>3</td></tr><tr><td>P(X=X)</td><td>k </td><td>2k</td><td>3k</td></tr></tbody></table><p>Leave your answer as a fraction (use a / between the numerator and denominator)</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="1/6"> <span class="review"></span> </p></div><div class="q-explanation"><p>The sum of probabilities in a probability distribution must be one (i.e. 100%). Therefore, <span class="math-tex">\(k+2k+3k=1\)</span> and so <span class="math-tex">\(6k=1\)</span>, which means <span class="math-tex">\(k=\frac{1}{6}\)</span>.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following table shows the probability distribution of a discrete random variable, A.</p><table border="0" cellpadding="0" cellspacing="0" height="45" width="452"><tbody><tr><td>a</td><td>2 </td><td>3</td><td>4</td><td>5 </td></tr><tr><td>P(A=a)</td><td>x</td><td>y </td><td>0.1</td><td>0.2</td></tr></tbody></table><p>Given that E(A) = 3.4, find the value of x and y.</p></div><div class="q-answer"><p>x = <input type="text" style="height: auto;" data-c="0.1"> <span class="review"></span></p><p>y = <input type="text" style="height: auto;" data-c="0.6"> <span class="review"></span></p></div><div class="q-explanation"><p>Given that we have two unknowns, we will create 2 simultneous equations to solve.</p><p>Equation 1: <span class="math-tex">\(x+y+0.2+0.1=1\)</span>, and thus <span class="math-tex">\(x+y=0.7\)</span></p><p>Equation 2: <span class="math-tex">\(2\times x+3\times y+4 \times 0.1+5\times 0.2=3.4\)</span>, and thus <span class="math-tex">\(2x+3y=2\)</span></p><p>Solving these linear simultaneous equations on the GDC, we obtain <span class="math-tex">\(x=0.1\)</span> and <span class="math-tex">\(y=0.6\)</span>.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Based on last season's statistics, a football player has the following probability distribution for goal scoring, with the discrete random variable, Y, representing the number of goals per game. </p><table border="0" cellpadding="0" cellspacing="0" height="45" width="312"><tbody><tr><td>y</td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr><td>P(Y=y)</td><td>0.3</td><td>0.45</td><td>0.2</td><td>0.05</td></tr></tbody></table><p>Find the expected number of goals he will score in the next game. Leave your answer to the nearest positive integer.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> </p></div><div class="q-explanation"><p>The expected value, <span class="math-tex">\(E(X)=0\times0.3+1\times0.45+2\times0.2+3\times0.05=1 \)</span>.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The random variable <span class="math-tex">\(X\)</span> has a probability distribution given by the function, <span class="math-tex">\(P(X=x)=\frac{2x}{k}, x=1,2,3\)</span>.</p><p>Find the value of the constant, <span class="math-tex">\(k\)</span>.</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 12</label></p><p><label class="radio"><input type="radio"> 1</label></p><p><label class="radio"><input type="radio"> 6</label></p><p><label class="radio"><input type="radio"> 10</label></p></div><div class="q-explanation"><p>If we substitute the given values of <span class="math-tex">\(x \)</span> into the function, we obtain the following table:</p><table border="0" cellpadding="0" cellspacing="0" height="45" width="168"><tbody><tr><td><span class="math-tex">\(x \)</span></td><td>1 </td><td>2 </td><td>3 </td></tr><tr><td><span class="math-tex">\(P(X=x)\)</span></td><td><span class="math-tex">\(\frac{2}{k}\)</span></td><td><span class="math-tex">\(\frac{4}{k}\)</span></td><td><span class="math-tex">\(\frac{6}{k}\)</span></td></tr></tbody></table><p>Since the probabilities in a probability distribution add up to one, then <span class="math-tex">\(\frac{2}{k}+\frac{4}{k}+\frac{6}{k}=1\)</span>, so that <span class="math-tex">\(\frac{12}{k}=1\)</span> and hence <span class="math-tex">\(k=12\)</span>.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>A survey was taken in a school asking how many hours, to the nearest hour, IB Diploma students studied for each night. Relative frequencies were calculated, and the table below shows the probability distribution. The random variable, X, is how many hours per night students study for.</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><span class="math-tex">\(x \)</span></td><td>0</td><td>1</td><td>2</td><td>3</td></tr><tr><td><span class="math-tex">\(P(X=x)\)</span></td><td>0.05</td><td>0.3</td><td>0.5</td><td>0.15</td></tr></tbody></table><p>Find <span class="math-tex">\(P\left(2\le X\le3\right)\)</span>.</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 0.65</label></p><p><label class="radio"><input type="radio"> 0.3</label></p><p><label class="radio"><input type="radio"> 0.8</label></p><p><label class="radio"><input type="radio"> 0.35</label></p></div><div class="q-explanation"><p><span class="math-tex">\(P\left(2\le X\le3\right)\)</span> is the probability that a student studies between 2 and 3 hours a night, including those values. That is 0.5 + 0.15 = 0.65.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following table shows the probability distribution of a discrete random variable, A.</p><table border="0" cellpadding="0" cellspacing="0" height="45" width="452"><tbody><tr><td>a</td><td>2 </td><td>3</td><td>4</td><td>5 </td></tr><tr><td>P(A=a)</td><td>x</td><td>0.2 </td><td>y</td><td>0.2</td></tr></tbody></table><p>Given that E(A) = 3.8, find the value of x and y.</p></div><div class="q-answer"><p>x = <input type="text" style="height: auto;" data-c="0.1"> <span class="review"></span></p><p>y = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>Given that we have two unknowns, we will create 2 simultaneous equations to solve.</p><p>Equation 1: <span class="math-tex">\(x+0.2+y+0.2=1\)</span>, and thus <span class="math-tex">\(x+y=0.6\)</span></p><p>Equation 2: <span class="math-tex">\(2\times x+3\times 0.2+4 \times y+5\times 0.2=3.8\)</span>, and thus <span class="math-tex">\(2x+4y=2.2\)</span></p><p>Solving these linear simultaneous equations on the GDC, we obtain <span class="math-tex">\(x=0.1\)</span> and <span class="math-tex">\(y=0.5\)</span>.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following table shows the probability distribution of a discrete random variable, A.</p><table border="0" cellpadding="0" cellspacing="0" height="45" width="452"><tbody><tr><td>a</td><td>2 </td><td>3</td><td>4</td><td>5 </td></tr><tr><td>P(A=a)</td><td>0.3</td><td>x</td><td>y</td><td>0.2</td></tr></tbody></table><p>Given that E(A) = 3.4, find the value of x and y.</p></div><div class="q-answer"><p>x = <input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span></p><p>y = <input type="text" style="height: auto;" data-c="0.3"> <span class="review"></span></p></div><div class="q-explanation"><p>Given that we have two unknowns, we will create 2 simultaneous equations to solve.</p><p>Equation 1: <span class="math-tex">\(0.3+x+y+0.2=1\)</span>, and thus <span class="math-tex">\(x+y=0.5\)</span></p><p>Equation 2: <span class="math-tex">\(2\times 0.3+3\times x+4 \times y+5\times 0.2=3.4\)</span>, and thus <span class="math-tex">\(3x+4y=1.8\)</span></p><p>Solving these linear simultaneous equations on the GDC, we obtain <span class="math-tex">\(x=0.2\)</span> and <span class="math-tex">\(y=0.3\)</span>.</p></div><div class="remove-section" style="display: none;" title="Remove question field"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-lightgreen"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Central Tendency and Dispersion</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-522-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the mean and median of this set of numbers?</p><p>8, 5, 6, 9, 2, 5, 3, 6</p></div><div class="q-answer"><p>Mean = <input type="text" style="height: auto;" data-c="5.5"> <span class="review"></span></p><p>Median = <input type="text" style="height: auto;" data-c="5.5"> <span class="review"></span></p></div><div class="q-explanation"><p>Use lists and 1 variable data analysis on the GDC</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the the standard deviation of this set of numbers? (<em>Exact or 3.s.f</em>)</p><p>42, 63, 62, 59, 48, 33, 41</p></div><div class="q-answer"><p>Standard deviation = <input type="text" style="height: auto;" data-c="10.9"> <span class="review"></span> </p></div><div class="q-explanation"><p>Use lists and 1 variable data analysis on the GDC</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out the mean and the standard deviation of the data from the following frequency table (<em>exact or 3.s.f</em>):</p><p><img alt="" src="../../images/rn/frequency-1.png" style="width: 300px; height: 199px;"></p></div><div class="q-answer"><p>Mean = <input type="text" style="height: auto;" data-c="3.23"> <span class="review"></span> </p><p>Standard deviation = <input type="text" style="height: auto;" data-c="1.32"> <span class="review"></span> </p></div><div class="q-explanation"><p>Use lists and 1 variable data analysis on the GDC - make sure list 2 is set as frequency</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A set of newborn babies in a hospital were weighed in kilograms. Calculate an estimation of the mean and standard deviation from the data collected in the frequency table below (<em>Exact or 3.s.f</em>)</p><p><img alt="" src="../../images/rn/frequency-20.png" style="width: 200px; height: 163px;"></p></div><div class="q-answer"><p>Mean: <input type="text" style="height: auto;" data-c="3.42"> <span class="review"></span> </p><p>Standard deviation: <input type="text" style="height: auto;" data-c="0.841"> <span class="review"></span> </p></div><div class="q-explanation"><p>Use lists and 1 variable data analysis on the GDC - make sure list 2 is set as frequency</p><p>To estimate <span class="math-tex">\(x\)</span>, calculate the midpoint between the limits</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-lightgreen"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Normal Distribution</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-321-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Tick the boxes next to statements you think are true. </p></div><div class="q-answer"><p>If a data set is normally distributed then we would expect it...</p><p><label class="radio"><span class="checkbox"><input class="c" type="checkbox"> To be symmetrical about the mean</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> 99% of the data to be within 3 standard deviations of the mean</label></p><p><label class="checkbox"><input type="checkbox">68% of results to be within 2 standard deviations of the mean</label></p><p><label class="checkbox"><input type="checkbox"> 95% of results to be within 1 standard deviation of the mean</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 68% of results to be within 1 standard deviation of the mean</label></p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>The ditribution below shows a data set that is normally distributed with a mean of 50 and standard deviation of 10. For each region (labelled A - D) say the percentage of results you would expect to find.</p><p style="text-align: center;"><img alt="" src="../../files/Stats/nd/ndq3.jpg" style="width: 500px; height: 378px;"></p></div><div class="q-answer"><p>A <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> %, B <input type="text" style="height: auto;" data-c="13.5"> <span class="review"></span> %, C <input type="text" style="height: auto;" data-c="68"> <span class="review"></span> %, D <input type="text" style="height: auto;" data-c="16"> <span class="review"></span> %</p></div><div class="q-explanation"><p>Based on the approximations, 68% +/-1sd, 95% +/-2sd, 99% +/-3sd and the symmetry of the diagram</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows the normal distribution curve for a data set with a mean of 30 and standard deviation of 5. What pecentage of the data set should be represented by the shaded area.</p><p style="text-align: center;"><img alt="" src="../../files/Stats/nd/ndq1.jpg" style="width: 500px; height: 354px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="47.5"> <span class="review"></span> %</p></div><div class="q-explanation"><p>20 is 2 standard deviations below 30. Between 20 and 40 we would expect 95% of the results. As such between 20 and 30 should be half of that.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows the normal distribution curve for a data set with a mean of 30 and standard deviation of 5. What pecentage of the data set should be represented by the shaded areas A and B?</p><p style="text-align: center;"><img alt="" src="../../files/Stats/nd/ndq2.jpg" style="width: 500px; height: 362px;"></p></div><div class="q-answer"><p>A <input type="text" style="height: auto;" data-c="2.5"> <span class="review"></span> %, B <input type="text" style="height: auto;" data-c="50"> <span class="review"></span> %</p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider a data set that is normally distributed with a mean of 40 and standard devisation of 7. Calculate the probability that a data item selected at random will be in the categories listed below. (give answers correct to 3 sf)</p></div><div class="q-answer"><p>a) Less than 30, <input type="text" style="height: auto;" data-c="0.0766"> <span class="review"></span> </p><p>b) Greater than 45, <input type="text" style="height: auto;" data-c="0.238"> <span class="review"></span> </p><p>c) Between 42 and 46, <input type="text" style="height: auto;" data-c="0.192"> <span class="review"></span> </p></div><div class="q-explanation"><p>Enter the Normal CDF function using the Mean, Standard deviation, Lower limit and upper limit.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider a data set that is normally distributed with a mean of 4.5 and standard devisation of 0.6. Calculate the probability that a data item selected at random will be in the categories listed below. (give answers correct to 3 sf)</p></div><div class="q-answer"><p>a) Less than 3, <input type="text" style="height: auto;" data-c="0.00621"> <span class="review"></span> </p><p>b) Greater than 4.7, <input type="text" style="height: auto;" data-c="0.369"> <span class="review"></span> </p><p>c) Between 4 and 6, <input type="text" style="height: auto;" data-c="0.791"> <span class="review"></span> </p></div><div class="q-explanation"><p>Enter the Normal CDF function using the Mean, Standard deviation, Lower limit and upper limit.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The weights of a sample of new born babies are normally distributed with a mean of 3.1kg and a standard deviation of 0.8 kg. Find the probability that a baby selected at random from that sample weighed.</p></div><div class="q-answer"><p>a) More than 4kg, <input type="text" style="height: auto;" data-c="0.130"> <span class="review"></span> </p><p> If the sample contained 1500 babies, how many babies would you expect to be more than 4kg?</p><p> <input type="text" style="height: auto;" data-c="195"> <span class="review"></span> babies (nearest whole number)</p></div><div class="q-explanation"><p>Enter the Normal CDF function using the Mean, Standard deviation, Lower limit and upper limit. Then mutiply the probability by 1500.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider a data set that is normally distributed with a mean of 40 and standard devisation of 7. The probability of a data item selected at random being less than x, is given by 0.7.</p></div><div class="q-answer"><p>What is the value of x? <input type="text" style="height: auto;" data-c="43.7"> <span class="review"></span> (3 sf)</p></div><div class="q-explanation"><p>Use the inverse normal function with Area/Probability = 0.7, Mean = 40, sd = 7. Round correctly</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider a data set that is normally distributed with a mean of 4.5 and standard devisation of 0.6. The probability of a data item selected at random being more than x, is given by 0.45.</p></div><div class="q-answer"><p>What is the value of x? <input type="text" style="height: auto;" data-c="4.58"> <span class="review"></span> (3 sf)</p></div><div class="q-explanation"><p>Use the inverse normal function with Area/Probability = 0.55, Mean = 40, sd = 7. Probability is 0.55 because that would leave 0.45 to the right. 0.55 is the probability on the left. Round correctly</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The weights of a sample of new born babies are normally distributed with a mean of 3.1kg and a standard deviation of 0.8 kg. The bottom 10% of the sample were all below x kg.</p></div><div class="q-answer"><p>What is the value of x? <input type="text" style="height: auto;" data-c="2.07"> <span class="review"></span>(3 sf)</p></div><div class="q-explanation"><p>Use the inverse normal function with Area/Probability = 0.1, Mean = 3.1, sd = 0.8. Round correctly</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>......</p><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-lightgreen"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Probability</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-322-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Question zone</p></div><div class="q-answer"><p>Answers zone</p></div><div class="q-explanation"><p>Hidden explanation (optional)</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-lightgreen"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Binomial Distribution</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-524-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>For <span class="math-tex">\(X\sim B(n,p)\)</span>, describe what each element of this probability distribution represents. Use the following statements within the input boxes: discrete random variable, number of trials, probability of success, binomial distribution.</p></div><div class="q-answer"><p><span class="math-tex">\(X \)</span> is a <input type="text" style="height: auto;" data-c="discrete random variable"> <span class="review"></span></p><p><span class="math-tex">\(B\)</span> represents the <input type="text" style="height: auto;" data-c="binomial distribution"> <span class="review"></span></p><p><span class="math-tex">\(n\)</span> is the <input type="text" style="height: auto;" data-c="number of trials "> <span class="review"></span></p><p><span class="math-tex">\(p\)</span> is the <input type="text" style="height: auto;" data-c="probability of success"> <span class="review"></span></p></div><div class="q-explanation"><p>This is clear from the answers above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(10,0.3)\)</span>, find <span class="math-tex">\(P(X=4)\)</span>. Give your answer to 1 decimal place.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span> </p></div><div class="q-explanation"><p>Using the binomial distribution (pdf) on your GDC, you instantly arrive at the answer 0.2.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(10,0.3)\)</span>, find <span class="math-tex">\(P\left(X\le3\right)\)</span>. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 0.650</label></p><p><label class="radio"><input type="radio"> 0.382</label></p><p><label class="radio"><input type="radio"> 0.649</label></p><p><label class="radio"><input type="radio"> 0.383</label></p></div><div class="q-explanation"><p>Using the binomial distribution (cdf) on your GDC, ensuring that you input "3" into your calculator as this value must be included, you instantly arrive at the answer 0.6496107..., which is 0.650 written to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(8,0.25)\)</span>, find <span class="math-tex">\(P\left(X<5\right)\)</span>. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 0.996</label></p><p><label class="radio"><input type="radio"> 0.995</label></p><p><label class="radio"><input type="radio"> 0.972</label></p><p><label class="radio"><input class="c" type="radio"> 0.973</label></p></div><div class="q-explanation"><p>Using the binomial distribution (cdf) on your GDC, ensuring that you input "4" into your calculator given that "5" is not included, you instantly arrive at the answer 0.97270..., which is 0.973 written to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>There are 20 students in a class. Assuming that being born on any day of the week is equally likely, find the probability that exactly 3 of the students were born on a Monday. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.242"> <span class="review"></span></p></div><div class="q-explanation"><p>This is a binomial distribution, with <span class="math-tex">\(X\)</span> representing the number of students born on a Monday. Hence, <span class="math-tex">\(X\sim B(20,\frac { 1 }{ 7 } )\)</span>, since there are 20 students and the probability of being born on a Monday is <span class="math-tex">\(\frac { 1 }{ 7 } \)</span>. The question is then, find <span class="math-tex">\(P(X=3)\)</span>, which using binomial (pdf) on the GDC, gives 0.242 to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(X\sim B(20,0.1)\)</span>. Identify which sum below is equivalent to <span class="math-tex">\(P\left(X>18\right)\)</span>.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(P\left(X>18\right)=P(X=19)+P(X=20)+P(X=21)\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(P\left(X>18\right)=P(X=19)+P(X=20)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(P\left(X>18\right)=P(X=18)+P(X=19)+P(X=20)\)</span></label></p></div><div class="q-explanation"><p>See answer given above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(8,0.25)\)</span>, find <span class="math-tex">\(P\left(X=5\right)\)</span>. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 0.231</label></p><p><label class="radio"><input type="radio"> 0.973</label></p><p><label class="radio"><input type="radio"> 0.230</label></p><p><label class="radio"><input type="radio"> 0.996</label></p></div><div class="q-explanation"><p>Use the binomial distribution (pdf) on your GDC, and make sure to round carefully to 3sf</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(9,0.3)\)</span>, find <span class="math-tex">\(P\left(X\le3\right)\)</span>. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 0.729</label></p><p><label class="radio"><input type="radio"> 0.270</label></p><p><label class="radio"><input type="radio"> 0.463</label></p><p><label class="radio"><input class="c" type="radio"> 0.730</label></p></div><div class="q-explanation"><p>Using the binomial distribution (cdf) on your GDC, ensuring that you input "3" into your calculator as this value must be included, you instantly arrive at the answer 0.7296..., which is 0.730 written to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(10,0.3)\)</span>, find <span class="math-tex">\(P(X=3)\)</span>. Give your answer to 3sf</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.267"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the binomial distribution (pdf) on your GDC, you instantly arrive at the answer 0.267. See calculator skills page for more help with this!</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Geometry and Trigonometry</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-blue"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Voronoi and Perpendicular Bisectors</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-574-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the gradient of the perpendicular bisector of the points (1,1) and (7,3)?</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{3}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-3\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac{1}{2}\)</span></label></p></div><div class="q-explanation"><p>The gradient between the two points is <span class="math-tex">\(\frac{1}{3}\)</span>. Hence, the perpendiular gradient is <span class="math-tex">\(-\frac{3}{1}=-3\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the equation of the perpendicular bisector between the points (-2, 5) and (4, 3).</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=\frac{1}{3}x-\frac{1}{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=\frac{1}{3}x+3\frac{2}{3}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(y=3x+1\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=3x-11\)</span></span></label> </p></div><div class="q-explanation"><p>Midpoint: <span class="math-tex">\(\left(\frac{-2+4}{2},\frac{5+3}{2}\right)=\left(1,4\right)\)</span></p><p>Gradient: <span class="math-tex">\(\frac{3-5}{4--2}=-\frac{2}{6}=-\frac{1}{3}\)</span></p><p>Perpendicular Gradient: <span class="math-tex">\(\frac{3}{1}=3\)</span></p><p>Equation: <span class="math-tex">\(y=3x+c\)</span></p><p>Substitute <span class="math-tex">\((1,4)\)</span> into the equation: <span class="math-tex">\(4=3\times1+c\)</span>, so <span class="math-tex">\(c=1\)</span>.</p><p>Thus, <span class="math-tex">\(y=3x+1\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the equation of the perpendicular bisector between the points given on the graph below. </p><p><img alt="" src="../../images/picture1(1).png" style="width: 250px; height: 256px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=0x+4\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=4\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(x =4\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=x+4\)</span></span></label> </p></div><div class="q-explanation"><p>A vertical line which passes through <span class="math-tex">\(4\)</span> on the x-axis has equation, <span class="math-tex">\(x =4\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A line segment has a gradient of <span class="math-tex">\(3\)</span>. What is the gradient of the perpendiular line?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(-\frac{1}{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-0.3\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(-3\)</span></span></label> </p></div><div class="q-explanation"><p>If the gradient is <span class="math-tex">\(3\)</span>, we can write the gradient as <span class="math-tex">\(\frac{3}{1}\)</span> so that the perpendicular gradient is <span class="math-tex">\(-\frac{1}{3}\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Below is a Voronoi diagram with an edge missing. Find the equation of the edge, and write your answer in the form <span class="math-tex">\(ax+by+c=0\)</span>, where <span class="math-tex">\(a,b,c \in \mathbb{Z}\)</span> and <span class="math-tex">\(a,b>0\)</span>.</p><p><img alt="" src="../../images/voronoipi.png" style="width: 250px; height: 260px;"></p></div><div class="q-answer"><p>Equation: <input type="text" style="height: auto;" data-c="10x+4y-55=0"> <span class="review"></span> </p></div><div class="q-explanation"><p>The missing edge seperates sites A(1,4) and B(6,6). </p><p>Midpoint: <span class="math-tex">\(\left(\frac{1+6}{2},\frac{4+6}{2}\right)=\left(3.5,5\right)\)</span></p><p>Gradient: <span class="math-tex">\(\frac{6-4}{6-1}=\frac{2}{5}\)</span></p><p>Perpendicular Gradient: <span class="math-tex">\(-\frac{5}{2}\)</span></p><p>Equation: <span class="math-tex">\(y=-\frac{5}{2}x+c\)</span></p><p>Substitute <span class="math-tex">\((3.5,5)\)</span> into the equation: <span class="math-tex">\(5=3.5\times-\frac{5}{2}+c\)</span>, so <span class="math-tex">\(c=13.75\)</span>.</p><p>Thus, <span class="math-tex">\(y=-\frac{5}{2}x+13.75\)</span></p><p>Since <span class="math-tex">\(a,b,c \)</span> must be integers with <span class="math-tex">\(a,b>0\)</span>, we can multiply te equation by <span class="math-tex">\(4\)</span> and rearrange to obtain </p><p><span class="math-tex">\(10x+4y-55=0\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>The rainfall in three nearby cities, A, B, C and D, over the course of the previous month, is 15 cm, 22 cm, 18 cm and 20 cm respectively. Use nearest neighbour interpolation to estimate the rainfall at (2, 3).</p><p><img alt="" src="../../images/picture1(2).png" style="width: 300px; height: 242px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="19"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>The point (2, 3) is located on an edge between the sites C and D. We therefore take the mean of the rainfalls at C and D: <span class="math-tex">\(\frac{18+20}{2}=19\ cm\)</span></p><p>If the point was located inside a cell, then we would estimate the rainfall to be the same as the site within that cell. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The rainfall in three nearby cities, A, B, C and D, over the course of the previous month, is 15 cm, 22 cm, 18 cm and 20 cm respectively. Use nearest neighbour interpolation to estimate the rainfall at (5, 3).</p><p><img alt="" src="../../images/picture1(3).png" style="width: 300px; height: 242px;"></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>The point (5, 3) is located on a vertex between the sites B, C and D. We therefore take the mean of the rainfalls at B, C and D: <span class="math-tex">\(\frac{22+18+20}{3}=20\ cm\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>In the Voronoi diagram below, a new site D has been added. </p><p><img alt="" src="../../images/newsite.png" style="width: 300px; height: 205px;"> </p><p>Choose which statement below is the correct process to create a new Voronoi diagram for sites A, B, C and D.</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> Find a perpendicular bisector between D and A and ensure it hits te vertex already in the current Voronoi diagram. </label></p><p><label class="checkbox"><input type="checkbox"> Find a perpendicular bisector between D and C and ensure it stops at the edge between A and C.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Find a perpendicular bisector between D and C, ensuring it stops at the edge between A and C, then find a second prpendicular bisector between D and A and construct the new Voronoi diagram accordingly. </label></p></div><div class="q-explanation"><p>When adding a new site, a perpendicular bistector needs to be found between te new site, and the site which already exists in the cell. Then a new perpendicular bistector must be found between the new site, and any sites in cells which are directly next to the new site. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the co-ordinates of the missing site in the Voronoi diagram below.</p><p><img alt="" src="../../images/picture1(4).png" style="width: 300px; height: 245px;"></p></div><div class="q-answer"><p>( <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span> , <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span> )</p></div><div class="q-explanation"><p>The missing site is contained in the cell which currently does not have a site. </p><p>If we start by looking at the edge between the missing site and site D, it has gradient of 3 right and 1 down, <span class="math-tex">\(-\frac{1}{3}\)</span>. Therefore the perpendicular gradient must be <span class="math-tex">\(3\)</span>. Utilising the same idea with the edge between site B and the missing site, the gradient is <span class="math-tex">\(-\frac{1}{2}\)</span>, which means the perpendicular gradient must be <span class="math-tex">\(2\)</span>. Starting at site D and B, and drawing a line backwards with the perpendicular gradients found will ascertain the location of the missing site. </p><p><img alt="" src="../../images/picture1(5).png" style="width: 300px; height: 242px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the centre and radius of the largest empty circle in the Voronoi diagram below. </p><p><img alt="" src="../../images/picture1(6).png" style="width: 300px; height: 232px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\((5, 6); 4\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\((5, 6); \sqrt{34}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\((9, 5); \sqrt{15}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\((9, 5); \sqrt{17}\)</span></span></label> </p></div><div class="q-explanation"><p>Since there are two vertices, we will need to check the distance between each vertex and a randomly picked site that each vertex seperates. </p><p>For the vertex <span class="math-tex">\((9,5)\)</span>, if I randomly pick site A, the distance is given by, <span class="math-tex">\(d=\sqrt{(9-13)^2+(5-4)^2}=\sqrt{(-4)^2+1^2}=\sqrt{17} \)</span></p><p>For the vertex (5, 6), if I randomly pick site D, the distance is given by,</p><p><span class="math-tex">\(d=\sqrt{(5-0)^2+(6-9)^2}=\sqrt{5^2+3^2}=\sqrt{34} \)</span></p><p>Therefore, the largest empty circle has centre <span class="math-tex">\((5, 6)\)</span> and radius <span class="math-tex">\(\sqrt{34} \)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-blue"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Applications of Trigonometry</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-572-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Identify the angle of depression in the figure below: </p><p><img alt="" src="../../images/depression-1.png" style="width: 300px; height: 135px;"></p></div><div class="q-answer"><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input class="c" type="radio"> C</label></p></div><div class="q-explanation"><p>The angle of depression is the angle between the horizontal line of sight and the line of sight <em>down to</em> an object. It is therefore angle C. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the angle of elevation, <span class="math-tex">\(x\)</span>, from the persons head to the top of the tree. </p><p><img alt="" src="../../images/elevation-1.png" style="width: 300px; height: 186px;"></p></div><div class="q-answer"><p>x = <input type="text" style="height: auto;" data-c="48.2"> <span class="review"></span> </p></div><div class="q-explanation"><p>Using right-angle triangle trigonometry, we can find the angle of elevation: </p><p><span class="math-tex">\(cos(\theta)=\frac{10}{15}\)</span></p><p>So, <span class="math-tex">\(\theta=\cos^{-1}\left(\frac{10}{15}\right)\)</span></p><p><span class="math-tex">\(\theta=48.2^{o}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>A person is standing near a tree. If the angle of elevation is given by 43 degrees, and the person is half the height of the tree, find the height of the tree. </p><p><img alt="" src="../../images/elevation-2.png" style="width: 300px; height: 168px;"></p></div><div class="q-answer"><p>Height of Tree =<input type="text" style="height: auto;" data-c="18.7"> <span class="review"></span> </p></div><div class="q-explanation"><p>First finding the height of the right-angled triangle: </p><p><span class="math-tex">\(\tan\left(43^{o}\right)=\frac{h}{10}\)</span></p><p><span class="math-tex">\(h=10\cdot\tan\left(43^{o}\right)\)</span></p><p><span class="math-tex">\(h=9.3251...m\)</span></p><p>Given that the person is half the height of the tree, we can now double the previous answer to obtain the height of the tree. </p><p>Height of Tree <span class="math-tex">\(=18.7 \space m\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A lighthouse keeper measures the angle of depression between himself and a ship as <span class="math-tex">\(3^{o}\)</span>. If the keeper is 20 metres above the water level, how far is the ship from the lighthouse?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>381 m</span></label> </p><p><label class="radio"> <input type="radio"> <span>103 m</span></label> </p><p><label class="radio"> <input type="radio"> <span>20 m</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>382 m</span></label> </p></div><div class="q-explanation"><p>Here is the worded statement translated into a picture: </p><p><img alt="" src="../../images/q4.png" style="width: 300px; height: 96px;"></p><p>Using right-triangle trigonometry: SOH CAH TOA</p><p><span class="math-tex">\(\tan\left(87\right)=\frac{x}{20}\)</span></p><p>So <span class="math-tex">\(x=20\cdot\tan\left(87\right)\)</span></p><p><span class="math-tex">\(x= 382 \space{m} (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Write the bearing of T from P in each diagram below:</p><p><img alt="" src="../../images/q5a.png" style="width: 180px; height: 249px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(30^{o}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(030^{o}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(150^{o}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(330^{o}\)</span></span></label> </p></div><div class="q-explanation"><p>A bearing is a 3-digit angle which is measured clockwise from North. i.e. you start at the North line and identify the angle turned. Therefore, in this case the bearing is <span class="math-tex">\(030^{o}\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Write the bearing of T from P in each diagram below:</p><p><img alt="" src="../../images/q5b.png" style="width: 180px; height: 187px;"></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(290^{o}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(110^{o}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(070^{o}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(220^{o}\)</span></span></label> </p></div><div class="q-explanation"><p>A bearing is a 3-digit angle which is measured clockwise from North. i.e. you start at the North line and identify the angle turned. Therefore, in this case the bearing is <span class="math-tex">\(180^{o}+110^{o}=290^{o}\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The bearing of Ship A to Ship B is <span class="math-tex">\(080^{o}\)</span>. Give the bearing of Ship A from Ship B. You do not need to put the degree sign in the textbox provided. </p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="260"> <span class="review"></span> </p></div><div class="q-explanation"><p>By drawing a sketch, we first see that finding the supplementary angle (<span class="math-tex">\(100^{o}\)</span>) then allows us to calculate the bearing of A from B (or B to A). </p><p><img alt="" src="../../images/q6.png" style="width: 200px; height: 102px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Ships A and B leave port at the same time. Ship A travels at 20 kmh on a bearing of <span class="math-tex">\(045^{o}\)</span> and Ship B travels at 15 kmh on a bearing of <span class="math-tex">\(110^{o}\)</span>. How far are the ships away from each other after 3 hours of travelling?</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="57.8"> <span class="review"></span> </p></div><div class="q-explanation"><p>Starting with a sketch: </p><p><img alt="" src="../../images/q8.png" style="width: 300px; height: 321px;"></p><p>Once the sketch has been drawn, use the cosine rule to find the missing length. </p><p><span class="math-tex">\(x^{2}=60^{2}+45^{2}-2\cdot60\cdot45\cdot\cos\left(65\right)\)</span></p><p><span class="math-tex">\(x = 57.8 \space{km}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>From a lamp post, a skateboarder travels 200 m east before turning on a bearing of <span class="math-tex">\(162^{o}\)</span> for 160 m. She then turns on a bearing of <span class="math-tex">\(242^{o}\)</span> for 380 m, and finally in a straight line back to the original lamp post. </p><p>Which of sketches below represent this information?</p></div><div class="q-answer"><p><label class="radio"> </label></p><p><label class="radio"> <input type="radio"> <span><img alt="" src="../../images/q9b.png" style="width: 200px; height: 163px;"></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><img alt="" src="../../images/q9a.png" style="width: 200px; height: 189px;"></span></label> </p></div><div class="q-explanation"><p>The bearings in the correct sketch both turn clockwise from North, whereas the second bearing in the incorrect skecth turns anti-clockwise. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the dotted line distance in the diagram below: </p><p><img alt="" src="../../images/q10.png" style="width: 200px; height: 182px;"></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="292"> <span class="review"></span> </p></div><div class="q-explanation"><p>Start by finding the included angle between 200 m and 160 m. This is <span class="math-tex">\(360^{o}-90^{o}-162^{o}=108^{o}\)</span>. Then use the cosine rule to find the missing length: </p><p><span class="math-tex">\(x^{2}=200^{2}+160^{2}-2\cdot200\cdot160\cdot\cos\left(108\right)\)</span></p><p><span class="math-tex">\(x = 292 \space{m}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-blue"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Circles</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-516-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which formula from the formula booklet finds the length of a ciruclar arc, where <span class="math-tex">\(\theta\)</span> is measured in degrees and r is the radius of the arc?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(l= 2\pi r\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(l=\frac { \theta }{ 360 } \times 2\pi r\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(l=\frac { \theta }{ 360 } \times \pi { r }^{ 2 }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(l=\frac { \theta }{ 360 } \times r\)</span></span></label> </p></div><div class="q-explanation"><p>The angle of the arc as a fraction of a whole circle (360 degrees), multiplied by the circumference of a circle (<span class="math-tex">\(2\pi r\)</span>), gives the arc length.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the length of a cicular arc which has a radius of 5 cm and a central angle of 30°.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>5.25 cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>31.4 cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>2.61 cm</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>2.62 cm</span></label> </p></div><div class="q-explanation"><p>Using the arc length formula:</p><p><span class="math-tex">\(l=\frac { 30° }{ 360° } \times 2\times \pi \times 5=2.62\quad cm\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>If the area of a sector is <span class="math-tex">\(50{c m }^{ 2 }\)</span> and the central angle is 45 degrees, find the radius of the sector.</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>11.3cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>11.2cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>1.41cm</span></label> </p><p><label class="radio"> <input type="radio"> <span>63.7cm</span></label> </p></div><div class="q-explanation"><p>Using the formula for the area of a sector:</p><p><span class="math-tex">\(50=\frac { 45° }{ 360° } \times \pi { r }^{ 2 }\)</span></p><p>Dividing both sides by <span class="math-tex">\(\frac { 45° }{ 360° }\)</span> and by <span class="math-tex">\(\pi\)</span>:</p><p><span class="math-tex">\({r}^2=127.324...\)</span></p><p><span class="math-tex">\(\therefore \quad r=11.3cm\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A semi-circlular logo has a radius of 10cm. Find the area of the logo, giving your answer to 1 decimal place.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>157.0 <span class="math-tex">\({cm}^2\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>31.4 <span class="math-tex">\({cm}^2\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>180 <span class="math-tex">\({cm}^2\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>157.1 <span class="math-tex">\({cm}^2\)</span></span></label> </p></div><div class="q-explanation"><p>We could use the formula for the area of a sector, with the angle, <span class="math-tex">\(\theta\)</span>, being 180 degrees. However, it may be more efficient to use the formula for the area of a circle and then divide that area by 2 to get the area of the semi-circle.</p><p><span class="math-tex">\(A=\frac { \pi \times { 10 }^{ 2 } }{ 2 } =157.0796{ cm }^{ 2 }=157.1{ cm }^{ 2 }\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the arc length of a sector with radius 6cm and a central angle of 120 degrees. Give your answer in terms of <span class="math-tex">\(\pi\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(4\pi\)</span></span></label> cm </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(36\pi\)</span></span></label> cm </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{2\pi}{3}\)</span></span></label> cm </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(12\pi\)</span></span></label> cm </p></div><div class="q-explanation"><p>Using the formula for the arc length of a sector:</p><p><span class="math-tex">\(A=\frac { 120° }{ 360° } \times 2\times \pi \times 6=\frac { 1 }{ 3 } \times 12\pi =\frac { 12\pi }{ 3 } =4\pi \)</span> cm</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the total perimeter of the shape below, giving your answer to 3 significant figures. Do not include units.</p><p><img alt="" height="165" src="../../picture1(2).png" width="163"></p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="53.1"> <span class="review"></span> </p></div><div class="q-explanation"><p>There are a number of ways to do this, one way being to find the interior angle of the sector, which is 320°, and then utilize the arc length formula to obtain 39.0954 cm. To then find the total perimeter, we must add 14 cm for the two radii, giving 53.1 cm to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the area of a sector of a circle with radius 8 cm and central angle 10°. Give your answer to 3 significant figures. Do not include units.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="5.59"> <span class="review"></span></p></div><div class="q-explanation"><p>The area of the sector is given by:</p><p><span class="math-tex">\(A=\frac { 10° }{ 360° } \times \pi \times { 8 }^{ 2 }=5.5851{ cm }^{ 2 }=5.59{cm}^2\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>A garden sprinkler is set at an angle of 160° and a spray distance of 4 metres. Find the area of grass that the sprinkler covers, leaving your answer to 3 significant figures. Do not include units.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="22.3"> <span class="review"></span> </p></div><div class="q-explanation"><p>Using the area of a sector formula:</p><p><span class="math-tex">\(A=\frac { 160° }{ 360° } \times \pi \times { 4 }^{ 2 }=22.3{ m }^{ 2 }\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which formulae below find the circumference of a circle, C, with radius, r, and diameter, d?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(C=\pi {r}^2\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(C=\frac { \theta }{ 360 } \times 2\pi r\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(C=2\pi r\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(C=\pi d\)</span></label></p></div><div class="q-explanation"><p>See answers.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The Area of a sector formula is given by, <span class="math-tex">\(A=\frac { \theta }{ 360 } \times \pi { r }^{ 2 }\)</span>. Rearrange this formula to write <span class="math-tex">\(\theta\)</span> in terms of <span class="math-tex">\(A\)</span> and <span class="math-tex">\(r\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\theta =360A-\pi { r }^{ 2 }\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\theta =\sqrt { \frac { 360A }{ \pi { r } } } \)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\theta =360A\pi { r }^{ 2 }\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\theta =\frac { 360A }{ \pi { r }^{ 2 } } \)</span></span></label> </p></div><div class="q-explanation"><p>Given that <span class="math-tex">\(\theta\)</span> is divided by 360°, we must multiply both sides by 360°. Also, since we are multiplying <span class="math-tex">\(\theta\)</span> by <span class="math-tex">\(\pi{r}^2\)</span>, we must divide both sides by <span class="math-tex">\(\pi{r}^2\)</span>. Essentially we are conducting the inverse operations when rearranging.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-blue"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Volume and Surface Area</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-323-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>a) What is the volume of the cuboid in the diagram below?</p><p>b) What is its surface area?</p><p style="text-align: center;"><img alt="" src="../../files/GT/vsa/q1.jpg" style="width: 250px; height: 240px;"></p></div><div class="q-answer"><p>Write a number only</p><p>a) Volume = <input type="text" style="height: auto;" data-c="60"> <span class="review"></span> cm<sup>3</sup></p><p>b) Surface area = <input type="text" style="height: auto;" data-c="104"> <span class="review"></span> cm<sup>2</sup></p></div><div class="q-explanation"><p>The volume of the base is 5 x 2, then multiply by the height, 6, giveives volume = 5 x 2 x 6</p><p>b) 2 faces are 5 x 2, 2 faces are 6 x 2 and 2 faces are 5 x 6, giving a total of 20 + 24 + 60 = 104cm<sup>2</sup></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>....</p><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Drag the correct name of the shape in to the right box.</p><p style="text-align: center;"><img alt="" src="../../files/GT/vsa/which-shape.jpg" style="width: 400px; height: 455px;"></p></div><div class="q-answer"><table border="1" cellpadding="0" cellspacing="0" style="width: 100%;"><tbody><tr><td> Shape A <input type="text" style="height: auto;" data-c="Triangular Prism"> <span class="review"></span></td><td><span class="q-text-draggable draggable" draggable="true">Triangular prism</span> <span class="q-text-draggable draggable" draggable="true">Pyramid</span></td></tr><tr><td>Shape B <input type="text" style="height: auto;" data-c="Cylinder"> <span class="review"></span> </td><td><span class="q-text-draggable draggable" draggable="true">Cylinder</span> <span class="q-text-draggable draggable" draggable="true">Cone</span></td></tr><tr><td>Shape C<span style="background-color: rgb(235, 244, 250);"> <input type="text" style="height: auto;" data-c="Cuboid"> <span class="review"></span> </span></td><td><span class="q-text-draggable draggable" draggable="true">Cuboid</span> <span class="q-text-draggable draggable" draggable="true">Octangonal based pyramid</span></td></tr><tr><td><table border="1" cellpadding="0" cellspacing="0" style="width: 100%;"><tbody><tr><td>Shape D <input type="text" style="height: auto;" data-c="Rectangular based Pyramid"> <span class="review"></span> </td><td> </td></tr></tbody></table></td><td><span class="q-text-draggable draggable" draggable="true">Rectangular based Pyramid</span> <span class="q-text-draggable draggable" draggable="true">Sphere</span></td></tr></tbody></table></div><div class="q-explanation"> </div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>....</p><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The following right angled triangle is one face of a triangular prism whose length is 12 cm....</p><p><img alt="" src="../../files/GT/vsa/q3.jpg" style="width: 200px; height: 177px; float: left;"></p><p>a) Calculate the length of the hypotenuse of the triangle</p><p>b) What is the area of the triangular cross section?</p><p>c) What is the volume of the prism?</p><p>d) What is the total surface area of the prism?</p><p> </p></div><div class="q-answer"><p>Write numbers only</p><p>a) Length of hypotenuse is <input type="text" style="height: auto;" data-c="5"> <span class="review"></span> cm</p><p>b) The area of the triangular cross section is <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> cm<sup>2</sup></p><p>c) The volume of the prism is <input type="text" style="height: auto;" data-c="72"> <span class="review"></span> cm<sup>3</sup></p><p>d)The total surface area of the prism is <input type="text" style="height: auto;" data-c="156"> <span class="review"></span> cm<sup>2</sup></p></div><div class="q-explanation"><p>a) Using pythagoras theorem, h<sup>2</sup> = 3<sup>2</sup> + 4<sup>2</sup></p><p>b) Area = (3 x 4)/2 = 6</p><p>c) Volume is corss sectional area x length, so V = 6 x 12 = 72</p><p>d) There are 5 faces, 2 triangles on each end each 6, the bottom which is 3 x 12, the back which is 4 x 12 and the sloping face which is 5 x 12, so 6 + 6 + 36 + 48 + 60 = 156</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>....</p><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>A triangular prism has a volume of 300 cm<sup>3</sup> and a length of 15cm.</p><p>a) What is the area of the triangular cross section?</p><p>b) If the triangular corss section has a height of 5cm, how long is its base?</p></div><div class="q-answer"><p>a) Area of triangular face is <input type="text" style="height: auto;" data-c="20"> <span class="review"></span> cm<sup>2</sup></p><p>b) the base of the triangular criss section is <input type="text" style="height: auto;" data-c="8"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>a) Volume = Cross section x length, so 300 = Cross section x 15. Cross section = 300/15</p><p>b) Area of truiangle is base x height / 2. If base x 5/2 = 20 then base must be 8</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>...</p><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following cylinder with a radius of 7 cm and a heigth of 15 cm</p><p><img alt="" src="../../files/GT/vsa/q5.jpg" style="width: 234px; height: 316px; float: left;"></p><p>a) Work out the volume of the cylinder</p><p>b) Work out the surface are of the cylinder</p><p>Give your answers to 3sf</p></div><div class="q-answer"><p>a) Volume = <input type="text" style="height: auto;" data-c="2040"> <span class="review"></span> cm<sup>3</sup></p><p>b) Surface area = <input type="text" style="height: auto;" data-c="4930"> <span class="review"></span> cm<sup>2</sup></p></div><div class="q-explanation"><p>a) Volume of a cylinder = (pi)r<sup>2</sup>h. in this case pi x 7<sup>2 </sup>x 15 = 2039.07060 which is 2040 to 3sf</p><p>b) Surface area of a cylinder = 2 (pi)r<sup>2</sup> + 2(pi)rh, substitute r = 7 and h = 15 and you get SA = 4926.01728 which rounds to 4930 to 3sf</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>....</p><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following rectangular based Pyramid and its net. CD = 4cm, BC = 2cm, EF = 3cm</p><p style="text-align: center;"><img alt="" src="../../files/GT/vsa/q5b.jpg" style="width: 307px; height: 275px;"><img alt="" src="../../files/GT/vsa/q5c.jpg" style="width: 325px; height: 275px;"></p><p>a) Work out the volume of the pyramid</p><p>b) What is the perpendicualr height, h, of triangle ECD? (give answer to 3sf)</p><p>c) What is the perpendicualr height, p, of triangle EBC? (give answer to 3sf)</p><p>d) Calculate the surface area of the Pyramid using rounded answers to the previous questions</p></div><div class="q-answer"><p>a) V = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span> cm<sup>3</sup></p><p>b) h = <input type="text" style="height: auto;" data-c="3.16"> <span class="review"></span> cm</p><p>c) p = <input type="text" style="height: auto;" data-c="3.61"> <span class="review"></span> cm</p><p>d) SA = <input type="text" style="height: auto;" data-c="27.9"> <span class="review"></span> cm<sup>2</sup></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(V=\frac { 1 }{ 3 } Area\quad of\quad base\quad \times \quad heigth\\ V\quad =\quad \frac { 2\times 4\times 3 }{ 3 } =8{ cm }^{ 3 }\)</span></p><p>b) Consider m to be the midpoint of CD then we have the folllowing triangle</p><p><img alt="" src="../../files/GT/vsa/q5d.jpg" style="width: 150px; height: 130px; float: left;">The vertical height of triangle ECD is</p><p><span class="math-tex">\(h\quad =\quad \sqrt { { 3 }^{ 2 }+{ 1 }^{ 2 } } \\ h\quad =\quad 3.162277\\ h\quad =\quad 3.16\quad cm\quad (3sf)\)</span></p><hr class="hidden-separator"><p>c) Consider n to be the midpoint of BC then we have the folllowing triangle</p><p><img alt="" src="../../files/GT/vsa/q5e.jpg" style="float: left; width: 130px; height: 130px;">The vertical height of triangle EBC is</p><p><span class="math-tex">\(p\quad =\quad \sqrt { { 3 }^{ 2 }+{ 2 }^{ 2 } } \\ p\quad =\quad 3.605551\\ p\quad =\quad 3.61\quad cm\quad (3sf)\)</span></p><hr class="hidden-separator"><p>d) <span class="math-tex">\(Base\quad area\quad =\quad 2\quad x\quad 4\quad =\quad 8{ cm }^{ 2 }\\ \\ Triangle\quad ECD\quad =\quad triangle\quad EBA\quad \\ Area\quad =\quad \frac { 3.16\times 4 }{ 2 } =\quad 6.32\\ \\ Triangle\quad EBC\quad =\quad triangle\quad EAD\quad \\ Area\quad =\quad \frac { 3.61\times 2 }{ 2 } =\quad 3.61\\ \\ Total\quad surface\quad area\quad =\quad 8\quad +\quad 2\times 6.32\quad +\quad 2\times 3.61\quad =\quad 27.86\quad { cm }^{ 2 }\quad \\ \\ \)</span></p><p>SA = 27.9 (3sf)</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><p>...</p><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Calculate a) the volume and b) the surface area of a cone with a base radius of 5cm and a height 12cm</p></div><div class="q-answer"><p>a) Volume = <input type="text" style="height: auto;" data-c="314"> <span class="review"></span> cm<sup>3</sup></p><p>b) Surface area = <input type="text" style="height: auto;" data-c="283"> <span class="review"></span> cm<sup>2</sup></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(Volume\quad of\quad a\quad cone\quad =\quad \frac { \pi { r }^{ 2 }h }{ 3 } \\ V\quad =\quad \frac { \pi { \quad \times \quad 5 }^{ 2 }\quad \times \quad 12 }{ 3 } \\ V\quad =\quad 314.15926\\ V\quad =\quad 314\quad { cm }^{ 3 }\\ \\ \)</span></p><p>b)<img alt="" src="../../files/GT/vsa/q7.jpg" style="float: left; width: 130px; height: 141px;"> The profile of the cone looks like the diagram to the left. the slant height, l is calculated....</p><p><span class="math-tex">\(Slant\quad height\quad of\quad a\quad cone,\quad l\\ l\quad =\quad \sqrt { { 5 }^{ 2 }+{ 12 }^{ 2 } } \\ l\quad =\quad \sqrt { 169 } \\ l\quad =\quad 13\quad cm\\ \\ \\ \)</span></p><p><span class="math-tex">\(Surface\quad area\quad of\quad a\quad cone\quad =\quad \pi { r }^{ 2 }+\pi rl\\ SA\quad =\quad \pi \quad \times \quad 25\quad +\quad \pi \quad \times \quad 5\quad \times \quad 13\\ SA\quad =\quad 282.7433\\ SA\quad =\quad 283\quad { cm }^{ 2 }\)</span></p><hr class="hidden-separator"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p><img alt="" src="../../files/GT/vsa/q3.jpg" style="float: left; width: 200px; height: 177px;">The right angled triangle on the left is the base of a right pyramid whose volume is 90 cm<sup>3</sup>. What is the height of the pyramid?</p></div><div class="q-answer"><p>Height = <input type="text" style="height: auto;" data-c="45"> <span class="review"></span> cm</p></div><div class="q-explanation"><p><span class="math-tex">\(Volume\quad of\quad pyramid\quad =\quad \frac { 1 }{ 3 } Area\quad of\quad base\quad \times \quad height\quad \\ 90\quad =\quad \frac { 1 }{ 3 } \quad \times \quad 6\quad \times \quad h\\ 270\quad =\quad 6h\\ h\quad =\quad \frac { 270 }{ 6 } \quad =\quad 45cm\\ \\ \\ \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is a) the volume and b) the surface area of a Sphere with a radius of 10cm? Give answers to 3sf</p></div><div class="q-answer"><p>a) Volume = <input type="text" style="height: auto;" data-c="4190"> <span class="review"></span> cm<sup>3</sup></p><p>b) Surface Area = <input type="text" style="height: auto;" data-c="1260"> <span class="review"></span> cm<sup>2</sup></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(Volume\quad of\quad sphere\quad =\quad \frac { 4 }{ 3 } \pi { r }^{ 3 }\\ V\quad =\quad \frac { 4 }{ 3 } \pi \quad \times \quad 10^{ 3 }\\ V\quad =\quad 4188.7902\\ V\quad =\quad 4190\quad { cm }^{ 3 }\quad (3sf)\\ \\ \\ \)</span></p><p>b) <span class="math-tex">\(Surface\quad Area\quad of\quad sphere\quad =\quad 4\pi { r }^{ 2 }\\ SA\quad =\quad 4\pi { \quad \times \quad 10 }^{ 2 }\\ SA\quad =\quad 1256.6370\\ SA\quad =\quad 1260\quad { cm }^{ 2 }\quad (3sf)\\ \\ \\ \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>A sphere and a cone have the same radius, 7cm, and the same volume. What is the height of the cone? Do not round until your final answer then give your answer to 3sf.</p></div><div class="q-answer"><p>Height of cone = <input type="text" style="height: auto;" data-c="28.0"> <span class="review"></span> cm</p></div><div class="q-explanation"><p><span class="math-tex">\(Volume\quad of\quad sphere\quad =\quad \frac { 4 }{ 3 } \pi { r }^{ 3 }\\ { V }_{ sphere }\quad =\quad \frac { 4 }{ 3 } \pi \quad \times \quad 7^{ 3 }\\ { V }_{ sphere }\quad =\quad 1436.7550402\quad (Keep\quad calculator\quad display)\\ \\ { V }_{ sphere }\quad =\quad { V }_{ cone\quad }=\quad \frac { 1 }{ 3 } \pi { r }^{ 2 }h\quad =\quad \frac { 1 }{ 3 } \pi { \quad \times \quad 7 }^{ 2 }h\\ (\times 3)\quad \quad \quad 3\quad \times \quad { V }_{ sphere }\quad =\quad \pi { \quad \times \quad 49 }h\\ (\div 49\pi )\quad \frac { 3\quad \times \quad { V }_{ sphere } }{ 49\pi } \quad =\quad h\\ h\quad =\quad 27.99999999\\ h\quad =\quad 28.0\quad cm\quad (3sf)\\ \\ \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-blue"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Right Angled Trigonometry</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-326-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use your calculator to work out solve the following equations. Give your answers as decimals correct to 3sf.</p><p><em><span style="font-family:times new roman,times,serif;">a) x = sin 35</span></em></p><p><em><span style="font-family:times new roman,times,serif;">b) y = tan 75</span></em></p><p><em><span style="font-family:times new roman,times,serif;">c) z = cos 26</span></em></p><p><em><span style="font-family:times new roman,times,serif;">d) Sinp = 0.7</span></em></p><p><em><span style="font-family:times new roman,times,serif;">e) Cosq = 0.1</span></em></p><p><em><span style="font-family:times new roman,times,serif;">f) Tanr = 2.5</span></em></p></div><div class="q-answer"><p><em><span style="font-family:times new roman,times,serif;">a) x = <input type="text" style="height: auto;" data-c="0.574"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">b) y = <input type="text" style="height: auto;" data-c="3.73"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">c) z = <input type="text" style="height: auto;" data-c="0.899"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">d) p = <input type="text" style="height: auto;" data-c="44.4"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">e) q = <input type="text" style="height: auto;" data-c="84.3"> <span class="review"></span> </span></em></p><p><em><span style="font-family:times new roman,times,serif;">f) r = <input type="text" style="height: auto;" data-c="68.2"> <span class="review"></span> </span></em></p></div><div class="q-explanation"><p>This is really just about correctly using the buttons on your calculator. For the last three parts when you have the sine of an angle, you need the inverse sine of that number to get the angle. For example, </p><p>part d) <span class="math-tex">\(sin\quad p\quad =\quad 0.7\\ so\quad p\quad =\quad { sin }^{ -1 }(0.7)\quad =\quad 44.4\\ \\ \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use your calculator to work out solve the following equations. Give your answers as decimals correct to 3sf</p></div><div class="q-answer"><p><span style="font-family:times new roman,times,serif;"><em>a) <span class="math-tex">\(sin\quad 30\quad =\quad \frac { x }{ 10 } \)</span> , so x = <input type="text" style="height: auto;" data-c="5.00"> <span class="review"></span> </em></span></p><p><span style="font-family:times new roman,times,serif;"><em>b) <span class="math-tex">\(Cos\quad 40\quad =\quad \frac { 5 }{ x } \)</span> , so x <input type="text" style="height: auto;" data-c="6.53"> <span class="review"></span> </em></span></p><p><span style="font-family:times new roman,times,serif;"><em>c) <span class="math-tex">\(Tan\quad x\quad =\quad \frac { 7 }{ 14 } \)</span>, so x = <input type="text" style="height: auto;" data-c="26.6"> <span class="review"></span> </em></span></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(sin\quad 30\quad =\quad \frac { x }{ 10 } \\ so\quad x\quad =\quad 10\quad \times \quad sin\quad 30\quad =\quad 5.00\)</span></p><p>b) <span class="math-tex">\(Cos\quad 40\quad =\quad \frac { 5 }{ x } \\ so\quad x\quad =\quad \frac { 5 }{ cos\quad 40 } =\quad 6.53\)</span></p><p>c) <span class="math-tex">\(Tan\quad x\quad =\quad \frac { 7 }{ 14 } \\ so\quad x\quad =\quad { tan }^{ -1 }\left( \frac { 7 }{ 14 } \right) =\quad 26.6\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the folloiwng diagram</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q3.jpg" style="width: 600px; height: 272px;"></p><p>Capital letters (in green) are angles. Lower case letters (in blue) are side lengths.</p><p>Next to each of the statements below, enter either 'T' for True of 'F' for False</p></div><div class="q-answer"><ol><li><span class="math-tex">\(Sin\quad A\quad =\quad \frac { f }{ b } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Cos\quad B\quad =\quad \frac { f }{ a } \)</span> <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> </li><li><span class="math-tex">\(Tan\quad P\quad =\quad \frac { f }{ d } \)</span> <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> </li><li><span class="math-tex">\(Cos\quad Q\quad =\quad \frac { e }{ a } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Tan\quad B\quad =\quad \frac { f }{ e } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Sin\quad Q\quad =\quad \frac { e }{ a } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Sin\quad A\quad =\quad Cos\quad P\)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </li><li><span class="math-tex">\(Tan\quad A\quad =\quad Tan\quad B\)</span> <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> </li></ol></div><div class="q-explanation"><p>This question is really about encouraging the making of statements that are true about a given situation. In practice you can then pick which ones will be mist hepful to you. Clearly it depends on remmebering the trig ratios - SOHCAHTOA and identifying the Hypotenuse as the longest side and the positions of the Opposite and adject sides relative to the angle in question.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the triangle in the image below.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q4.jpg" style="width: 300px; height: 224px;"></p><p>Use trigonometry to work out the lengths of sides a and b in the diagram. Give your answer correct to 3 sf.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="6.43"> <span class="review"></span> </p><p>b = <input type="text" style="height: auto;" data-c="7.66"> <span class="review"></span> </p></div><div class="q-explanation"><p>a, <span class="math-tex">\(Sin40\quad =\quad \frac { a }{ 10 } \\ so\quad a\quad =\quad 10\times sin40\\ a\quad =\quad 6.43\quad (3sf)\)</span></p><p>b, <span class="math-tex">\(Cos40\quad =\quad \frac { b }{ 10 } \\ so\quad b\quad =\quad 10\times cos40\\ a\quad =\quad 7.66\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the triangle in the image below.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q5.jpg" style="width: 300px; height: 220px;"></p><p>Use trigonometry to work out the lengths of sides a and b in the diagram. Give your answer correct to 3 sf.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="32.0"> <span class="review"></span> </p><p>b = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> </p></div><div class="q-explanation"><p>a, <span class="math-tex">\(sin28\quad =\quad \frac { 15 }{ a } \\ so\quad a\quad =\quad \frac { 15 }{ sin28 } \\ a\quad =\quad 32.0\quad (3sf)\)</span></p><p>b, <span class="math-tex">\(tan28\quad =\quad \frac { b }{ 15 } \\ so\quad b\quad =\quad 15\times tan28\\ a\quad =\quad 7.98\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the triangle in the image below.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q6.jpg" style="width: 300px; height: 225px;"></p><p>Use trigonometry to work out the lengths of angles A and B in the diagram. Give your answer correct to 3 sf.</p></div><div class="q-answer"><p>A = <input type="text" style="height: auto;" data-c="42.9"> <span class="review"></span> </p><p>B = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> </p></div><div class="q-explanation"><p>A, <span class="math-tex">\(cos\quad A\quad =\quad \frac { 41 }{ 56 } \\ so,\quad A\quad =\quad { cos }^{ -1 }\left( \frac { 41 }{ 56 } \right) \\ A\quad =\quad 42.9\quad (3sf)\)</span></p><p>B, <span class="math-tex">\(sin\quad B\quad =\quad \frac { 41 }{ 56 } \\ so,\quad B\quad =\quad { sin }^{ -1 }\left( \frac { 41 }{ 56 } \right) \\ A\quad =\quad 47.1\quad (3sf)\)</span></p><p>NOTE, A + B = 90 as they should!</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows a ladder put against a house for cleaning the gutters. The yop of the ladder is 5m above the ground and the angle the ladder makes with the horizontal must be 65°.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q7.jpg" style="width: 500px; height: 361px;"></p><p>a) How far away (<em>x</em>) from the wall must the base of the ladder be put to make the angle 65°</p><p>b) How long does the ladder need to be? (Use trigonometry and unrounded numbers to answer this question)</p><p>c) If the length of the ladder is fixed at 6m and the height against the house remains at 5m, what would the angle between the ladder and the gorund become?</p><p>d) What is the precentage error in the new angle from the required 65°? (Use the rounded answer to part c)</p></div><div class="q-answer"><p>a) x = <input type="text" style="height: auto;" data-c="2.33"> <span class="review"></span> m</p><p>b) length of ladder = <input type="text" style="height: auto;" data-c="5.52"> <span class="review"></span> m</p><p>c) New angle betwen the ladder and the ground = <input type="text" style="height: auto;" data-c="56.4"> <span class="review"></span> °</p><p>d) Percentage error = <input type="text" style="height: auto;" data-c="13.2"> <span class="review"></span> %</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(tan\quad 65\quad =\quad \frac { 5 }{ x } \\ so,\quad x\quad =\quad \frac { 5 }{ tan\quad 65 } \\ A\quad =\quad 2.33\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(sin\quad 65\quad =\quad \frac { 5 }{ l } \\ so,\quad l\quad =\quad \frac { 5 }{ sin\quad 65 } \\ A\quad =\quad 5.52\quad (3sf)\\ \\ OR\quad { l }^{ 2 }\quad =\quad { 5 }^{ 2 }+{ x }^{ 2 }\\ Using\quad the\quad unrounded\quad answer\quad for\quad x\quad gives\quad the\quad same\quad result\quad for\quad l\)</span></p><p>NOTE, you could have used cosine with x and 65°. Again if you used the unrounded anser for x you would have got the same result.</p><p>c) <img alt="" src="../../files/GT/RAT/q7a.jpg" style="width: 200px; height: 259px;"> <span class="math-tex">\(sin\quad A\quad =\quad \frac { 5 }{ 6 } \\ so,\quad A\quad =\quad { sin }^{ -1 }\left( \frac { 5 }{ 6 } \right) \\ A\quad =\quad 56.4\quad (3sf)\)</span></p><p>d) = <span class="math-tex">\(\frac { 65\quad -\quad 56.4 }{ 65 } \times \quad 100\quad =\quad 13.2\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows part of a metal structure and lists some its geometrical properties.</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q8.jpg" style="width: 500px; height: 319px;"></p></div><div class="q-answer"><p>Find the folllowing giving all answers correct to 3sf</p><p>a) The length AC = <input type="text" style="height: auto;" data-c="5.00"> <span class="review"></span> m</p><p>b) <span class="math-tex">\(\hat { CAD } \)</span> = <input type="text" style="height: auto;" data-c="53.1"> <span class="review"></span> °</p><p>c) <span class="math-tex">\(\hat { mAD } \)</span> = <input type="text" style="height: auto;" data-c="45.0"> <span class="review"></span> </p><p>In changes to the design it is decided that <span class="math-tex">\(\hat { mAD } \)</span> needs to be exactly halved </p><p>d) What is the new length AM? <input type="text" style="height: auto;" data-c="3.25"> <span class="review"></span> m</p></div><div class="q-explanation"><p>a) <span class="math-tex">\({ AC }^{ 2 }\quad =\quad { 4 }^{ 2 }+{ 5 }^{ 2 }\\ AC\quad =\sqrt { 25 } =5.00\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(tan\hat { CAD } =\frac { 4 }{ 3 } \\ so,\quad \hat { CAD } \quad =\quad { tan }^{ -1 }\left( \frac { 4 }{ 3 } \right) \\ \hat { CAD } \quad =\quad 53.1°\)</span></p><p>c) <span class="math-tex">\(tan\hat { mAD } =\frac { 2 }{ 3 } \\ so,\quad \hat { mAD } \quad =\quad { tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) \\ \hat { mAD } \quad =\quad 45.0°\)</span></p><p>d) <img alt="" src="../../files/GT/RAT/q8a.jpg" style="width: 200px; height: 134px;"> <span class="math-tex">\(cos22.5\quad =\quad \frac { 3 }{ mA } \\ so,\quad mA\quad =\quad \frac { 3 }{ cos\quad 22.5 } \\ mA\quad =\quad 3.25m\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>USe the diagram below for the following questions.</p><p>The angle of elevation from Point A to some looking out of the window at point B is 60°</p><p>a) What is the height, h, of point B above the ground?</p><p>A person looking at point A from point C is 35m above the ground.</p><p>b) What is the angle of depression, D, they are looking at?</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q9.jpg" style="width: 500px; height: 390px;"></p></div><div class="q-answer"><p>a) Height h = <input type="text" style="height: auto;" data-c="20.9"> <span class="review"></span> </p><p>b) Angle of depression, a = <input type="text" style="height: auto;" data-c="71.1"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) <img alt="" src="../../files/GT/RAT/q9-answer.jpg" style="width: 250px; height: 186px;"> <span class="math-tex">\(tan60\quad =\quad \frac { h }{ 12 } \\ so,\quad h\quad =\quad 12\times tan60\\ h\quad =\quad 20.9m\quad (3sf)\)</span></p><p>b) <img alt="" src="../../files/GT/RAT/q9a2.jpg" style="width: 300px; height: 233px;"> <span class="math-tex">\(tanD\quad =\quad \frac { 35 }{ 12 } \\ so,\quad D\quad =\quad { tan }^{ -1 }\left( \frac { 35 }{ 12 } \right) \\ h\quad =\quad 71.1°\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows ta narrow chanel of water between the cliff coasts of two islands. Some distances are shown.</p><p style="text-align: center;"><img alt="" src="../../files/LSP/PB/q10.jpg" style="width: 600px; height: 445px;"></p><p>The diagram also shows.....</p><ul><li>The top of a lighthouse at point <strong>A</strong></li><li>The view from the base of the light house at point <strong>B</strong></li><li>A boat at point <strong>C</strong></li><li>The view from the top of the cliff at point <strong>D</strong></li></ul><p>The angle of elevation of point D from point C is 65°. The angle of depression of point C from point D is 40°.</p><p>a) Find the width of the water chanel by finding the respective distances x and y to 3sf. (Use rounded answers to get the total width)</p><p>The angle of elevation from point C to point A is 75° .</p><p>b) What is the height of the lighthouse above the the ground? (Use rounded ansers to any distances you have calculated)</p><p>c) What is the angle, p, of depression from the top of the light house to point D? (Use rounded ansers to any distances you have calculated)</p></div><div class="q-answer"><p>a) x = <input type="text" style="height: auto;" data-c="11.9"> <span class="review"></span> m to (3sf), y = <input type="text" style="height: auto;" data-c="8.39"> <span class="review"></span> m (3sf),</p><p>Total distance = <input type="text" style="height: auto;" data-c="20.3"> <span class="review"></span> m</p><p>b) h = <input type="text" style="height: auto;" data-c="34.4"> <span class="review"></span> m</p></div><div class="q-explanation"><p>a) <img alt="" src="../../files/LSP/PB/q10a2.jpg" style="width: 200px; height: 136px;"> <span class="math-tex">\(tan40\quad =\quad \frac { 10 }{ x } \\ x\quad =\quad \frac { 10 }{ tan40 } \\ x\quad =\quad 11.9\quad (3sf)\)</span></p><p> <img alt="" src="../../files/LSP/PB/q10a1.jpg" style="width: 200px; height: 193px;"> <span class="math-tex">\(tan65\quad =\quad \frac { 18 }{ y } \\ y\quad =\quad \frac { 18 }{ tan65 } \\ y\quad =\quad 8.39\quad (3sf)\)</span></p><p>Total distance = 8.39 + 11.9 = 20.29 , 20.3 (3sf)</p><p>b) <img alt="" src="../../files/LSP/PB/q10a3.jpg" style="width: 200px; height: 395px;"> <span class="math-tex">\(tan75\quad =\quad \frac { h+10 }{ 11.9 } \\ h\quad +\quad 10\quad =\quad 11.9\times tan75\\ h\quad =\quad 11.9\times tan75\quad -\quad 10\\ h\quad =\quad 34.4\quad m\quad (3sf)\)</span></p><p>c) <img alt="" src="../../files/LSP/PB/q10a4.jpg" style="width: 200px; height: 100px;"> <span class="math-tex">\(tanp\quad =\quad \frac { 10\quad +\quad 34.4\quad -\quad 18 }{ 11.9 } =\frac { 26.4 }{ 20.3 } \\ p\quad =\quad { tan }^{ -1 }\left( \frac { 26.4 }{ 20.3 } \right) \\ p\quad =\quad 52.4\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-blue"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Non Right Angled Trigonometry</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz" data-stats="15-327-1094"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve the following equations, giving your answers correct to 3sf</p><p>a) <span class="math-tex">\(\frac { a }{ Sin20 } =\frac { 10 }{ sin30 } \)</span> b) <span class="math-tex">\(\frac { 18 }{ sinA } =\frac { 27 }{ sin40 } \)</span> c) <span class="math-tex">\({ a }^{ 2 }={ 7 }^{ 2 }+{ 10 }^{ 2 }-2\times 7\times 10\times cos48\)</span> d) <span class="math-tex">\(cosA\quad =\quad \frac { { 13 }^{ 2 }+{ 20 }^{ 2 }-{ 8 }^{ 2 } }{ 2\times 13\times 20 } \)</span></p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="6.84"> <span class="review"></span> b) b = <input type="text" style="height: auto;" data-c="25.4"> <span class="review"></span> ° </p><p>c) a = <input type="text" style="height: auto;" data-c="7.44"> <span class="review"></span> d) A = <input type="text" style="height: auto;" data-c="13.8"> <span class="review"></span> °</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(\frac { a }{ sin20 } =\frac { 10 }{ sin30 } \\ a\quad =\quad \frac { 10 }{ sin30 } \times sin20\\ a\quad =\quad 6.84\quad (3sf)\)</span> </p><p>b) <span class="math-tex">\(\frac { 18 }{ sinA } =\frac { 27 }{ sin40 } \\ \frac { sinA }{ 18 } =\frac { sin40 }{ 27 } \\ sinA\quad =\quad \frac { sin40 }{ 27 } \times 18\\ A\quad =\quad { sin }^{ -1 }\left( \frac { sin40 }{ 27 } \times 18 \right) \\ A\quad =\quad 25.4°\quad (3sf)\)</span> </p><p>c) <span class="math-tex">\({ a }^{ 2 }={ 7 }^{ 2 }+{ 10 }^{ 2 }-2\times 7\times 10\times cos\quad 48\\ a=\sqrt { 149\quad -\quad 140cos48 } \\ a\quad =\quad 7.44\quad (3sf)\)</span> </p><p>d) <span class="math-tex">\(cosA\quad =\quad \frac { { 13 }^{ 2 }+{ 20 }^{ 2 }-{ 8 }^{ 2 } }{ 2\times 13\times 20 } \\ A\quad =\quad { cos }^{ -1 }\left( \frac { { 13 }^{ 2 }+{ 20 }^{ 2 }-{ 8 }^{ 2 } }{ 2\times 13\times 20 } \right) \\ A\quad =\quad 13.8° (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following diagram</p><p style="text-align: center;"><img alt="" src="../../files/GT/RAT/q3.jpg" style="width: 600px; height: 272px;"></p><p>Mark each of the following statements about the diagram as either true (T) or false (F)</p></div><div class="q-answer"><p>Enter either 'T' for true or 'F' for false</p><p>a) <span class="math-tex">\(\frac { sinA }{ a } =\frac { sinB }{ b } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </p><p>b) <span class="math-tex">\({ b }^{ 2 }={ a }^{ 2 }+{ c }^{ 2 }-2ac\times cosQ\)</span> <input type="text" style="height: auto;" data-c="F"> <span class="review"></span> </p><p>c) <span class="math-tex">\(cosB\quad =\quad \frac { { a }^{ 2 }+{ c }^{ 2 }-{ b }^{ 2 } }{ 2ac } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </p><p>d) <span class="math-tex">\(\frac { c }{ sin(P+Q) } =\frac { a }{ sinA } \)</span> <input type="text" style="height: auto;" data-c="T"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) T - Angles and sides in each fraction are opposite eachother</p><p>b) F - Q is not the angle between sides a and c</p><p>c) T - B is the angle between a and c</p><p>d) T - P + Q is the angle opposite side c and likewise for A and a</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the diagram below.</p><p>a) What is the exact value of z?</p><p>b) Find the values of x and y (give your answer to 3sf)</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q3a.jpg" style="width: 400px; height: 165px;"></p></div><div class="q-answer"><p>z = <input type="text" style="height: auto;" data-c="100"> <span class="review"></span> °</p><p>x = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> cm</p><p>y = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>a) Since angles in a triangle add up to 180°, z = 100</p><p>b) <span class="math-tex">\(\frac { x }{ sin50 } =\frac { 10 }{ sin30 } \\ x\quad =\quad \frac { 10 }{ sin30 } \times sin50\\ x\quad =\quad 15.3\quad cm\quad (3sf)\)</span></p><p>c) <span class="math-tex">\(\frac { y }{ sin100 } =\frac { 10 }{ sin30 } \\ y\quad =\quad \frac { 10 }{ sin30 } \times sin100\\ y\quad =\quad 19.7\quad cm\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>For the triangle below</p><p>a) Calculate the value of z to 3sf</p><p>b) Use that rounded value for z and the sine rule to work out the value of x</p><p>c) Again, use the rounded value of z to calculate the area of the triangle</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q4.jpg" style="width: 400px; height: 336px;"></p></div><div class="q-answer"><p>a) z = <input type="text" style="height: auto;" data-c="67.0"> <span class="review"></span> ° , b) x = <input type="text" style="height: auto;" data-c="24.2"> <span class="review"></span> m, c) Area = <input type="text" style="height: auto;" data-c="479"> <span class="review"></span> <span class="math-tex">\({ cm }^{ 2 }\)</span></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(\frac { sinz }{ 57 } =\frac { sin44 }{ 43 } \\ sinz\quad =\quad \frac { sin44 }{ 43 } \times 57\\ z\quad =\quad { sin }^{ -1 }\left( \frac { sin44 }{ 43 } \times 57 \right) \\ z\quad =\quad 67.0°\quad (3sf)\)</span></p><p>b) Since z = 67.0° to 3sf, we can calculate that the thrid angle (opposite side x) is 23°</p><p> <span class="math-tex">\(\frac { x }{ sin23 } =\frac { 43 }{ sin44 } \\ x\quad =\quad \frac { 43 }{ sin44 } \times sin23\\ y\quad =\quad 24.2\quad cm\quad (3sf)\)</span></p><p>c) <span class="math-tex">\(Area\quad =\quad \frac { 1 }{ 2 } \times 43\times 57\times sin23\\ Area\quad =\quad 479{ cm }^{ 2 }\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the following diagram</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q5.jpg" style="width: 400px; height: 330px;"></p><p>a) Work out the value of x</p><p>b) Work out the area of the triangle</p></div><div class="q-answer"><p>a) x = <input type="text" style="height: auto;" data-c="85.0"> <span class="review"></span> cm</p><p>b) Area = <input type="text" style="height: auto;" data-c="1680"> <span class="review"></span> <span class="math-tex">\({ cm }^{ 2 }\)</span></p></div><div class="q-explanation"><p>a) <span class="math-tex">\({ x }^{ 2 }={ 52 }^{ 2 }+{ 65 }^{ 2 }-2\times 52\times 65\times cos95\\ a=\sqrt { 6929\quad -\quad 3380cos95 } \\ a\quad =\quad 85.0cm\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(Area\quad =\quad \frac { 1 }{ 2 } \times 52\times 65\times sin95\\ Area\quad =\quad 1683.56904\\ Area\quad =\quad 1680\quad { cm }^{ 2 }\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Answer questions referring to the diagram below.</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q6.jpg" style="width: 400px; height: 187px;"></p><p>a) Calculate the value of x to 3sf.</p><p>b) Calculate the value of y to 3sf.</p><p>c) Use the rounded value of x to work out the area of the triangle</p></div><div class="q-answer"><p>a) x = <input type="text" style="height: auto;" data-c="91.1"> <span class="review"></span> °</p><p>b) y = <input type="text" style="height: auto;" data-c="54.0"> <span class="review"></span> °</p><p>c) Area = <input type="text" style="height: auto;" data-c="102"> <span class="review"></span> <span class="math-tex">\({ cm }^{ 2 }\)</span></p></div><div class="q-explanation"><p>a) <span class="math-tex">\(cosx\quad =\quad \frac { { 12 }^{ 2 }+{ 17 }^{ 2 }-{ 21 }^{ 2 } }{ 2\times 12\times 17 } \\ x\quad =\quad { cos }^{ -1 }\left( \frac { { 12 }^{ 2 }+{ 17 }^{ 2 }-{ 21 }^{ 2 } }{ 2\times 12\times 17 } \right) \\ x\quad =\quad 91.1°\)</span></p><p>b) <span class="math-tex">\(cosy\quad =\quad \frac { { 12 }^{ 2 }+{ 21 }^{ 2 }-{ 17 }^{ 2 } }{ 2\times 12\times 21 } \\ y\quad =\quad { cos }^{ -1 }\left( \frac { { 12 }^{ 2 }+{ 21 }^{ 2 }-{ 17 }^{ 2 } }{ 2\times 12\times 21 } \right) \\ y\quad =\quad 54.0°\)</span></p><p>c) <span class="math-tex">\(Area\quad =\quad \frac { 1 }{ 2 } \times 12\times 17\times sin91.1\\ Area\quad =\quad 101.9812026\\ Area\quad =\quad 102\quad { cm }^{ 2 }\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the square based pyramid ABCDE</p><p><img alt="" src="../../files/GT/NRAT/q7.jpg" style="width: 400px; height: 285px;"> </p><p>EF = 4cm, GF = 3cm, CD = 6cm, EC = <span class="math-tex">\(\sqrt { 34 } cm\)</span></p><p>a) What is the angle <span class="math-tex">\(\hat { DEC } \)</span> (to 3sf)</p><p>b) What is the angle, y, between FG and the plane ECD</p><p>c) What is the angle, z, between EC and the plane ABCD</p></div><div class="q-answer"><p>a) <span class="math-tex">\(\hat { DEC } \)</span> = <input type="text" style="height: auto;" data-c="61.9"> <span class="review"></span> °</p><p>b) Angle = <input type="text" style="height: auto;" data-c="53.1"> <span class="review"></span> °</p><p>c) Angle = <input type="text" style="height: auto;" data-c="43.3"> <span class="review"></span> °</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(cos\hat { DEC } \quad =\quad \frac { { \left( \sqrt { 34 } \right) }^{ 2 }+{ \left( \sqrt { 34 } \right) }^{ 2 }-{ 6 }^{ 2 } }{ 2\times \sqrt { 34 } \times \sqrt { 34 } } \\ \hat { DEC } \quad =\quad { cos }^{ -1 }\left( \frac { 34+34-{ 6 }^{ 2 } }{ 2\times 34 } \right) \\ \hat { DEC } \quad =\quad 61.9°\quad (3sf)\)</span> </p><p>Could also be done by splitting the triangle in totwo congruent right angled triangles since triangle ECD is isosceles.</p><p>b) Extracting a 2D triangle from the 3D situation. Triangle EFG is right angled with sides EF and FG known.</p><p><img alt="" src="../../files/GT/NRAT/q7a1.jpg" style="width: 300px; height: 213px;"> <span class="math-tex">\(y\quad =\quad \hat { FGE } \\ tany\quad =\quad \frac { 4 }{ 3 } \\ y\quad =\quad { tan }^{ -1 }\left( \frac { 4 }{ 3 } \right) \\ y\quad =\quad 53.1°\quad (3sf)\)</span></p><p>c) Extracting a 2D triangle from the 3D situation. Triangle EFC is right angled with sides EF and EC known.</p><p><img alt="" src="../../files/GT/NRAT/q7a2.jpg" style="width: 300px; height: 204px;"> <span class="math-tex">\(z\quad =\quad \hat { FCE } \\ sinz\quad =\quad \frac { 4 }{ \sqrt { 34 } } \\ z\quad =\quad { sin }^{ -1 }\left( \frac { 4 }{ \sqrt { 34 } } \right) \\ z\quad =\quad 43.3°\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use the map below to answer the questions</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q8a.jpg" style="width: 600px; height: 337px;"></p><ul><li>The map shows the locations of four boats A, B, C and D</li><li>The shortest distance from A to B is 1km</li><li>The shortest distance from B to C is 800m</li><li>The angle ABC is 100°</li></ul><p>a) What is the shortest distance from A to C in km? (to 3sf)</p><p>b) Use the result to part a) and the sine rule to work out the angle BCA</p><ul><li>Angle CBD = 60°</li><li>The distance BD = 300m</li></ul><p>c) What is the distance AD in km? (to 3sf)</p><ul><li>Angle DAC is 30°</li><li>Using rounded answers to previous parts of the question</li></ul><p>d) What is the distance DC?</p></div><div class="q-answer"><p>a) AC = <input type="text" style="height: auto;" data-c="1.38"> <span class="review"></span> km</p><p>b) Angle BCA = <input type="text" style="height: auto;" data-c="45.5"> <span class="review"></span> °</p><p>c) DIstance AD = <input type="text" style="height: auto;" data-c="1.29"> <span class="review"></span> km</p><p>d) Distance DC = <input type="text" style="height: auto;" data-c="0.696"> <span class="review"></span> km</p></div><div class="q-explanation"><p>a) Looking at the the triangle ABC</p><p><img alt="" src="../../files/GT/NRAT/q8a1.jpg" style="width: 300px; height: 162px;"> <span class="math-tex">\({ AC }^{ 2 }={ 1 }^{ 2 }+{ 0.8 }^{ 2 }-2\times 1\times 0.8\times cos100\\ AC=\sqrt { { 1 }^{ 2 }+{ 0.8 }^{ 2 }-2\times 1\times 0.8\times cos100 } \\ AC\quad =\quad 1.38km\quad (3sf)\)</span></p><p>b) Using AC = 1.38 km </p><p><span class="math-tex">\(\frac { sin\hat { BCA } }{ 1 } =\frac { sin100 }{ 1.38 } \\ sin\hat { BCA } \quad =\quad \frac { sin100 }{ 1.38 } \times 1\\ \hat { BCA } \quad =\quad { sin }^{ -1 }\left( \frac { sin100 }{ 1.38 } \times 1 \right) \\ \hat { BCA } \quad =\quad 45.5°\quad (3sf)\)</span></p><p>c) Looking at Triangle ABD</p><p><img alt="" src="../../files/GT/NRAT/q8a2.jpg" style="width: 300px; height: 159px;"> <span class="math-tex">\({ AD }^{ 2 }={ 1 }^{ 2 }+0.3^{ 2 }-2\times 1\times 0.3\times cos100\\ AD=\sqrt { { 1 }^{ 2 }+{ 0.3 }^{ 2 }-2\times 1\times 0.3\times cos100 } \\ AD\quad =\quad 1.29km\quad (3sf)\)</span></p><p>d) Considering triangle ADC</p><p><img alt="" src="../../files/GT/NRAT/q8a3.jpg" style="width: 300px; height: 167px;"> <span class="math-tex">\({ DC }^{ 2 }={ 1 }.29^{ 2 }+1.38^{ 2 }-2\times 1.29\times 1.38\times cos30\\ DC=\sqrt { { 1 }.29^{ 2 }+1.38^{ 2 }-2\times 1.29\times 1.38\times cos30 } \\ DC\quad =\quad 0.696km\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>The diagram below shows a metal structure.</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q9a.jpg" style="width: 500px; height: 297px;"></p><ul><li>Give all answers correct to 3sf</li><li>Use rounded answers in subsequent calculations</li></ul><p>a) What is the length AB?</p><p>b) What is angle CDB?</p><p>c) Which is longer AD or AC?</p></div><div class="q-answer"><p>a) AB = <input type="text" style="height: auto;" data-c="4.77"> <span class="review"></span> m</p><p>b) Angle CDB = <input type="text" style="height: auto;" data-c="61.1"> <span class="review"></span> °</p><p>c) Which is longer AD or DC? <input type="text" style="height: auto;" data-c="DC"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) <span class="math-tex">\(\frac { AB }{ sin50 } =\frac { 4 }{ sin40 } \\ AB\quad =\quad \frac { 4 }{ sin40 } \times sin50\\ AB\quad =\quad 4.77\quad m\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(\frac { sin\hat { CDB } }{ 4 } =\frac { sin50 }{ 3.5 } \\ sin\hat { CDB } \quad =\quad \frac { sin50 }{ 3.5 } \times 4\\ \hat { CDB } \quad =\quad { sin }^{ -1 }\left( \frac { sin50 }{ 3.5 } \times 4 \right) \\ \hat { CDB } \quad =\quad 61.1°\quad (3sf)\)</span></p><p>c) For the length AD, since angle CDB = 61.1° then ADB must equal 118.9° and so angle DBA must be 21.1°</p><p><span class="math-tex">\(\frac { AD }{ sin21.1 } =\frac { 3.5 }{ sin40 } \\ AD\quad =\quad \frac { 3.5 }{ sin40 } \times sin21.1\\ AD\quad =\quad 1.96\quad m\quad (3sf)\)</span></p><p>For the length AC, since angle CDB is 61.1°, then DBC must be 68.9°</p><p><span class="math-tex">\(\frac { DC }{ sin68.9 } =\frac { 3.5 }{ sin50 } \\ DC\quad =\quad \frac { 3.5 }{ sin50 } \times sin68.9\\ DC\quad =\quad 4.26\quad m\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Triangle ABC is the base of a triangular prism as shown in the diagram below</p><p style="text-align: center;"><img alt="" src="../../files/GT/NRAT/q10.jpg" style="width: 700px; height: 251px;"></p><ul><li>The length AB is 10cm</li><li>The length AC is 13cm</li><li>The Length BC is 7cm</li><li>The prism has a depth of 5cm</li></ul><p>a) Calculate the angle BAC</p><p>b) What is the area of triangle ABC</p><p>A line is drawn from the point E down to the mid point of AC, M.</p><p>c) How long is that line EM?</p><p>d) What angle does the line EM make with the plane ABC</p></div><div class="q-answer"><p>a) Angle BAC = <input type="text" style="height: auto;" data-c="32.2"> <span class="review"></span> °</p><p>b) Area = <input type="text" style="height: auto;" data-c="34.6"> <span class="review"></span> cm<sup>2</sup></p><p>c) The line EM has length <input type="text" style="height: auto;" data-c="5.68"> <span class="review"></span> cm</p><p>d) The angle EM makes with the plane ABC = <input type="text" style="height: auto;" data-c="41.4"> <span class="review"></span> °</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(cos\hat { BAC } \quad =\quad \frac { { 10 }^{ 2 }+{ 13 }^{ 2 }-{ 7 }^{ 2 } }{ 2\times 10\times 13 } \\ \hat { BAC } \quad =\quad { cos }^{ -1 }\left( \frac { { 10 }^{ 2 }+{ 13 }^{ 2 }-{ 7 }^{ 2 } }{ 2\times 10\times 13 } \right) \\ \hat { BAC } \quad =\quad 32.2°\quad (3sf)\)</span></p><p>b) <span class="math-tex">\(Area\quad =\quad \frac { 1 }{ 2 } \times 10\times 13\times sin32.2\\ Area\quad =\quad 34.6\quad (3sf)\)</span></p><p>c) and d) Consider the triangle EBM</p><p><img alt="" src="../../files/GT/NRAT/q10a.jpg" style="width: 600px; height: 251px;"></p><p>The angle required is the angle EMB (The angle the line EM makes with the plane ABC)</p><p>EB is known as 5cm (depth of prism)</p><p>BM can be calculated using the cosine rule as follows.....</p><p><span class="math-tex">\({ BM }^{ 2 }=10^{ 2 }+6.5^{ 2 }-2\times 10\times 6.5\times cos32.2\\ BM=\sqrt { 10^{ 2 }+6.5^{ 2 }-2\times 10\times 6.5\times cos32.2 } \\ BM\quad =\quad 5.68cm\quad (3sf)\)</span></p><p>Now using triangle EBM and BM = 5.68</p><p><span class="math-tex">\(tan\hat { EMB } \quad =\quad \frac { 5 }{ 5.68 } \\ EMB\quad =\quad { tan }^{ -1 }\left( \frac { 5 }{ 5.68 } \right) \\ EMB\quad =\quad 41.4°\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-blue"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a></div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-purple"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Calculus</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Concept of Differentiating</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1094" data-mtime="1675505848.2165"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Another way to find the rate of change of a particular function using a graph is to find the <input type="text" style="height: auto;" data-c=" gradient"> <span class="review"></span> at that point.</p></div><div class="q-answer"><p> </p></div><div class="q-explanation"><p> </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question">Please complete the sentence.</div><div class="q-answer"><p>The instantaneous rate of change of a function <span class="math-tex">\(f(x)\)</span> at a point A on a curve is given by the <input type="text" style="height: auto;" data-c=" tangent"> <span class="review"></span> to the curve at point A</p></div><div class="q-explanation"></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find:</p><p><span class="math-tex">\(\lim _{ x\rightarrow 3 }{ (x+4) } \)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p></div><div class="q-explanation"><p>This is asking us to find the limit, as x approaches 3, for the expression x+4.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find:</p><p><span class="math-tex">\(\lim _{ x\rightarrow 3 }{ ({ x }^{ 2 }-3) } \)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>This is asking us to find the limit, as x approaches 3, for the expression <span class="math-tex">\(x^2-3\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Identify the gradient/rate of change of the following functions</p></div><div class="q-answer"><p>y = 3x +2 <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>y= -5x <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span> </p><p>y = 7 <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> </p><p>y = 0.5x - 4 <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>These are all straight line (linear) functions in the form y = mx + c, where m is the gradient. In the case of y = 7, we can think of this as y = 0x + 7 and so the gradient is 0. Graphs like this are horizontal lines so it is logical to conclude that they have gradient = 0.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <img align="middle" alt="f left parenthesis x right parenthesis equals 2 x to the power of 5" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-9.php?formula=cfe5d309eb8164946c463240c3dc5b82.png"> for which <img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals a x to the power of b" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»a«/mi»«msup»«mi»x«/mi»«mi»b«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-10.php?formula=e671c34160ee0d23060d2d296a1eabf1.png">. What are the values of a and b?</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="i f space f left parenthesis x right parenthesis space equals space a x to the power of n comma space f apostrophe left parenthesis x right parenthesis space equals space a n x to the power of n minus 1 end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«mo»,«/mo»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«msup»«mi»x«/mi»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-11.php?formula=50dfafd9e6d11660b4838c85ab299f98.png"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <img align="middle" alt="f left parenthesis x right parenthesis equals 1 over x plus fraction numerator 3 over denominator 4 x squared end fraction" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mfrac»«mn»1«/mn»«mi»x«/mi»«/mfrac»«mo»+«/mo»«mfrac»«mn»3«/mn»«mrow»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-15.php?formula=3bdce127c358f891c8528925318b0adb.png">, for which <img align="middle" alt="f apostrophe left parenthesis x right parenthesis equals a x to the power of b minus c over 2 x to the power of negative d end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»a«/mi»«msup»«mi»x«/mi»«mi»b«/mi»«/msup»«mo»-«/mo»«mfrac»«mi»c«/mi»«mn»2«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mi»d«/mi»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-16.php?formula=a16ac5da8502af7cedadcfee3d0f4c07.png">. What are the values of a, b, c and d? (do it on paper and be careful with the signs)</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> , b= <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> , d = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="i f space f left parenthesis x right parenthesis space equals space x to the power of negative 1 end exponent plus 3 over 4 x to the power of negative 2 end exponent space s o space f apostrophe left parenthesis x right parenthesis space equals space minus x to the power of negative 2 end exponent minus 6 over 4 x to the power of negative 3 end exponent space space space space a n d space 6 over 4 equals 3 over 2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»i«/mi»«mi»f«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«mo»+«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»2«/mn»«/mrow»«/msup»«mo»§#160;«/mo»«mi»s«/mi»«mi»o«/mi»«mo»§#160;«/mo»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»2«/mn»«/mrow»«/msup»«mo»-«/mo»«mfrac»«mn»6«/mn»«mn»4«/mn»«/mfrac»«msup»«mi»x«/mi»«mrow»«mo»-«/mo»«mn»3«/mn»«/mrow»«/msup»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»a«/mi»«mi»n«/mi»«mi»d«/mi»«mo»§#160;«/mo»«mfrac»«mn»6«/mn»«mn»4«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-17.php?formula=c2448d84731771fe1458cfb514785f49.png"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the gradient of the function <img align="middle" alt="f left parenthesis x right parenthesis space equals space 4 x squared plus 3 x" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»3«/mn»«mi»x«/mi»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-15.php?formula=1740d4882b338149f0242a7d8ae4127d.png"> when x = 2?</p></div><div class="q-answer"><p>Gradient = <input type="text" style="height: auto;" data-c="19"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="f apostrophe left parenthesis x right parenthesis space equals space 8 x plus 3 comma space s u b s t i t u t e space x space equals space 2f apostrophe left parenthesis 2 right parenthesis space equals space 19" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»8«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»s«/mi»«mi»u«/mi»«mi»b«/mi»«mi»s«/mi»«mi»t«/mi»«mi»i«/mi»«mi»t«/mi»«mi»u«/mi»«mi»t«/mi»«mi»e«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mspace linebreak=¨newline¨/»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»19«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-16.php?formula=849a1c49afd0a00461f15b26b389254b.png"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the gradient of the function <img align="middle" alt="f left parenthesis x right parenthesis space equals space minus 2 x cubed minus 4 x squared plus 5 x minus 3" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»4«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»5«/mn»«mi»x«/mi»«mo»-«/mo»«mn»3«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-17.php?formula=a24857962e9bb2ca0fe4a34fee38233c.png"> when x = -3?</p></div><div class="q-answer"><p>Gradient = <input type="text" style="height: auto;" data-c="-25"> <span class="review"></span> </p></div><div class="q-explanation"><p><img align="middle" alt="f apostrophe left parenthesis x right parenthesis space equals space minus 6 x squared minus 8 x plus 5 comma space w h e n space x space equals space minus 3f apostrophe left parenthesis negative 3 right parenthesis space equals space minus 54 space minus space minus space 24 space plus space 5 space equals space minus 25" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»8«/mn»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mi»w«/mi»«mi»h«/mi»«mi»e«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»x«/mi»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»3«/mn»«mspace linebreak=¨newline¨/»«mi»f«/mi»«mo»`«/mo»«mo»(«/mo»«mo»-«/mo»«mn»3«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»54«/mn»«mo»§#160;«/mo»«mo»-«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mo»§#160;«/mo»«mn»24«/mn»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»25«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-18.php?formula=d0f74f080faa5734a3256fb187fac8c6.png"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Identify the gradient/rate of change of the following functions</p></div><div class="q-answer"><p>y = 7x +2 <input type="text" style="height: auto;" data-c="7"> <span class="review"></span> </p><p>y= 3x <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>y = 4 <input type="text" style="height: auto;" data-c="0"> <span class="review"></span> </p><p>y = 1.5x - 4 <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span> </p></div><div class="q-explanation"><p>These are all straight line (linear) functions in the form y = mx + c, where m is the gradient. In the case of y = 7, we can think of this as y = 0x + 7 and so the gradient is 0. Graphs like this are horizontal lines so it is logical to conclude that they have gradient = 0.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Increasing and Decreasing Functions</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1094" data-mtime="1675505848.2192"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Fill in the missing parts in the following sentences.</p></div><div class="q-answer"><p>If <span class="math-tex">\(f'(x)>0, \)</span>the graph of <span class="math-tex">\(f(x) \)</span> is <input type="text" style="height: auto;" data-c="increasing"> <span class="review"></span></p></div><div class="q-explanation"><p>Use this optional field for teacher explanations.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Fill in the missing parts in the following sentences.</p></div><div class="q-answer"><p>If <span class="math-tex">\(f'(x)=0, \)</span>the graph of <span class="math-tex">\(f(x) \)</span> is <input type="text" style="height: auto;" data-c="stationary"> <span class="review"></span></p></div><div class="q-explanation"><p>Use this optional field for teacher explanations.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Fill in the missing parts in the following sentences.</p></div><div class="q-answer"><p>If <span class="math-tex">\(f'(x)<0, \)</span>the graph of <span class="math-tex">\(f(x) \)</span> is <input type="text" style="height: auto;" data-c="decreasing"> <span class="review"></span></p></div><div class="q-explanation"><p>Use this optional field for teacher explanations.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Tangents and Normals</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1094" data-mtime="1675505848.2216"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="tib-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn></math></span>, The tangent to this curve at x = -2 is given by y = ax +b. Find the values of a and b</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="tib-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>s</mi><mi>o</mi><mo> </mo><mi>f</mi><mo>'</mo><mo>(</mo><mo>-</mo><mn>2</mn><mo>)</mo><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn></math></span> Therefore the gradient, a = -2. So you know y = -2x + b, <span class="tib-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mo>-</mo><mn>2</mn><mo>)</mo><mo> </mo><mo>=</mo><mo> </mo><mn>12</mn></math></span> which means the line goes through the point (-2, 12). N<span style="background-color: initial;">ow substitute x = -2 and y = 12 (from the gievn point) to get the value of b.</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the function <span class="tib-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></math></span> , The tangent to this curve at x = 2 is given by y = ax +b. Find the values of a and b.</p></div><div class="q-answer"><p>a = <input type="text" style="height: auto;" data-c="1.35"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="-0.85"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="tib-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>,</mo><mo> </mo><mi>s</mi><mi>o</mi><mo> </mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo> </mo><mi>a</mi><mi>n</mi><mi>d</mi><mo> </mo><mi>f</mi><mo>'</mo><mo>(</mo><mn>2</mn><mo>)</mo><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo>.</mo><mn>35</mn></math></span></p><p>You can read this from the table function on your GDC. <img align="middle" alt="f left parenthesis 2 right parenthesis equals 1.85" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mn»2«/mn»«mo»)«/mo»«mo»=«/mo»«mn»1«/mn»«mo».«/mo»«mn»85«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-29.php?formula=2aff32144589066b9558ff792a609dbc.png">, so the line goes through the point (2, 1.85). Use this to work out the value of c when y = 1.35x + c (since you know that the gradient is 1.35)</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the tangent of <span class="math-tex">\(f(x)=\frac { 8{ x }^{ 2 } }{ 6 } +\frac { { x }^{ 4 } }{ 3 } -5{ x }^{ 2 }+9\)</span> when x = 3.5. If the tangent is <span class="math-tex">\(y=mx+c\)</span>, find the values of m and c (<em>with calculator</em>) <em>(Exact or 3.s.f)</em></p></div><div class="q-answer"><p><span class="math-tex">\(m=\)</span> <input type="text" style="height: auto;" data-c="31.5"> <span class="review"></span> </p><p><span class="math-tex">\(c=\)</span> <input type="text" style="height: auto;" data-c="-96.1"> <span class="review"></span> </p></div><div class="q-explanation"><p>On GDC; graph; sketch; tangent; x=3.5</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the tangent of <span class="math-tex">\(f(x)=\frac { { x }^{ 3 } }{ 2 } +8{ x }^{ 2 }-x+7\)</span> when x = 2. If the tangent is <span class="math-tex">\(y=mx+c\)</span>, find the values of m and c <em>(without a calculator - you may be asked to 'show') (Exact or 3.s.f)</em></p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="37"> <span class="review"></span> </p><p>c = <input type="text" style="height: auto;" data-c="-33"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=\frac { { x }^{ 3 } }{ 2 } +8{ x }^{ 2 }-x+7\\ f'(x)=\frac { 3{ x }^{ 2 } }{ 2 } +16x-1\\ at\quad x=2,\quad f(2)=41,\quad f'(2)=37\\ so,\quad at\quad (2,41),\quad m=37\\ Tangent\quad is\quad y=mx+c\\ so\quad y=37x+c,\\ Substitute\quad x=2\quad and\quad y=41\\ 41=37\times 2+c\\ so,\quad c=-33\\ tangent\quad is\quad y=37x-33\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the tangent of <span class="math-tex">\(f(x)=\frac { { 3x }^{ 2 } }{ 2 } -6{ x }+9\)</span> when x = 4. If the tangent is <span class="math-tex">\(y=mx+c\)</span>, find the values of m and c <em>(without a calculator - you may be asked to 'show') (Exact or 3.s.f)</em></p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> </p><p>c = <input type="text" style="height: auto;" data-c="-15"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=\frac { { 3x }^{ 2 } }{ 2 } -6{ x }+9\\ \)</span></p><p><span class="math-tex">\(f'(x)=3x-6\)</span></p><p>at <span class="math-tex">\( x=4,\quad f(4)=9,\quad f'(4)=6\)</span></p><p>so <span class="math-tex">\((4,9),\quad m=6\\\)</span></p><p>tangent is <span class="math-tex">\( y=mx+c\)</span></p><p>so<span class="math-tex">\(y=6x+c,\)</span></p><p>substitute <span class="math-tex">\( x=4\quad and\quad y=9\)</span></p><p><span class="math-tex">\(\ 9=6\times 4+c\)</span></p><p>so <span class="math-tex">\(c=-15\)</span></p><p>therefore tangent is <span class="math-tex">\(y=6x-15\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the normal of <span class="math-tex">\(f(x)=3{ x }^{ -3 }+\frac { x }{ 4 } -2\)</span> when x = 8. If the normal is <span class="math-tex">\(y=mx+c\)</span>, find the values of m and c (<em>with calculator</em>) <em>(3.s.f)</em></p></div><div class="q-answer"><p><span class="math-tex">\(m=\)</span> <input type="text" style="height: auto;" data-c="-4.04"> <span class="review"></span> </p><p><span class="math-tex">\(c=\)</span> <input type="text" style="height: auto;" data-c="32.3"> <span class="review"></span> </p></div><div class="q-explanation"><p>On GDC; graph; sketch; normal; x=8</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the normal of <span class="math-tex">\(f(x)=2{ x }^{ 2 }+3x-5\)</span> when x = 3. If the normal is <span class="math-tex">\(y=mx+c\)</span>, find the values of m and c (<em>show your working</em>) <em>(3.s.f)</em></p></div><div class="q-answer"><p><span class="math-tex">\(m=\)</span> <input type="text" style="height: auto;" data-c="-0.0667"> <span class="review"></span> </p><p><span class="math-tex">\(c=\)</span> <input type="text" style="height: auto;" data-c="22.2"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=2{ x }^{ 2 }+3x-5\\ f'(x)=4x+3\\ at\quad x=3,\quad f(3)=22,\quad f'(3)=15\\ so,\quad at\quad (3,22),\quad m(tangent)=15\\ Normal\quad is\quad y=-\frac { 1 }{ m } x+c\\ so\quad y=-\frac { 1 }{ 15 } x+c\\ Substitute\quad x=3\quad and\quad y=22\\ 22=-\frac { 1 }{ 15 } \times 3+c\\ so,\quad c=22.2\\ Normal\quad is\quad y=-0.0667x+22.2\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the normal of <span class="math-tex">\(f(x)=4{ x }^{ 3 }+2{ x }^{ 2 }+6\)</span> when x = 2. If the normal is <span class="math-tex">\(y=mx+c\)</span>, find the values of m and c (<em>show your working)</em>) <em>(3.s.f)</em></p></div><div class="q-answer"><p><span class="math-tex">\(m=\)</span> <input type="text" style="height: auto;" data-c="-0.0179"> <span class="review"></span> </p><p><span class="math-tex">\(c=\)</span> <input type="text" style="height: auto;" data-c="46.0"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=4{ x }^{ 3 }+2{ x }^{ 2 }+6\\ f'(x)=12{ x }^{ 2 }+4x\\ at\quad x=2,\quad f(2)=46,\quad f'(2)=56\\ so,\quad at\quad (2,46),\quad m(tangent)=56\\ Normal\quad is\quad y=-\frac { 1 }{ m } x+c\\ so\quad y=-\frac { 1 }{ 56 } x+c\\ Substitute\quad x=2\quad and\quad y=46\\ 46=-\frac { 1 }{ 56 } \times 2+c\\ so,\quad c=46.0\\ Normal\quad is\quad y=-0.0179x+46.0\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the normal of <span class="math-tex">\(f(x)=\frac { 2 }{ { x }^{ 3 } } +3{ x }^{ 2 }-\frac { 4 }{ x } \)</span> when x = 2. If the normal is <span class="math-tex">\(y=mx+c\)</span>, find the values of m and c (<em>show your working)</em>) <em>(3.s.f)</em></p></div><div class="q-answer"><p><span class="math-tex">\(m=\)</span> <input type="text" style="height: auto;" data-c="-0.0792"> <span class="review"></span> </p><p><span class="math-tex">\(c=\)</span> <input type="text" style="height: auto;" data-c="10.4"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=\frac { 2 }{ { x }^{ 3 } } +3{ x }^{ 2 }-\frac { 4 }{ x } \\ f'(x)=-6{ x }^{ -4 }+6x+4{ x }^{ -2 }\\ at\quad x=2,\quad f(2)=\frac { 41 }{ 4 } ,\quad f'(2)=\frac { 101 }{ 8 } \\ so,\quad at\quad (2,\frac { 41 }{ 4 } ),\quad m(tangent)=\frac { 101 }{ 8 } \\ Normal\quad is\quad y=-\frac { 1 }{ m } x+c\\ so\quad y=-\frac { 1 }{ (\frac { 101 }{ 8 } ) } x+c\\ Substitute\quad x=2\quad and\quad y=\frac { 41 }{ 4 } \\ \frac { 41 }{ 4 } =-\frac { 1 }{ (\frac { 101 }{ 8 } ) } \times 2+c\\ so,\quad c=10.4\\ Normal\quad is\quad y=-0.0792x+10.4\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the normal of <span class="math-tex">\(f(x)=\frac { 1 }{ { x }^{ 2 } } -3{ x }^{ -4 }+{ x }^{ 3 }\)</span> when x = 1. If the normal is <span class="math-tex">\(y=mx+c\)</span>, find the values of m and c (<em>show your working)</em>) <em>(3.s.f)</em></p></div><div class="q-answer"><p><span class="math-tex">\(m=\)</span> <input type="text" style="height: auto;" data-c="-0.0769"> <span class="review"></span> </p><p><span class="math-tex">\(c=\)</span> <input type="text" style="height: auto;" data-c="-0.923"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(f(x)=\frac { 1 }{ { x }^{ 2 } } -3{ x }^{ -4 }+{ x }^{ 3 }\\\)</span></p><p><span class="math-tex">\( f'(x)=-2{ x }^{ -3 }+12{ x }^{ -5 }+3{ x }^{ 2 }\\\)</span></p><p>at <span class="math-tex">\( x=1,\quad f(1)=-1,\quad f'(1)=13\)</span></p><p><span class="math-tex">\(\\ so,\quad at\quad (1,-1),\quad m(tangent)=13\)</span></p><p><span class="math-tex">\(\\ Normal\quad is\quad y=-\frac { 1 }{ m } x+c\)</span></p><p><span class="math-tex">\(\\ so\quad y=-\frac { 1 }{ 13 } x+c\)</span></p><p>Substituting <span class="math-tex">\(x=1\quad and\quad y=-1\)</span></p><p><span class="math-tex">\(\\ -1=-\frac { 1 }{ 13 } \times 1+c\)</span></p><p>so <span class="math-tex">\(c=-0.923\)</span></p><p><span class="math-tex">\( Normal\quad is\quad y=-0.0769x-0.923\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Stationary Points</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1094" data-mtime="1675505848.2239"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question">The x value at the minimum point on the curve <span class="math-tex">\(y=4x^{2}+kx+38\)</span> is 3. Find the value of k.</div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="-24"> <span class="review"></span> </p></div><div class="q-explanation"><p>First differentiate to find the gradient function:</p><p><span class="math-tex">\(\frac{dy}{dx}=8x+k\)</span></p><p>As the minimum point exists when x=3, set the derivative equal to zero and then solve to find k:</p><p><span class="math-tex">\(0=8\left(3\right)+k\)</span></p><p><span class="math-tex">\(k=-24\)</span></p><p><br> </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question">The x value at the maximum point on the curve <span class="math-tex">\(y=kx^{2}-36x-160\)</span> is -9. Find the value of k.</div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></p></div><div class="q-explanation"><p>First differentiate to find the gradient function:</p><p><span class="math-tex">\(\frac{dy}{dx}=2kx-36\)</span></p><p>As the maximum point exists when x=-9, substitute in <span class="math-tex">\(x=-9\)</span> ,set the derivative equal to zero and then solve to find k:</p><p><span class="math-tex">\(0=-18k-36\)</span></p><p><span class="math-tex">\(k=-2\)</span></p><p><br> </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Second Derivative</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1094" data-mtime="1675505848.2257"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>In which ways can you classify a stationary point?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> By substituting the x-value of the stationary point into the second derivative and identifying whether it is negative, positive or equal to zero. </label></p><p><label class="checkbox"><input class="c" type="checkbox"> By sketching the graph and verifying that the point is a minimum, maximum or stationary inflexion point.</label></p><p><label class="checkbox"><input class="c" type="checkbox"> By looking at the gradient of points either side of the stationary point - i.e. using a sign diagram. </label></p></div><div class="q-explanation"><p>These are all correct responses. The most common method in an exam scenario is to use the second derivative. If you find that the second derivative is equal to zero, then do be aware that you also need to check the sign of either the first derivative or second derivative either side of the point to be sure that you have a stationary inflexion point.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>A model rocket follows a quadratic path, given by the function, <span class="math-tex">\(H(t)=16t-4{ t }^{ 2 }+2\)</span>, where H is the height of the rocket in metres and t is the time in seconds. Find the maximum height of the rocket during its path.</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="18 metres"> <span class="review"></span></p></div><div class="q-explanation"><p>Step 1: Differente</p><p><span class="math-tex">\(H'(t)=16-8t\)</span></p><p>Step 2: Find the maximum point when the gradient is equal to zero. It is a maximum because it is a negative quadratic curve.</p><p><span class="math-tex">\(16-8t=0\\ \therefore \quad t=2\)</span></p><p>Step 3: The maximum height occurs 2 seconds after launch. Substitute <span class="math-tex">\(t=2\)</span> into the original function.</p><p><span class="math-tex">\(H(2)=16(2)-4({ 2 }^{ 2 })+2\\H(2)=18\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Optimisation</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1094" data-mtime="1675505848.2276"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>A 30cm long string is shaped to create a rectangle. What width should the shape have to create the biggest possible area? </p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="7.5"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>Length=x</p><p>Width=15-x</p><p><span class="math-tex">\(A=\left(15-x\right)x\)</span></p><p>Differentiating:</p><p><span class="math-tex">\(\frac{dA}{dx}=15-2x\)</span></p><p>Where the Area is the maximum, the derivative is equal to zero:</p><p><span class="math-tex">\(0=15-2x\)</span></p><p>At the maximum area, the length is therefore 7.5cm</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Integration</p> </div> </div> <div class="panel-body"> <div> <div class="tib-quiz dyn-quiz" data-pid="1094" data-mtime="1675505848.2301"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Estimate the area between the curve, <span class="math-tex">\(y=2^x-1\)</span>, and the x-axis between <span class="math-tex">\(2 \le x\le5\)</span>, using 3 trapezoids. </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>x</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>y</td><td>3</td><td>8</td><td>15</td><td>31</td></tr></tbody></table></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="40"> <span class="review"></span> </p></div><div class="q-explanation"><p>First find the height of each trapezoid, <span class="math-tex">\(h\)</span> using the trapezoid formula, where <span class="math-tex">\(a=2,b=5 \space and \space n=3\)</span></p><p><span class="math-tex">\(h=\frac{b-a}{n}=\frac{5-2}{3}=1\)</span> (or simply notice that the height of each trapezoid is increasing by 1 in the top row of the table).</p><p>Now using the trapezoid rule formula: </p><p><span class="math-tex">\(\int _{ 2 }^{ 5 }{ { 2 }^{ x }-1\quad dx=\frac { 1 }{ 2 } \times 1\times (3+2(8+15)+31)=\space 40\space { units }^{ 2 } } \)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(\int { x²\quad dx } \)</span></p></div><div class="q-answer"><p><label class="checkbox"><span class="checkbox"><input type="checkbox"><span class="math-tex">\({ x }^{ 3 }\)</span></span></label><label class="checkbox"><span class="checkbox"><span class="checkbox"></span></span></label></p><p><label class="checkbox"><span class="checkbox"><span class="checkbox"><input class="c" type="checkbox"> <span class="radio"> <span class="math-tex">\(\frac { 1 }{ 3 } { x }^{ 3 }\)</span></span></span></span></label><label class="checkbox"><span class="radio"></span></label><label class="radio"></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac { 1 }{ 2 } { x }\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(0.\overline { 3 } { x }^{ 3 }\)</span></label></p></div><div class="q-explanation"><p>Add one to the power and divide by the power</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default 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