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HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2910/statistics-and-probability-examination-questions-hl.html">Statistics and Probability Examination Questions HL</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2912/calculus-examination-questions-hl.html">Calculus Examination Questions HL</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"> Functions Examination Questions HL <a href="#" class="mark-page-favorite pull-right" data-pid="2906" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../2902/examination-questions.html">Examination Questions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Functions Examination Questions HL</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p>On this page you can find examination questions from the topic of functions</p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Equation of a Straight Line</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="874"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>The line <span class="math-tex">\(l_1\)</span> has equation 2x+5y+6=0</p> <p>The line <span class="math-tex">\(l_2\)</span> is perpendicular to the line <span class="math-tex">\(l_1\)</span>and passes through the point (2 , -2)</p> <p>a) Find the equation of <span class="math-tex">\(l_2\)</span>in the form ax+by+d=0, where a, b and d are constants.</p> </div> <p>b) Find the coordinates where <span class="math-tex">\(l_2\)</span> meets the y axis</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content>If the gradient of a line is <em><strong>m</strong></em> , then the gradient of a perpendicular line is <span class="math-tex">\({-1 \over m}\)</span></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/straight-line-graphs/esq_functions_stlines1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/straight-line-graphs/esq_functions_stlines1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="872"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The point A has coordinates (a , 3) and the point B has coordinates (7 , b).</p> <p>The line AB has equation 2x + 3y = 11.</p> <p>a) Find the values of a and b</p> <p>The line AC is perpendicular to the line AB.</p> <p>b) Find the equation of the line AC in the form ax + by + d = 0, where a, b and d are constants</p> <p>c) Given that C lies on the x axis, find its coordinates.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">b) the line AC is perpendicular to AB and passes through the point A (found in part a)<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/straight-line-graphs/esq_functions_stlines2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/straight-line-graphs/esq_functions_stlines2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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> Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="873"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The line <span class="math-tex">\(l_1 \)</span> passes through the point P(3k , 2k) with gradient = -2.</p> <p><span class="math-tex">\(l_1 \)</span> meets the x axis at A and the y axis at B.</p> <p>a) Find the equation of the line <span class="math-tex">\(l_1 \)</span> and show that A(4k , 0)</p> <p>b) Find the area of the triangle AOB in terms of k</p> <p>The line <span class="math-tex">\(l_2\)</span> passes through P and is perpendicular to <span class="math-tex">\(l_1 \)</span></p> <p>c) Find the equation of <span class="math-tex">\(l_2\)</span></p> <p><span class="math-tex">\(l_2\)</span>meets the x axis at C</p> <p>d) Show that the midpoint of PC lies on the line y = x</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>The key to this type of question is to be able to visualize it.</p> <p><content>Draw a sketch to help you:</content></p> <p><img alt="" src="../../files/functions/straight-line-graphs/esq3-hint.png" style="width: 400px; height: 353px;"></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/straight-line-graphs/esq_functions_stlines3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/straight-line-graphs/esq_functions_stlines3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="871"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Point A has coordinates (<em><strong>a</strong></em> , 6) and point B has coordinates (5 , <em><strong>b</strong></em>).</p> <p>The line 8x - 6y + 3 = 0 is the perpendicular bisector of AB.</p> <p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Find the gradient of AB in terms of <strong><em>a</em></strong> and <strong><em>b</em></strong>.</p> <p><content>Gradient of AB is perpendicular to line 8x - 6y + 3 = 0 </content></p> <hr> <p> </p> <p><content>Find the midpoint of AB</content></p> <p>Midpoint should lie on line 8x - 6y + 3 = 0</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/straight-line-graphs/esq_functions_stlines4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/straight-line-graphs/esq_functions_stlines4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Functions - The Basics</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="864"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The function <strong><em>f</em></strong> and<strong><em> g</em></strong> are defined as follows</p> <p><span class="math-tex">\(f(x)=3x^2 \ ,\ x\in\mathbb{R}\)</span></p> <p><span class="math-tex">\(g\left(x\right)=\frac{2}{x}-4\ ,x\in\mathbb{R},\ x>0\)</span></p> <p>a) Find the range of <strong><em>g</em></strong></p> <p>b) Find g<sup>-1</sup></p> <p>c) Solve gf(x) = 2</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) To find the range, consider the graph of the function. What are all the possible y values?</p> <p><content>c) Careful x > 0</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/the-basics/esq_functions_basics1a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics1a.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="863"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <strong><em>f </em></strong>and <strong><em>g</em></strong> be the functionsLet <strong><em>f </em></strong>and <strong><em>g</em></strong> be the functions</p> <p><span class="math-tex">\(f(x)=ln(1-2x),x<\frac{1}{2}\)</span></p> <p><span class="math-tex">\(g(x)=\frac{4}{x-2},x\neq2\)</span></p> <p>a) Find the exact value of fg(-2)</p> <p>b) Find f<sup>-1</sup>(x), stating its domain</p> <p>c) Show that <span class="math-tex">\(gg(x)=\frac{4x-8}{8-2x}\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>b) domain of f<sup>-1 </sup>is the range of f</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/the-basics/esq_functions_basics2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="865"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>The graph of the function <strong><em>f</em></strong> is plotted below</p> </div> <p>Use the graph to find values of</p> <p>a) <span class="math-tex">\(ff(-2)\)</span></p> <p>b) <span class="math-tex">\(f^{-1}(0)\)</span></p> <p>Let <strong><em>g</em></strong> be the function such that g(x) = 4x + 2</p> <p>c) Work out fg(1)</p> <p>d) Draw a sketch of <span class="math-tex">\(y=f^{-1}(x)\)</span></p> <p><img alt="" src="../../files/functions/the-basics/esq3" style="width: 500px; height: 397px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Think of this question in terms of inputs and outputs</p> <p><content>f(-2) is the output from the function f when we input 2. From the graph, what is the y value, if the x value is -2?</content></p> <p>f(-2) = 1</p> <p>ff(-2) is the value that we get when the output from f(-2) becomes the new input into the function f</p> <p>ff(-2) = 2</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/the-basics/esq_functions_basics3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="866"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(f(x)=\frac{ax-1}{x-a},\ x\neq\ a\)</span></p> <p>a) Show that <span class="math-tex">\(f^{-1}(x)=f(x)\)</span></p> <p>b) Hence, find ff(b)</p> <p>c) The graph of y = g(x) is shown below. Find the values of <strong><em>a</em></strong> if fg(3) = a + 5</p> <p><img alt="" src="../../files/functions/the-basics/esq4" style="width: 500px; height: 385px;"></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">b) Notice the word 'hence'. This means that you should use what you found out in part a to answer this question<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/the-basics/esq_functions_basics4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics4.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="893"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let the function <strong><em>f</em></strong> be defined by</p> <p><span class="math-tex">\(f(x)=\frac{x-k}{2-x},x\neq2\)</span></p> <p>a) Find <span class="math-tex">\(f^{-1}(x)\)</span> in terms of <strong><em>k</em></strong></p> <p>b) Find <strong><em>k</em></strong> if <span class="math-tex">\(y=f(x)\)</span> and <span class="math-tex">\(y=f^{-1}(x)\)</span> intersect at (-2 , -2) and (3 , 3)</p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Solve <span class="math-tex">\(f(x)=f^{-1}(x)\)</span><content></content></p> <p><span class="math-tex">\(\frac{x-k}{2-x}=\frac{2x+k}{1+x}\)</span></p> </section> <h4>Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Print from <a href="../../files/functions/the-basics/esq_functions_basics5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/the-basics/esq_functions_basics5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Quadratics</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="497"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>The graph below is the function f(x)=a(x-h)²+k</p> <p>The graph pass through the point (0,-9) and has vertex (2,3)</p> <p><img alt="" height="357" src="../../files/functions/quadratics/quadratics1image.jpg" width="295"></p> <div> <p>a) Write down the value of a, h and k</p> </div> <p>b) Find f(x) giving your answer in the form f(x) = Ax²+Bx+C</p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">h and k give the coordinates of the vertex (h,k)<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/quadratics/esq_quadratics1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="496"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let f(x) = 2x²+12x +11</p> <p>The function can also be expressed in the form f(x) = a(x-h)² +k</p> <p>a) Find the equation of the axis of symmetry</p> <p>b) Write down the value of h</p> <p>c) Write down the value of k</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">h and k give the coordinates of the vertex (h,k)<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/quadratics/esq_quadratics2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="498"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let f(x) = x² + 2px + (3p+4)</p> <p>Find the value of p so that f(x)=0 has two <strong>equal </strong>roots.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>You can use the discriminant to solve this question</p> <p><span class="math-tex">\(\triangle =b^ 2-4ac\)</span></p> <p><content>For two <strong>equal </strong>roots , b² - 4ac = 0</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/quadratics/esq_quadratics3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="501"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL hard"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>Let <strong>f(x)=2x² + kx + 1</strong> and <strong>g(x) = -x – 1</strong>.</p> <p>The graphs of <strong>f</strong> and <strong>g</strong> intersect at two distinct points.</p> <p>Find the possible values of <strong>k</strong></p> </div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" height="340" src="../../files/functions/quadratics/esq-quadratics4-image.png" width="284"></p> <p>If <strong>f</strong> and <strong>g</strong> intersect at two distinct points, then <strong>f(x)=g(x) </strong>has two roots (discriminant >0)</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/quadratics/esq_quadratics4hl.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics4hl.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="499"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A quadratic function <strong><em>f</em></strong> can be written in the form <strong><em>f(x) = a(x - h)² + k </em></strong>. The graph of <strong><em>f</em></strong> has vertex <strong><em>(-1 , 4)</em></strong> and has y-intercept at <strong><em>(0 , 5)</em></strong>.</p> <p>a) Find the value of <strong><em>a</em></strong> , <strong><em>h</em></strong> and <strong><em>k</em></strong>.</p> <p>b) The line <strong><em>y = mx + 1</em></strong> is a tangent to the curve <strong><em>f</em></strong>. Find the value of <strong><em>m</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/functions/quadratics/esq5-image.png" style="width: 350px; height: 452px;"></p> <p>b) If the line y = mx + 1 is a tangent to the curve to the curve of <strong><em>f</em></strong>, then the solution to</p> <p>mx+1=a(x-h)²+k</p> <p>has one repeated root.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/quadratics/esq_quadratics5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/functions/quadratics/esq_quadratics5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Rational Functions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="868"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let f(x) = 2x + 1 and <span class="math-tex">\(g(x)=\frac{x}{1-x} \ ,x\neq1\)</span></p> <p>a) Show that <span class="math-tex">\(f\circ g(x)=\frac{x+1}{1-x}\)</span></p> <p>b) Let <span class="math-tex">\(h(x)=\frac{x+1}{1-x}\)</span> , for x < 1</p> <p>c) Sketch the graph of <strong><em>h</em></strong></p> <p>d) Sketch the graph of <span class="math-tex">\(h^{-1}\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c) & d) In a sketch of a graph include</p> <ul> <li>the equations of the asymptotes</li> <li>x intercept</li> <li>y intercept</li> </ul> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/rational-functions/esq_rational_functions1.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="869"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(f(x)=\frac{3x-2}{x-a},x\neq\ a\)</span></p> <p>a) Find the inverse function <span class="math-tex">\(f^{-1}(x)\)</span> in terms of <em>a</em></p> <p>b) Find the value of <em>a</em> such that <strong><em>f</em> </strong>is a self-inverse function<em> </em></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">If <strong><em>f</em> </strong>is a self inverse function, then <span class="math-tex">\(f(x)=f^{-1}(x)\)</span><content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/rational-functions/esq_rational_functions2.pdf" target="_blank">here</a></p> <p> </p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="870"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The function <strong><em>f</em></strong> is defined by <span class="math-tex">\(f(x)=\frac{6x+1}{2x-1},x\in\mathbb{R},x\neq\frac{1}{2}\)</span></p> <p>a) Write <strong><em>f(x)</em></strong> in the form <span class="math-tex">\(A+\frac{B}{2x-1}\)</span> where A and B are constants</p> <p>b) Sketch the graph of <strong><em>f(x)</em></strong> stating the equations of any asymptotes and the coordinates of any intercepts with the axes</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">The horizontal asymptote is at x = A<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/rational-functions/esq_rational_functions3.pdf" target="_blank">here</a></p> <p> </p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="876"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">Sketch the graph of <span class="math-tex">\(f\left(x\right)=\frac{x^2+x-1}{x+2}\)</span> <span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">giving the equations of any asymptotes and the coordinates of the x and y intercepts as well as any stationary points</span> </span></p> <hr class="hidden-separator"> <p><span calibri="" style="font-size:11.0pt;line-height:115%; font-family:"></span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Before trying to sketch this function</p> <ul> <li>Find the equation of the vertical asymptote</li> <li><content></content>Check for 'the hole' by looking for common factpr in numerator and denominator</li> <li>Find the equation of the oblique asymptote</li> <li>Find the y intercept</li> <li>Find any x intercepts</li> <li>Find any stationary points</li> </ul> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/rational-functions/esq_rational_functions4.pdf" target="_blank">here</a></p> <p> </p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="875"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the value(s) of <strong><em>a</em></strong> such that <span class="math-tex">\(f(x)=\frac{x+1}{ax^2+3x+2}\)</span> has only one vertical asymptote.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>If f(x) has only one vertical asymptote, there are two possibilities</p> <p>1) ax² + 3x + 2 has 2 equal factors</p> <p>2) the numerator and denominator have a common factor</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/rational-functions/esq_rational_functions5.pdf" target="_blank">here</a></p> <p> </p> </section> <h4> </h4> </div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Factor and Remainder Theorem</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="476"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The cubic polynomial <span class="math-tex">\(3x^3+ax^2+bx-12\)</span> has a factor <span class="math-tex">\((x-2)\)</span> and leaves a remainder of <strong>-20</strong> when divided by <span class="math-tex">\((x-1)\)</span>.</p> <p>Find the value of <strong><em>a</em></strong> and <strong><em>b</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Use the Factor and Remainder Theorem</p> <p><content>f(2)=0</content></p> <p>f(1)=-20</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/factor-and-remainder-theorem/esq_functions_remainder_th1.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="475"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Given that <span class="math-tex">\(ax^3+bx^2+17x-6\)</span> is exactly divisible by <span class="math-tex">\((x-1)(x-2)\)</span>, find the value of <strong><em>a</em></strong> and <strong><em>b</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Use the Factor and Remainder Theorem</p> <p><content>f(1)=0</content></p> <p>f(2)=0</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/factor-and-remainder-theorem/esq_functions_remainder_th2.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="474"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>It is given that <span class="math-tex">\(f(x)=x^3+ax^2+bx+8\)</span></p> <p>a. Given that <span class="math-tex">\(x^2-4\)</span> is a factor of <span class="math-tex">\(f(x)\)</span>, find the values of <strong><em>a </em></strong> and <strong><em>b</em></strong>.</p> <p>b. Factorize <span class="math-tex">\(f(x)\)</span> into a product of linear factors.</p> <p>c. Sketch the graph of <span class="math-tex">\(y=f(x)\)</span> labelling any stationary points and the x and y intercepts.</p> <p>d. <strong>Hence</strong> state the range of values of <strong><em>k</em></strong> for which <span class="math-tex">\(f(x)=k\)</span> has exactly one root.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c. Notice that you are allowed to use your calculator in this question. You should be able to find the stationary points from the calculator.</p> <p>d. Think of <span class="math-tex">\(f(x)=k\)</span> as a horizontal line. What values of <strong><em>k</em></strong> makes it cross the graph in only one place.<content></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/factor-and-remainder-theorem/esq_functions_remainder_th3.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </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-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Sums and Products of Roots</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="480"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The quadratic equation <span class="math-tex">\(3x^{ 2 }−8x+2=0\)</span> has roots <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span>.</p> <p>a. Without solving the equation, find the value of <span class="math-tex">\(\alpha + \beta\)</span> and <span class="math-tex">\(\alpha \beta\)</span>.</p> <p>b. Another quadratic equation <span class="math-tex">\(3x^{ 2 }+bx+c=0\quad ,\quad b,c\in \mathbb{Z}\)</span> has roots <span class="math-tex">\(\frac{\alpha}{\beta}\)</span> and <span class="math-tex">\(\frac{\beta}{\alpha}\)</span>. Find the value of <em><strong>b</strong></em> and <em><strong>c</strong></em>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content>b. Write the sum and product of <span class="math-tex">\(\frac{\alpha}{\beta}\)</span> and <span class="math-tex">\(\frac{\beta}{\alpha}\)</span> in terms of <span class="math-tex">\(\alpha + \beta\)</span> and <span class="math-tex">\(\alpha \beta\)</span>.</section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/viete/esq_viete1a.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="481"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>In the quadratic equation <span class="math-tex">\(px^{ 2 }−45x+25=0\quad ,\quad p\in \mathbb{Z}\)</span> ,one root is two times the other.</p> <p>Find the value of <em><strong>p</strong></em>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Let the roots be <span class="math-tex">\(\alpha, 2\alpha\)</span><content> </content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/viete/esq_viete2.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="482"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Let <span class="math-tex">\(f(x)=2x^{ 4 }+x^{ 3 }−14x^{ 2 }+5x+6\quad ,\quad x\in \mathbb{R}\)</span></p> <p>a. For the polynomial equation f(x) = 0 , find the value of</p> <ol> <li>the sum of the roots</li> <li>the product of the roots</li> </ol> <p>b. A new polynomial is defined by g(x) = f(x - 2). Find the sum of the roots of the equation g(x) = 0</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">b. What transformation is represented by g(x) = f(x - 2)? What would happen to the position of the roots?<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/viete/esq_viete3a.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="483"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the equation <span class="math-tex">\(8x^{ 3 }−42x^{ 2 }+px−27=0.\)</span></p> <ol style="list-style-type:lower-alpha;"> <li>State <ol> <li>the sum of the roots of the equation</li> <li>the product of the roots of the equation</li> </ol> </li> <li>The roots of this equation are three consecutive terms of a geometric sequence. Taking the roots to be <span class="math-tex">\(\frac{\alpha}{\beta},\alpha,\alpha\beta\)</span> show that one of the roots is <span class="math-tex">\(\frac{3}{2}\)</span></li> <li>Solve the equation.</li> <li>Find the value of <strong><em>p</em></strong>.</li> </ol> <p><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>b. What is the product of the three roots <span class="math-tex">\(\frac{\alpha}{\beta},\alpha,\alpha\beta\)</span>?</p> <p>c. What is the sum of the three roots <span class="math-tex">\(\frac{\alpha}{\beta},\alpha,\alpha\beta\)</span>?</p> <p>d. The equation is <span class="math-tex">\(a(x-\frac{\alpha}{\beta})(x-\alpha)(x-\alpha\beta)=0\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/functions/viete/esq_viete4.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="484"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>One root of the equation <em>z² + bz + c = 0</em> is <em>2+3i</em> where <span class="math-tex">\(b,c\in\mathbb{Z}\)</span>.</p> <p>Find the value of <strong><em>b</em></strong> and the value of <strong><em>c</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">If z=2+3i is a root of the equation, then the complex conjugate z=2-3i is also a root of the equation.<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/roots-of-polynomials/esq_complex_roots_polynomials1.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="485"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The equation <span class="math-tex">\(2z^{ 4 }−9z^{ 3 }+pz^{ 2 }+qz−174=0 \quad,\quad p,q\in\mathbb{Z}\)</span> has two real roots <span class="math-tex">\(\alpha\)</span> and <span class="math-tex">\(\beta\)</span> and two complex roots <span class="math-tex">\(\gamma\)</span> and <span class="math-tex">\(\delta\)</span> where <span class="math-tex">\(\gamma=2-5i\)</span>.</p> <p>a. Show that <span class="math-tex">\(\alpha+\beta=\frac{1}{2}.\)</span></p> <p>b. Find <span class="math-tex">\(\alpha\beta\)</span>.</p> <p>c. <strong>Hence</strong> find the two real roots α and β.</p> <p>d. Find the values of <strong><em>p</em></strong> and <strong><em>q</em></strong>.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a. If <em>γ=2−5i</em> is a root…then <em>δ = 2+5i</em> is also a root. Work out γ+δ and sum of 4 roots</p> <p>b. Work out γδ and the product of the 4 roots</p> <p>c. Work out α and β using the two equations for <span class="math-tex">\(\alpha+\beta\)</span> and <span class="math-tex">\(\alpha\beta\)</span>.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/complex-numbers/roots-of-polynomials/esq_complex_roots_polynomials2.pdf" target="_blank">here</a></p> <p> </p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2906"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of 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