SL Paper 2
The following diagram shows a pole BT 1.6 m tall on the roof of a vertical building.
The angle of depression from T to a point A on the horizontal ground is \({35^ \circ }\) .
The angle of elevation of the top of the building from A is \({30^ \circ }\) .
Find the height of the building.
Consider the following circle with centre O and radius r .
The points P, R and Q are on the circumference, \({\rm{P}}\widehat {\rm{O}}{\rm{Q}} = 2\theta \) , for \(0 < \theta < \frac{\pi }{2}\) .
Use the cosine rule to show that \({\rm{PQ}} = 2r\sin \theta \) .
Let l be the length of the arc PRQ .
Given that \(1.3{\rm{PQ}} - l = 0\) , find the value of \(\theta \) .
Consider the function \(f(\theta ) = 2.6\sin \theta - 2\theta \) , for \(0 < \theta < \frac{\pi }{2}\) .
(i) Sketch the graph of f .
(ii) Write down the root of \(f(\theta ) = 0\) .
Use the graph of f to find the values of \(\theta \) for which \(l < 1.3{\rm{PQ}}\) .
Consider the triangle ABC, where AB =10 , BC = 7 and \({\rm{C}}\widehat {\rm{A}}{\rm{B}}\) = \({30^ \circ }\) .
Find the two possible values of \({\rm{A}}\widehat {\rm{C}}{\rm{B}}\) .
Hence, find \({\rm{A}}\widehat {\rm{B}}{\rm{C}}\) , given that it is acute.
Let \(f(x) = a\cos (b(x - c))\) . The diagram below shows part of the graph of f , for \(0 \le x \le 10\) .
The graph has a local maximum at P(3, 5) , a local minimum at Q(7, − 5) , and crosses the x-axis at R.
Write down the value of
(i) \(a\) ;
(ii) \(c\) .
Find the value of b .
Find the x-coordinate of R.
The height, \(h\) metros, of a seat on a Ferris wheel after \(t\) minutes is given by
\[h(t) = - 15\cos 1.2t + 17,{\text{ for }}t \geqslant 0{\text{.}}\]
Find the height of the seat when \(t = 0\).
The seat first reaches a height of 20 m after \(k\) minutes. Find \(k\).
Calculate the time needed for the seat to complete a full rotation, giving your answer correct to one decimal place.
The diagram below shows a circle centre O, with radius r. The length of arc ABC is \(3\pi {\text{ cm}}\) and \({\rm{A}}\widehat {\rm{O}}{\rm{C}} = \frac{{2\pi }}{9}\).
Find the value of r.
Find the perimeter of sector OABC.
Find the area of sector OABC.
Triangle ABC has a = 8.1 cm, b = 12.3 cm and area 15 cm2. Find the largest possible perimeter of triangle ABC.
The following diagram shows three towns A, B and C. Town B is 5 km from Town A, on a bearing of 070°. Town C is 8 km from Town B, on a bearing of 115°.
Find \({\rm{A\hat BC}}\).
Find the distance from Town A to Town C.
Use the sine rule to find \({\rm{A\hat CB}}\).
At Grande Anse Beach the height of the water in metres is modelled by the function \(h(t) = p\cos (q \times t) + r\), where \(t\) is the number of hours after 21:00 hours on 10 December 2017. The following diagram shows the graph of \(h\) , for \(0 \leqslant t \leqslant 72\).
The point \({\text{A}}(6.25,{\text{ }}0.6)\) represents the first low tide and \({\text{B}}(12.5,{\text{ }}1.5)\) represents the next high tide.
How much time is there between the first low tide and the next high tide?
Find the difference in height between low tide and high tide.
Find the value of \(p\);
Find the value of \(q\);
Find the value of \(r\).
There are two high tides on 12 December 2017. At what time does the second high tide occur?
The diagram shows a circle of radius \(8\) metres. The points ABCD lie on the circumference of the circle.
BC = \(14\) m, CD = \(11.5\) m, AD = \(8\) m, \(A\hat DC = {104^ \circ }\) , and \(B\hat CD = {73^ \circ }\) .
Find AC.
(i) Find \(A\hat CD\) .
(ii) Hence, find \(A\hat CB\) .
Find the area of triangle ADC.
(c) Find the area of triangle ADC.
(d) Hence or otherwise, find the total area of the shaded regions.
Hence or otherwise, find the total area of the shaded regions.
The following diagram shows the triangle ABC.
The angle at C is obtuse, \({\text{AC}} = 5{\text{ cm}}\), \({\text{BC}} =13.6{\text{ cm}}\) and the area is \(20{\text{ c}}{\text{m}}^2\) .
Find \({\rm{A}}\widehat {\rm{C}}{\rm{B}}\) .
Find AB.
The following diagram shows \(\Delta {\rm{PQR}}\) , where RQ = 9 cm, \({\rm{P\hat RQ}} = {70^ \circ }\) and \({\rm{P\hat QR}} = {45^ \circ }\) .
Find \({\rm{R\hat PQ}}\) .
Find PR .
Find the area of \(\Delta {\rm{PQR}}\) .
The following diagram shows triangle ABC.
Find AC.
Find \({\rm{B\hat CA}}\).
Two points P and Q have coordinates (3, 2, 5) and (7, 4, 9) respectively.
Let \({\mathop {{\text{PR}}}\limits^ \to }\) = 6i − j + 3k.
Find \(\mathop {{\text{PQ}}}\limits^ \to \).
Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|\).
Find the angle between PQ and PR.
Find the area of triangle PQR.
Hence or otherwise find the shortest distance from R to the line through P and Q.
The following diagram shows a circular play area for children.
The circle has centre O and a radius of 20 m, and the points A, B, C and D lie on the circle. Angle AOB is 1.5 radians.
Find the length of the chord [AB].
Find the area of triangle AOB.
Angle BOC is 2.4 radians.
Find the length of arc ADC.
Angle BOC is 2.4 radians.
Find the area of the shaded region.
Angle BOC is 2.4 radians.
The shaded region is to be painted red. Red paint is sold in cans which cost \(\$ 32\) each. One can covers \(140{\text{ }}{{\text{m}}^2}\). How much does it cost to buy the paint?
The graph of \(y = p\cos qx + r\) , for \( - 5 \le x \le 14\) , is shown below.
There is a minimum point at (0, −3) and a maximum point at (4, 7) .
Find the value of
(i) p ;
(ii) q ;
(iii) r.
The equation \(y = k\) has exactly two solutions. Write down the value of k.
The following diagram shows the graph of \(f(x) = a\sin bx + c\), for \(0 \leqslant x \leqslant 12\).
The graph of \(f\) has a minimum point at \((3,{\text{ }}5)\) and a maximum point at \((9,{\text{ }}17)\).
The graph of \(g\) is obtained from the graph of \(f\) by a translation of \(\left( {\begin{array}{*{20}{c}} k \\ 0 \end{array}} \right)\). The maximum point on the graph of \(g\) has coordinates \((11.5,{\text{ }}17)\).
The graph of \(g\) changes from concave-up to concave-down when \(x = w\).
(i) Find the value of \(c\).
(ii) Show that \(b = \frac{\pi }{6}\).
(iii) Find the value of \(a\).
(i) Write down the value of \(k\).
(ii) Find \(g(x)\).
(i) Find \(w\).
(ii) Hence or otherwise, find the maximum positive rate of change of \(g\).
The following diagram shows part of the graph of \(y = p\sin (qx) + r\).
The point \({\text{A}}\left( {\frac{\pi }{6},{\text{ }}2} \right)\) is a maximum point and the point \({\text{B}}\left( {\frac{\pi }{6},{\text{ }}1} \right)\) is a minimum point.
Find the value of
\(p\);
\(r\);
\(q\).
Note: In this question, distance is in millimetres.
Let \(f(x) = x + a\sin \left( {x - \frac{\pi }{2}} \right) + a\), for \(x \geqslant 0\).
The graph of \(f\) passes through the origin. Let \({{\text{P}}_k}\) be any point on the graph of \(f\) with \(x\)-coordinate \(2k\pi \), where \(k \in \mathbb{N}\). A straight line \(L\) passes through all the points \({{\text{P}}_k}\).
Diagram 1 shows a saw. The length of the toothed edge is the distance AB.
The toothed edge of the saw can be modelled using the graph of \(f\) and the line \(L\). Diagram 2 represents this model.
The shaded part on the graph is called a tooth. A tooth is represented by the region enclosed by the graph of \(f\) and the line \(L\), between \({{\text{P}}_k}\) and \({{\text{P}}_{k + 1}}\).
Show that \(f(2\pi ) = 2\pi \).
Find the coordinates of \({{\text{P}}_0}\) and of \({{\text{P}}_1}\).
Find the equation of \(L\).
Show that the distance between the \(x\)-coordinates of \({{\text{P}}_k}\) and \({{\text{P}}_{k + 1}}\) is \(2\pi \).
A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.
Let \(f(x) = 3\sin x + 4\cos x\) , for \( - 2\pi \le x \le 2\pi \) .
Sketch the graph of f .
Write down
(i) the amplitude;
(ii) the period;
(iii) the x-intercept that lies between \( - \frac{\pi }{2}\) and 0.
Hence write \(f(x)\) in the form \(p\sin (qx + r)\) .
Write down one value of x such that \(f'(x) = 0\) .
Write down the two values of k for which the equation \(f(x) = k\) has exactly two solutions.
Let \(g(x) = \ln (x + 1)\) , for \(0 \le x \le \pi \) . There is a value of x, between \(0\) and \(1\), for which the gradient of f is equal to the gradient of g. Find this value of x.
Let \(f(x) = 5\cos \frac{\pi }{4}x\) and \(g(x) = - 0.5{x^2} + 5x - 8\) for \(0 \le x \le 9\) .
On the same diagram, sketch the graphs of f and g .
Consider the graph of \(f\) . Write down
(i) the x-intercept that lies between \(x = 0\) and \(x = 3\) ;
(ii) the period;
(iii) the amplitude.
Consider the graph of g . Write down
(i) the two x-intercepts;
(ii) the equation of the axis of symmetry.
Let R be the region enclosed by the graphs of f and g . Find the area of R.
A ship is sailing north from a point A towards point D. Point C is 175 km north of A. Point D is 60 km north of C. There is an island at E. The bearing of E from A is 055°. The bearing of E from C is 134°. This is shown in the following diagram.
Find the bearing of A from E.
Finds CE.
Find DE.
When the ship reaches D, it changes direction and travels directly to the island at 50 km per hour. At the same time as the ship changes direction, a boat starts travelling to the island from a point B. This point B lies on (AC), between A and C, and is the closest point to the island. The ship and the boat arrive at the island at the same time. Find the speed of the boat.
The following diagram shows the quadrilateral \(ABCD\).
\[{\text{AD}} = 6{\text{ cm}},{\text{ AB}} = 15{\text{ cm}},{\rm{ A\hat BC}} = 44^\circ ,{\rm{ A\hat CB}} = 83^\circ {\rm{ and D\hat AC}} = \theta \]
Find \(AC\).
Find the area of triangle \(ABC\).
The area of triangle \(ACD\) is half the area of triangle \(ABC\).
Find the possible values of \(\theta \).
Given that \(\theta \) is obtuse, find \(CD\).
The following diagram shows a triangle ABC.
The area of triangle ABC is \(80\) cm2 , AB \( = 18\) cm , AC \( = x\) cm and \({\rm{B}}\hat {\rm{A}}{\rm{C}} = {50^ \circ }\) .
Find \(x\) .
Find BC.
The following diagram represents a large Ferris wheel at an amusement park.
The points P, Q and R represent different positions of a seat on the wheel.
The wheel has a radius of 50 metres and rotates clockwise at a rate of one revolution every 30 minutes.
A seat starts at the lowest point P, when its height is one metre above the ground.
Find the height of a seat above the ground after 15 minutes.
After six minutes, the seat is at point Q. Find its height above the ground at Q.
The height of the seat above ground after t minutes can be modelled by the function \(h(t) = 50\sin (b(t - c)) + 51\).
Find the value of b and of c .
The height of the seat above ground after t minutes can be modelled by the function \(h(t) = 50\sin (b(t - c)) + 51\).
Hence find the value of t the first time the seat is \(96{\text{ m}}\) above the ground.
The depth of water in a port is modelled by the function \(d(t) = p\cos qt + 7.5\), for \(0 \leqslant t \leqslant 12\), where \(t\) is the number of hours after high tide.
At high tide, the depth is 9.7 metres.
At low tide, which is 7 hours later, the depth is 5.3 metres.
Find the value of \(p\).
Find the value of \(q\).
Use the model to find the depth of the water 10 hours after high tide.
The following diagram shows triangle ABC .
AB = 7 cm, BC = 9 cm and \({\rm{A}}\widehat {\rm{B}}{\rm{C}} = {120^ \circ }\) .
Find AC .
Find \({\rm{B}}\widehat {\rm{A}}{\rm{C}}\) .
The following diagram shows a quadrilateral ABCD.
\[{\text{AD}} = {\text{7}}\;{\text{cm,}}\;{\text{BC}} = {\text{8}}\;{\text{cm,}}\;{\text{CD}} = {\text{12}}\;{\text{cm,}}\;{\rm{D\hat AB}} = {\text{1.75}}\;{\text{radians,}}\;{\rm{A\hat BD}} = {\text{0.82}}\;{\text{radians.}}\]
Find BD.
Find \({\rm{D\hat BC}}\).
The following diagram shows a circle with centre \(\rm{O}\) and radius \(5 \rm\,{cm}\).
The points \(\rm{A}\), \(rm{B}\) and \(rm{C}\) lie on the circumference of the circle, and \({\rm{A\hat OC}} = 0.7\) radians.
Find the length of the arc \({\text{ABC}}\).
Find the perimeter of the shaded sector.
Find the area of the shaded sector.
The diagram below shows a quadrilateral ABCD with obtuse angles \({\rm{A}}\widehat {\rm{B}}{\rm{C}}\) and \({\rm{A}}\widehat {\rm{D}}{\rm{C}}\).
AB = 5 cm, BC = 4 cm, CD = 4 cm, AD = 4 cm , \({\rm{B}}\widehat {\rm{A}}{\rm{C}} = {30^ \circ }\) , \({\rm{A}}\widehat {\rm{B}}{\rm{C}} = {x^ \circ }\) , \({\rm{A}}\widehat {\rm{D}}{\rm{C}} = {y^ \circ }\) .
Use the cosine rule to show that \({\rm{AC}} = \sqrt {41 - 40\cos x} \) .
Use the sine rule in triangle ABC to find another expression for AC.
(i) Hence, find x, giving your answer to two decimal places.
(ii) Find AC .
(i) Find y.
(ii) Hence, or otherwise, find the area of triangle ACD.
The diagram below shows a plan for a window in the shape of a trapezium.
Three sides of the window are \(2{\text{ m}}\) long. The angle between the sloping sides of the window and the base is \(\theta \) , where \(0 < \theta < \frac{\pi }{2}\) .
Show that the area of the window is given by \(y = 4\sin \theta + 2\sin 2\theta \) .
Zoe wants a window to have an area of \(5{\text{ }}{{\text{m}}^2}\). Find the two possible values of \(\theta \) .
John wants two windows which have the same area A but different values of \(\theta \) .
Find all possible values for A .
The following diagram shows a triangle ABC.
\({\rm{BC}} = 6\) , \({\rm{C}}\widehat {\rm{A}}{\rm{B}} = 0.7\) radians , \({\rm{AB}} = 4p\) , \({\rm{AC}} = 5p\) , where \(p > 0\) .
Consider the circle with centre B that passes through the point C. The circle cuts the line CA at D, and \({\rm{A}}\widehat {\rm{D}}{\rm{B}}\) is obtuse. Part of the circle is shown in the following diagram.
(i) Show that \({p^2}(41 - 40\cos 0.7) = 36\) .
(ii) Find p .
Write down the length of BD.
Find \({\rm{A}}\widehat {\rm{D}}{\rm{B}}\) .
(i) Show that \({\rm{C}}\widehat {\rm{B}}{\rm{D}} = 1.29\) radians, correct to 2 decimal places.
(ii) Hence, find the area of the shaded region.
Consider the following circle with centre O and radius 6.8 cm.
The length of the arc PQR is 8.5 cm.
Find the value of \(\theta \) .
Find the area of the shaded region.
The diagram shows a parallelogram ABCD.
The coordinates of A, B and D are A(1, 2, 3) , B(6, 4,4 ) and D(2, 5, 5) .
(i) Show that \(\overrightarrow {{\rm{AB}}} = \left( {\begin{array}{*{20}{c}}
5\\
2\\
1
\end{array}} \right)\) .
(ii) Find \(\overrightarrow {{\rm{AD}}} \) .
(iii) Hence show that \(\overrightarrow {{\rm{AC}}} = \left( {\begin{array}{*{20}{c}}
6\\
5\\
3
\end{array}} \right)\) .
Find the coordinates of point C.
(i) Find \(\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} \).
(ii) Hence find angle A.
Hence, or otherwise, find the area of the parallelogram.
The diagram below shows triangle PQR. The length of [PQ] is 7 cm , the length of [PR] is 10 cm , and \({\rm{P}}\widehat {\rm{Q}}{\rm{R}}\) is \(75^\circ \) .
Find \({\rm{P}}\widehat {\rm{R}}{\rm{Q}}\) .
Find the area of triangle PQR.
A ship leaves port A on a bearing of \(030^\circ \) . It sails a distance of \(25{\text{ km}}\) to point B. At B, the ship changes direction to a bearing of \(100^\circ \) . It sails a distance of \(40{\text{ km}}\) to reach point C. This information is shown in the diagram below.
A second ship leaves port A and sails directly to C.
Find the distance the second ship will travel.
Find the bearing of the course taken by the second ship.
The following diagram shows a circle, centre O and radius \(r\) mm. The circle is divided into five equal sectors.
One sector is OAB, and \({\rm{A\hat OB}} = \theta \).
The area of sector AOB is \(20\pi {\text{ m}}{{\text{m}}^2}\).
Write down the exact value of \(\theta \) in radians.
Find the value of \(r\).
Find AB.
The following diagram shows triangle \(ABC\).
\[{\text{BC}} = 10{\text{ cm}},{\rm{ A\hat BC}} = 80^\circ \;{\text{and}}\;{\rm{B\hat AC}} = 35^\circ .\]
Find \(AC\).
Find the area of triangle \(ABC\).
Let \(f(x) = p\cos \left( {q(x + r)} \right) + 10\), for \(0 \leqslant x \leqslant 20\). The following diagram shows the graph of \(f\).
The graph has a maximum at \((4, 18)\) and a minimum at \((16, 2)\).
Write down the value of \(r\).
Find \(p\).
Find \(q\).
Solve \(f(x) = 7\).
The following diagram shows a circle with centre O and radius 4 cm.
The points A, B and C lie on the circle. The point D is outside the circle, on (OC).
Angle ADC = 0.3 radians and angle AOC = 0.8 radians.
Find AD.
Find OD.
Find the area of sector OABC.
Find the area of region ABCD.
A circle centre O and radius \(r\) is shown below. The chord [AB] divides the area of the circle into two parts. Angle AOB is \(\theta \) .
Find an expression for the area of the shaded region.
The chord [AB] divides the area of the circle in the ratio 1:7. Find the value of \(\theta \) .
The population of deer in an enclosed game reserve is modelled by the function \(P(t) = 210\sin (0.5t - 2.6) + 990\), where \(t\) is in months, and \(t = 1\) corresponds to 1 January 2014.
Find the number of deer in the reserve on 1 May 2014.
Find the rate of change of the deer population on 1 May 2014.
Interpret the answer to part (i) with reference to the deer population size on 1 May 2014.
The following graph shows the depth of water, y metres , at a point P, during one day. The time t is given in hours, from midnight to noon.
Use the graph to write down an estimate of the value of t when
(i) the depth of water is minimum;
(ii) the depth of water is maximum;
(iii) the depth of the water is increasing most rapidly.
The depth of water can be modelled by the function \(y = \cos A(B(t - 1)) + C\) .
(i) Show that \(A = 8\) .
(ii) Write down the value of C.
(iii) Find the value of B.
A sailor knows that he cannot sail past P when the depth of the water is less than 12 m . Calculate the values of t between which he cannot sail past P.
The diagram below shows a triangle ABD with AB =13 cm and AD = 6.5 cm.
Let C be a point on the line BD such that BC = AC = 7 cm.
Find the size of angle ACB.
Find the size of angle CAD.
The following diagram shows a triangle ABC.
\({\text{AB}} = 5{\rm{ cm, C\hat AB}} = \) 50° and \({\rm{A\hat CB}} = \) 112°
Find BC.
Find the area of triangle ABC.
The diagram below shows a circle with centre O and radius 8 cm.
The points A, B, C, D, E and F are on the circle, and [AF] is a diameter. The length of arc ABC is 6 cm.
Find the size of angle AOC .
Hence find the area of the shaded region.
The area of sector OCDE is \(45{\text{ c}}{{\text{m}}^2}\).
Find the size of angle COE .
Find EF .
Let \(f(x) = \frac{{3x}}{2} + 1\) , \(g(x) = 4\cos \left( {\frac{x}{3}} \right) - 1\) . Let \(h(x) = (g \circ f)(x)\) .
Find an expression for \(h(x)\) .
Write down the period of \(h\) .
Write down the range of \(h\) .
The circle shown has centre O and radius 3.9 cm.
Points A and B lie on the circle and angle AOB is 1.8 radians.
Find AB.
Find the area of the shaded region.
The following diagram shows a circle with centre O and radius \(r\) cm.
Points A and B are on the circumference of the circle and \({\rm{A}}\hat {\rm{O}}{\rm{B}} = 1.4\) radians .
The point C is on [OA] such that \({\rm{B}}\hat {\rm{C}}{\rm{O}} = \frac{\pi }{2}\) radians .
Show that \({\rm{OC}} = r\cos 1.4\) .
The area of the shaded region is \(25\) cm2 . Find the value of \(r\) .
The following diagram shows a circle with centre \(O\) and radius \(3\) cm.
Points A, B, and C lie on the circle, and \({\rm{A\hat OC}} = 1.3{\text{ radians}}\).
Find the length of arc \(ABC\).
Find the area of the shaded region.
The following diagram shows a circle with centre O and radius 40 cm.
The points A, B and C are on the circumference of the circle and \({\rm{A\hat OC}} = 1.9{\text{ radians}}\).
Find the length of arc ABC.
Find the perimeter of sector OABC.
Find the area of sector OABC.
The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counter-clockwise) direction.
The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres above the water level. After t seconds, the height of the bucket above the water level is given by \(h = a\sin bt + 2\) .
Show that \(a = 4\) .
The wheel turns at a rate of one rotation every 30 seconds.
Show that \(b = \frac{\pi }{{15}}\) .
In the first rotation, there are two values of t when the bucket is descending at a rate of \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\) .
Find these values of t .
In the first rotation, there are two values of t when the bucket is descending at a rate of \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\) .
Determine whether the bucket is underwater at the second value of t .
In triangle \(\rm{ABC}\), \(\rm{AB} = 6\,\rm{cm}\) and \(\rm{AC} = 8\,\rm{cm}\). The area of the triangle is \(16\,\rm{cm}^2\).
Find the two possible values for \(\hat A\).
Given that \(\hat A\) is obtuse, find \({\text{BC}}\).
Let \(f(x) = \cos \left( {\frac{\pi }{4}x} \right) + \sin \left( {\frac{\pi }{4}x} \right),{\text{ for }} - 4 \leqslant x \leqslant 4.\)
Sketch the graph of \(f\).
Find the values of \(x\) where the function is decreasing.
The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(a\);
The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(c\).
The following diagram shows a circle with centre \(O\) and radius \(8\) cm.
The points \(A\), \(B\) and \(C\) are on the circumference of the circle, and \({\rm{A\hat OB}}\) radians.
Find the length of arc \(ACB\).
Find \(AB\).
Hence, find the perimeter of the shaded segment \(ABC\).
There is a vertical tower TA of height 36 m at the base A of a hill. A straight path goes up the hill from A to a point U. This information is represented by the following diagram.
The path makes a \({4^ \circ }\) angle with the horizontal.
The point U on the path is \(25{\text{ m}}\) away from the base of the tower.
The top of the tower is fixed to U by a wire of length \(x{\text{ m}}\).
Complete the diagram, showing clearly all the information above.
Find x .
The following diagram shows a square \(ABCD\), and a sector \(OAB\) of a circle centre \(O\), radius \(r\). Part of the square is shaded and labelled \(R\).
\[{\rm{A\hat OB}} = \theta {\text{, where }}0.5 \ \le \ \theta < \pi .\]
Show that the area of the square \(ABCD\) is \(2{r^2}(1 - \cos \theta )\).
When \(\theta = \alpha \), the area of the square \(ABCD\) is equal to the area of the sector \(OAB\).
(i) Write down the area of the sector when \(\theta = \alpha \).
(ii) Hence find \(\alpha \).
When \(\theta = \beta \), the area of \(R\) is more than twice the area of the sector.
Find all possible values of \(\beta \).
Consider a circle with centre \(\rm{O}\) and radius \(7\) cm. Triangle \(\rm{ABC}\) is drawn such that its vertices are on the circumference of the circle.
\(\rm{AB}=12.2\) cm, \(\rm{BC}=10.4\) cm and \(\rm{A}\hat{\rm{C}}\rm{B}=1.058\) radians.
Find \({\rm{B\hat AC}}\).
Find \({\text{AC}}\).
Hence or otherwise, find the length of arc \({\text{ABC}}\).
The following diagram shows the chord [AB] in a circle of radius 8 cm, where \({\text{AB}} = 12{\text{ cm}}\).
Find the area of the shaded segment.
At an amusement park, a Ferris wheel with diameter 111 metres rotates at a constant speed. The bottom of the wheel is k metres above the ground. A seat starts at the bottom of the wheel.
The wheel completes one revolution in 16 minutes.
After t minutes, the height of the seat above ground is given by \(h\left( t \right) = 61.5 + a\,{\text{cos}}\left( {\frac{\pi }{8}t} \right)\), for 0 ≤ t ≤ 32.
After 8 minutes, the seat is 117 m above the ground. Find k.
Find the value of a.
Find when the seat is 30 m above the ground for the third time.
The following diagram shows quadrilateral ABCD.
\({\text{AB}} = 11\,{\text{cm,}}\,\,{\text{BC}} = 6\,{\text{cm,}}\,\,{\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{D = 100}}^\circ {\text{, and C}}\mathop {\text{B}}\limits^ \wedge {\text{D = 82}}^\circ \)
Find DB.
Find DC.
A Ferris wheel with diameter \(122\) metres rotates clockwise at a constant speed. The wheel completes \(2.4\) rotations every hour. The bottom of the wheel is \(13\) metres above the ground.
A seat starts at the bottom of the wheel.
After t minutes, the height \(h\) metres above the ground of the seat is given by\[h = 74 + a\cos bt {\rm{ .}}\]
Find the maximum height above the ground of the seat.
(i) Show that the period of \(h\) is \(25\) minutes.
(ii) Write down the exact value of \(b\) .
(b) (i) Show that the period of \(h\) is \(25\) minutes.
(ii) Write down the exact value of \(b\) .
(c) Find the value of \(a\) .
(d) Sketch the graph of \(h\) , for \(0 \le t \le 50\) .
Find the value of \(a\) .
Sketch the graph of \(h\) , for \(0 \le t \le 50\) .
In one rotation of the wheel, find the probability that a randomly selected seat is at least \(105\) metres above the ground.
The diagram shows a circle, centre O, with radius 4 cm. Points A and B lie on the circumference of the circle and AÔB = θ , where 0 ≤ θ ≤ \(\pi \).
Find the area of the shaded region, in terms of θ.
The area of the shaded region is 12 cm2. Find the value of θ.
Let \(f\left( x \right) = 12\,\,{\text{cos}}\,x - 5\,\,{\text{sin}}\,x,\,\, - \pi \leqslant x \leqslant 2\pi \), be a periodic function with \(f\left( x \right) = f\left( {x + 2\pi } \right)\)
The following diagram shows the graph of \(f\).
There is a maximum point at A. The minimum value of \(f\) is −13 .
A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically.
The distance, d centimetres, of the centre of the ball from O at time t seconds, is given by
\(d\left( t \right) = f\left( t \right) + 17,\,\,0 \leqslant t \leqslant 5.\)
Find the coordinates of A.
For the graph of \(f\), write down the amplitude.
For the graph of \(f\), write down the period.
Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).
Find the maximum speed of the ball.
Find the first time when the ball’s speed is changing at a rate of 2 cm s−2.