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CAIE P3 2018 June Q6
7 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-10_499_922_262_607} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circle and \(A T\) is a tangent to the circle. Angle \(A O B = \theta\) radians and \(O B T\) is a straight line.
  1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 3\) and \(\theta = 1.2\), find the perimeter of the shaded region.
CAIE P3 2018 June Q4
6 marks Moderate -0.8
4 The function f is such that \(\mathrm { f } ( x ) = a + b \cos x\) for \(0 \leqslant x \leqslant 2 \pi\). It is given that \(\mathrm { f } \left( \frac { 1 } { 3 } \pi \right) = 5\) and \(\mathrm { f } ( \pi ) = 11\).
  1. Find the values of the constants \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-05_63_1566_397_328}
  2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution. \includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-06_622_878_260_632} The diagram shows a three-dimensional shape. The base \(O A B\) is a horizontal triangle in which angle \(A O B\) is \(90 ^ { \circ }\). The side \(O B C D\) is a rectangle and the side \(O A D\) lies in a vertical plane. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O B\) respectively and the unit vector \(\mathbf { k }\) is vertical. The position vectors of \(A , B\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i } , \overrightarrow { O B } = 5 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }\).
CAIE P3 2018 June Q6
6 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-08_454_684_255_726} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor \(\operatorname { arc } A B\) and the lines \(A T\) and \(B T\). Angle \(A O B\) is \(2 \theta\) radians.
  1. In the case where the area of the sector \(A O B\) is the same as the area of the shaded region, show that \(\tan \theta = 2 \theta\).
  2. In the case where \(r = 8 \mathrm {~cm}\) and the length of the minor \(\operatorname { arc } A B\) is 19.2 cm , find the area of the shaded region.
CAIE P3 2018 June Q11
12 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-18_643_969_258_587} The diagram shows part of the curve \(y = \frac { x } { 2 } + \frac { 6 } { x }\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).
  1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\).
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in terms of \(\pi\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2018 June Q5
5 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-06_323_775_260_685} The diagram shows a triangle \(O A B\) in which angle \(O A B = 90 ^ { \circ }\) and \(O A = 5 \mathrm {~cm}\). The arc \(A C\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(O B\) at \(C\). Find the area of the shaded region.
CAIE P3 2018 June Q7
9 marks Moderate -0.3
7
    1. Express \(\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 }\) in the form \(a \sin ^ { 2 } \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
    2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 } = \frac { 1 } { 4 }$$ for \(- 90 ^ { \circ } \leqslant \theta \leqslant 0 ^ { \circ }\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-11_549_796_267_717} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
    1. Find the \(x\)-coordinate of \(A\).
    2. Find the \(y\)-coordinate of \(B\).
CAIE P3 2018 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-14_670_857_260_644} The diagram shows a pyramid \(O A B C D\) with a horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(O B\) is such that \(O E = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O C\) and \(E D\) respectively.
  1. Show that \(\overrightarrow { O E } = 1.6 \mathbf { i } + 1.2 \mathbf { j }\).
  2. Use a scalar product to find angle \(B D O\).
CAIE P3 2018 June Q11
11 marks Standard +0.8
11 \includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-18_645_723_258_573} The diagram shows part of the curve \(y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.
  1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2 ( x + 1 ) ^ { 3 } = 1\) and find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2019 June Q1
3 marks Moderate -0.5
1 Use the trapezium rule with 3 intervals to estimate the value of $$\int _ { 0 } ^ { 3 } \left| 2 ^ { x } - 4 \right| \mathrm { d } x$$
CAIE P3 2019 June Q2
4 marks Standard +0.3
2 Showing all necessary working, solve the equation \(\ln ( 2 x - 3 ) = 2 \ln x - \ln ( x - 1 )\). Give your answer correct to 2 decimal places.
CAIE P3 2019 June Q3
4 marks Standard +0.3
3 Find the gradient of the curve \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 1\) at the point with coordinates \(( 1,3 )\).
CAIE P3 2019 June Q4
6 marks Standard +0.8
4 By first expressing the equation \(\cot \theta - \cot \left( \theta + 45 ^ { \circ } \right) = 3\) as a quadratic equation in \(\tan \theta\), solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2019 June Q5
8 marks Standard +0.3
5
  1. Differentiate \(\frac { 1 } { \sin ^ { 2 } \theta }\) with respect to \(\theta\).
  2. The variables \(x\) and \(\theta\) satisfy the differential equation $$x \tan \theta \frac { d x } { d \theta } + \operatorname { cosec } ^ { 2 } \theta = 0$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\) and \(x > 0\). It is given that \(x = 4\) when \(\theta = \frac { 1 } { 6 } \pi\). Solve the differential equation, obtaining an expression for \(x\) in terms of \(\theta\).
CAIE P3 2019 June Q6
8 marks Standard +0.3
6
  1. By first expanding \(\sin ( 2 x + x )\), show that \(\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x\).
  2. Hence, showing all necessary working, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \mathrm {~d} x\).
CAIE P3 2019 June Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-10_702_597_258_772} The diagram shows the curves \(y = 4 \cos \frac { 1 } { 2 } x\) and \(y = \frac { 1 } { 4 - x }\), for \(0 \leqslant x < 4\). When \(x = a\), the tangents to the curves are perpendicular.
  1. Show that \(a = 4 - \sqrt { } \left( 2 \sin \frac { 1 } { 2 } a \right)\).
  2. Verify by calculation that \(a\) lies between 2 and 3 .
  3. Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2019 June Q8
10 marks Standard +0.3
8 Let \(f ( x ) = \frac { 16 - 17 x } { ( 2 + x ) ( 3 - x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2019 June Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-14_666_703_260_721} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and four points \(A , B , C\) and \(D\) with position vectors \(\overrightarrow { O A } = 3 \mathbf { i } , \overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j } , \overrightarrow { O C } = \mathbf { i } + 3 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k }\).
  1. Find the equation of the plane \(B C D\), giving your answer in the form \(a x + b y + c z = d\).
  2. Calculate the acute angle between the planes \(B C D\) and \(O A B C\).
CAIE P3 2019 June Q10
13 marks Standard +0.3
10 Throughout this question the use of a calculator is not permitted.
The complex number \(( \sqrt { } 3 ) + \mathrm { i }\) is denoted by \(u\).
  1. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\). Hence or otherwise state the exact values of the modulus and argument of \(u ^ { 4 }\).
  2. Verify that \(u\) is a root of the equation \(z ^ { 3 } - 8 z + 8 \sqrt { } 3 = 0\) and state the other complex root of this equation.
  3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2019 June Q1
4 marks Standard +0.3
1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - x ) ( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
CAIE P3 2019 June Q2
4 marks Moderate -0.3
2 Showing all necessary working, solve the equation \(9 ^ { x } = 3 ^ { x } + 12\). Give your answer correct to 2 decimal places.
CAIE P3 2019 June Q3
5 marks Standard +0.3
3 Showing all necessary working, solve the equation \(\cot 2 \theta = 2 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2019 June Q4
7 marks Standard +0.3
4 Find the exact coordinates of the point on the curve \(y = \frac { x } { 1 + \ln x }\) at which the gradient of the tangent is equal to \(\frac { 1 } { 4 }\).
CAIE P3 2019 June Q6
8 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{772393d7-6e81-4b99-913a-63c9f87d1af2-08_492_812_260_664} In the diagram, \(A\) is the mid-point of the semicircle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the semicircle at \(B\) and \(C\). The angle \(O A B\) is equal to \(x\) radians. The area of the shaded region bounded by \(A B , A C\) and the arc with centre \(A\) is equal to half the area of the semicircle.
  1. Use triangle \(O A B\) to show that \(A B = 2 r \cos x\).
  2. Hence show that \(x = \cos ^ { - 1 } \sqrt { } \left( \frac { \pi } { 16 x } \right)\).
  3. Verify by calculation that \(x\) lies between 1 and 1.5.
  4. Use an iterative formula based on the equation in part (ii) to determine \(x\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2019 June Q7
8 marks Standard +0.3
7 The variables \(x\) and \(y\) satisfy the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { x + y }\). It is given that \(y = 0\) when \(x = 0\).
  1. Solve the differential equation, obtaining \(y\) in terms of \(x\).
  2. Explain why \(x\) can only take values that are less than 1 .
CAIE P3 2019 June Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 10 x + 9 } { ( 2 x + 1 ) ( 2 x + 3 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 2 } \ln \frac { 9 } { 5 } + \frac { 1 } { 5 }\).