Questions — CAIE (7646 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
CAIE P3 2016 June Q9
10 marks Standard +0.3
9 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\), \(\overrightarrow { O B } = 4 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }\). A fourth point \(D\) is such that the quadrilateral \(A B C D\) is a parallelogram.
  1. Find the position vector of \(D\) and verify that the parallelogram is a rhombus.
  2. The plane \(p\) is parallel to \(O A\) and the line \(B C\) lies in \(p\). Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2016 June Q10
10 marks Standard +0.3
10
  1. Showing all necessary working, solve the equation \(\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    1. On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation \(| z | = | z - 4 - 3 \mathrm { i } |\).
    2. Find the complex number represented by the point on the locus where \(| z |\) is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.
CAIE P3 2016 June Q1
4 marks Standard +0.3
1 Solve the inequality \(2 | x - 2 | > | 3 x + 1 |\).
CAIE P3 2016 June Q2
5 marks Moderate -0.8
2 The variables \(x\) and \(y\) satisfy the relation \(3 ^ { y } = 4 ^ { 2 - x }\).
  1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. State the exact value of the gradient of this line.
  2. Calculate the exact \(x\)-coordinate of the point of intersection of this line with the line with equation \(y = 2 x\), simplifying your answer.
CAIE P3 2016 June Q3
6 marks Standard +0.3
3
  1. Express \(( \sqrt { } 5 ) \cos x + 2 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$( \sqrt { } 5 ) \cos \frac { 1 } { 2 } x + 2 \sin \frac { 1 } { 2 } x = 1.2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P3 2016 June Q4
8 marks Moderate -0.3
4 The parametric equations of a curve are $$x = t + \cos t , \quad y = \ln ( 1 + \sin t )$$ where \(- \frac { 1 } { 2 } \pi < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec t\).
  2. Hence find the \(x\)-coordinates of the points on the curve at which the gradient is equal to 3 . Give your answers correct to 3 significant figures.
CAIE P3 2016 June Q5
8 marks Moderate -0.3
5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 y } \tan ^ { 2 } x$$ for \(0 \leqslant x < \frac { 1 } { 2 } \pi\), and it is given that \(y = 0\) when \(x = 0\). Solve the differential equation and calculate the value of \(y\) when \(x = \frac { 1 } { 4 } \pi\).
CAIE P3 2016 June Q6
8 marks Standard +0.3
6 The curve with equation \(y = x ^ { 2 } \cos \frac { 1 } { 2 } x\) has a stationary point at \(x = p\) in the interval \(0 < x < \pi\).
  1. Show that \(p\) satisfies the equation \(\tan \frac { 1 } { 2 } p = \frac { 4 } { p }\).
  2. Verify by calculation that \(p\) lies between 2 and 2.5.
  3. Use the iterative formula \(p _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 4 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2016 June Q7
8 marks Standard +0.3
7 Let \(I = \int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } } \mathrm {~d} x\).
  1. Using the substitution \(u = 1 + x ^ { 2 }\), show that \(I = \int _ { 1 } ^ { 2 } \frac { ( u - 1 ) ^ { 2 } } { 2 u ^ { 3 } } \mathrm {~d} u\).
  2. Hence find the exact value of \(I\).
CAIE P3 2016 June Q8
9 marks Standard +0.3
8 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 3 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).
  1. Show that the line passing through \(A\) and \(B\) does not intersect \(l\).
  2. Show that the length of the perpendicular from \(A\) to \(l\) is \(\frac { 1 } { \sqrt { 2 } }\).
CAIE P3 2016 June Q10
10 marks Standard +0.3
10 Let \(\mathrm { f } ( x ) = \frac { 10 x - 2 x ^ { 2 } } { ( x + 3 ) ( x - 1 ) ^ { 2 } }\).
  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 2017 June Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x + 1 | < 3 | x - 2 |\).
CAIE P3 2017 June Q2
4 marks Moderate -0.3
2 Expand \(\frac { 1 } { \sqrt [ 3 ] { } ( 1 + 6 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2017 June Q3
6 marks Standard +0.3
3 It is given that \(x = \ln ( 1 - y ) - \ln y\), where \(0 < y < 1\).
  1. Show that \(y = \frac { \mathrm { e } ^ { - x } } { 1 + \mathrm { e } ^ { - x } }\).
  2. Hence show that \(\int _ { 0 } ^ { 1 } y \mathrm {~d} x = \ln \left( \frac { 2 \mathrm { e } } { \mathrm { e } + 1 } \right)\).
CAIE P3 2017 June Q4
8 marks Standard +0.3
4 The parametric equations of a curve are $$x = \ln \cos \theta , \quad y = 3 \theta - \tan \theta ,$$ where \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan \theta\).
  2. Find the exact \(y\)-coordinate of the point on the curve at which the gradient of the normal is equal to 1 . \includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-08_378_689_260_726} The diagram shows a semicircle with centre \(O\), radius \(r\) and diameter \(A B\). The point \(P\) on its circumference is such that the area of the minor segment on \(A P\) is equal to half the area of the minor segment on \(B P\). The angle \(A O P\) is \(x\) radians.
CAIE P3 2017 June Q6
8 marks Standard +0.3
6 The plane with equation \(2 x + 2 y - z = 5\) is denoted by \(m\). Relative to the origin \(O\), the points \(A\) and \(B\) have coordinates \(( 3,4,0 )\) and \(( - 1,0,2 )\) respectively.
  1. Show that the plane \(m\) bisects \(A B\) at right angles.
    A second plane \(p\) is parallel to \(m\) and nearer to \(O\). The perpendicular distance between the planes is 1 .
  2. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2017 June Q7
8 marks Standard +0.3
7 Throughout this question the use of a calculator is not permitted.
The complex numbers \(u\) and \(w\) are defined by \(u = - 1 + 7 \mathrm { i }\) and \(w = 3 + 4 \mathrm { i }\).
  1. Showing all your working, find in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real, the complex numbers \(u - 2 w\) and \(\frac { u } { w }\).
    In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , w\) and \(u - 2 w\) respectively.
  2. Prove that angle \(A O B = \frac { 1 } { 4 } \pi\).
  3. State fully the geometrical relation between the line segments \(O B\) and \(C A\).
CAIE P3 2017 June Q8
8 marks Standard +0.3
8
  1. By first expanding \(2 \sin \left( x - 30 ^ { \circ } \right)\), express \(2 \sin \left( x - 30 ^ { \circ } \right) - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [5]
  2. Hence solve the equation $$2 \sin \left( x - 30 ^ { \circ } \right) - \cos x = 1$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2017 June Q9
9 marks Standard +0.3
9
  1. Express \(\frac { 1 } { x ( 2 x + 3 ) }\) in partial fractions.
  2. The variables \(x\) and \(y\) satisfy the differential equation $$x ( 2 x + 3 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y$$ and it is given that \(y = 1\) when \(x = 1\). Solve the differential equation and calculate the value of \(y\) when \(x = 9\), giving your answer correct to 3 significant figures.
CAIE P3 2017 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-18_324_677_259_734} The diagram shows the curve \(y = \sin x \cos ^ { 2 } 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) and its maximum point \(M\).
  1. Using the substitution \(u = \cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.
  2. Find the \(x\)-coordinate of \(M\). Give your answer correct to 2 decimal places.
CAIE P3 2017 June Q1
3 marks Standard +0.3
1 Prove the identity \(\frac { \cot x - \tan x } { \cot x + \tan x } \equiv \cos 2 x\).
CAIE P3 2017 June Q2
4 marks Moderate -0.8
2 Expand \(( 3 + 2 x ) ^ { - 3 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2017 June Q3
4 marks Moderate -0.8
3 Using the substitution \(u = \mathrm { e } ^ { x }\), solve the equation \(4 \mathrm { e } ^ { - x } = 3 \mathrm { e } ^ { x } + 4\). Give your answer correct to 3 significant figures.
CAIE P3 2017 June Q4
4 marks Moderate -0.3
4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \theta \sin \frac { 1 } { 2 } \theta \mathrm {~d} \theta\).
CAIE P3 2017 June Q5
6 marks Standard +0.3
5 A curve has equation \(y = \frac { 2 } { 3 } \ln \left( 1 + 3 \cos ^ { 2 } x \right)\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan x\).
  2. Hence find the \(x\)-coordinate of the point on the curve where the gradient is - 1 . Give your answer correct to 3 significant figures.