Questions — CAIE P3 (1070 questions)

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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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
CAIE P3 2020 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{5f80ae11-34c3-4d2f-89f8-71b4ac021c7d-16_426_908_262_616} The diagram shows the curve \(y = ( 2 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the area of the shaded region bounded by the curve and the axes. Give your answer in terms of e.
CAIE P3 2020 November Q11
11 Two lines have equations \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant.
  1. Given that the two lines intersect, find the value of \(a\) and the position vector of the point of intersection.
  2. Given instead that the acute angle between the directions of the two lines is \(\cos ^ { - 1 } \left( \frac { 1 } { 6 } \right)\), find the two possible values of \(a\).
    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 2020 November Q1
1 Solve the equation $$\ln \left( 1 + \mathrm { e } ^ { - 3 x } \right) = 2$$ Give the answer correct to 3 decimal places.
CAIE P3 2020 November Q2
2
  1. Expand \(\sqrt [ 3 ] { 1 + 6 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
  2. State the set of values of \(x\) for which the expansion is valid.
CAIE P3 2020 November Q3
3 The variables \(x\) and \(y\) satisfy the relation \(2 ^ { y } = 3 ^ { 1 - 2 x }\).
  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. Find the exact \(x\)-coordinate of the point of intersection of this line with the line \(y = 3 x\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.
CAIE P3 2020 November Q4
4
  1. Show that the equation \(\tan \left( \theta + 60 ^ { \circ } \right) = 2 \cot \theta\) can be written in the form $$\tan ^ { 2 } \theta + 3 \sqrt { 3 } \tan \theta - 2 = 0$$
  2. Hence solve the equation \(\tan \left( \theta + 60 ^ { \circ } \right) = 2 \cot \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2020 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{77a45360-8e1d-4f4f-9830-075d832a14cf-08_334_895_258_625} The diagram shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta$$ for \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).
  1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(- 2 \sin \theta \cos ^ { 3 } \theta\).
    The gradient of the curve has its maximum value at the point \(P\).
  2. Find the exact value of the \(x\)-coordinate of \(P\).
CAIE P3 2020 November Q6
6 The complex number \(u\) is defined by $$u = \frac { 7 + \mathrm { i } } { 1 - \mathrm { i } }$$
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Show on a sketch of an Argand diagram the points \(A , B\) and \(C\) representing \(u , 7 + \mathrm { i }\) and \(1 - \mathrm { i }\) respectively.
  3. By considering the arguments of \(7 + \mathrm { i }\) and \(1 - \mathrm { i }\), show that $$\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) = \tan ^ { - 1 } \left( \frac { 1 } { 7 } \right) + \frac { 1 } { 4 } \pi$$
CAIE P3 2020 November Q7
7 The variables \(x\) and \(t\) satisfy the differential equation $$\mathrm { e } ^ { 3 t } \frac { \mathrm {~d} x } { \mathrm {~d} t } = \cos ^ { 2 } 2 x$$ for \(t \geqslant 0\). It is given that \(x = 0\) when \(t = 0\).
  1. Solve the differential equation and obtain an expression for \(x\) in terms of \(t\).
  2. State what happens to the value of \(x\) when \(t\) tends to infinity.
CAIE P3 2020 November Q8
8 With respect to the origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2
1
5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 4
- 1
1 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 1
1
2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { l } 3
2
3 \end{array} \right)$$
  1. Show that \(A B = 2 C D\).
  2. Find the angle between the directions of \(\overrightarrow { A B }\) and \(\overrightarrow { C D }\).
  3. Show that the line through \(A\) and \(B\) does not intersect the line through \(C\) and \(D\).
CAIE P3 2020 November Q9
9 Let \(\mathrm { f } ( x ) = \frac { 7 x + 18 } { ( 3 x + 2 ) \left( x ^ { 2 } + 4 \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find the exact value of \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).
CAIE P3 2020 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{77a45360-8e1d-4f4f-9830-075d832a14cf-18_549_933_260_605} The diagram shows the curve \(y = \sqrt { x } \cos x\), for \(0 \leqslant x \leqslant \frac { 3 } { 2 } \pi\), and its minimum point \(M\), where \(x = a\). The shaded region between the curve and the \(x\)-axis is denoted by \(R\).
  1. Show that \(a\) satisfies the equation \(\tan a = \frac { 1 } { 2 a }\).
  2. The sequence of values given by the iterative formula \(a _ { n + 1 } = \pi + \tan ^ { - 1 } \left( \frac { 1 } { 2 a _ { n } } \right)\), with initial value \(x _ { 1 } = 3\), converges to \(a\). Use this formula to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  3. Find the volume of the solid obtained when the region \(R\) is rotated completely 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 2020 November Q1
1 Solve the inequality \(2 - 5 x > 2 | x - 3 |\).
CAIE P3 2020 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{19aff1b7-51b7-4d44-86e6-45dad32aa121-16_426_908_262_616} The diagram shows the curve \(y = ( 2 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the area of the shaded region bounded by the curve and the axes. Give your answer in terms of e.
CAIE P3 2021 November Q1
1 Solve the equation \(4 \left| 5 ^ { x } - 1 \right| = 5 ^ { x }\), giving your answers correct to 3 decimal places.
CAIE P3 2021 November Q2
2
  1. Express \(5 \sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
  2. Hence state the greatest and least possible values of \(( 5 \sin x - 3 \cos x ) ^ { 2 }\).
CAIE P3 2021 November Q3
3 The curve with equation \(y = x \mathrm { e } ^ { 1 - 2 x }\) has one stationary point.
  1. Find the coordinates of this point.
  2. Determine whether the stationary point is a maximum or a minimum.
CAIE P3 2021 November Q4
4 Using the substitution \(u = \sqrt { x }\), find the exact value of $$\int _ { 3 } ^ { \infty } \frac { 1 } { ( x + 1 ) \sqrt { x } } \mathrm {~d} x$$
CAIE P3 2021 November Q5
5
  1. Show that the equation $$\cot 2 \theta + \cot \theta = 2$$ can be expressed as a quadratic equation in \(\tan \theta\).
  2. Hence solve the equation \(\cot 2 \theta + \cot \theta = 2\), for \(0 < \theta < \pi\), giving your answers correct to 3 decimal places.
CAIE P3 2021 November Q6
6 When \(( a + b x ) \sqrt { 1 + 4 x }\), where \(a\) and \(b\) are constants, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 2 }\) are 3 and - 6 respectively. Find the values of \(a\) and \(b\).
CAIE P3 2021 November Q7
7
  1. Given that \(y = \ln ( \ln x )\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln x }$$ The variables \(x\) and \(t\) satisfy the differential equation $$x \ln x + t \frac { \mathrm {~d} x } { \mathrm {~d} t } = 0$$ It is given that \(x = \mathrm { e }\) when \(t = 2\).
  2. Solve the differential equation obtaining an expression for \(x\) in terms of \(t\), simplifying your answer.
  3. Hence state what happens to the value of \(x\) as \(t\) tends to infinity.
CAIE P3 2021 November Q8
8 The constant \(a\) is such that \(\int _ { 1 } ^ { a } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x = 6\).
  1. Show that \(a = \exp \left( \frac { 1 } { \sqrt { a } } + 2 \right)\).
    \(\left[ \exp ( x ) \right.\) is an alternative notation for \(\left. \mathrm { e } ^ { x } .\right]\)
  2. Verify by calculation that \(a\) lies between 9 and 11 .
  3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2021 November Q10
10 The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + 4 x + b\), where \(a\) and \(b\) are real constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\).
  1. Find the values of \(a\) and \(b\).
  2. State a second complex root of this equation.
  3. Find the real factors of \(\mathrm { p } ( x )\).
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant \sqrt { 5 }\) and \(\arg z \leqslant \frac { 1 } { 4 } \pi\).
    2. Find the least value of \(\operatorname { Im } z\) for points in the shaded region. Give your answer in an exact form.
      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 2021 November Q1
1 Find the value of \(x\) for which \(3 \left( 2 ^ { 1 - x } \right) = 7 ^ { x }\). Give your answer in the form \(\frac { \ln a } { \ln b }\), where \(a\) and \(b\) are integers.
CAIE P3 2021 November Q2
2 Solve the inequality \(| 3 x - a | > 2 | x + 2 a |\), where \(a\) is a positive constant.