Questions — CAIE P2 (699 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 P2 2017 June Q3
5 marks
3 Given that \(\int _ { 0 } ^ { a } 4 \mathrm { e } ^ { \frac { 1 } { 2 } x + 3 } \mathrm {~d} x = 835\), find the value of the constant \(a\) correct to 3 significant figures. [5]
CAIE P2 2017 June Q4
4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + x _ { n } + 9 } { \left( x _ { n } + 1 \right) ^ { 2 } }$$ with \(x _ { 1 } = 2\), converges to \(\alpha\).
  1. Find the value of \(\alpha\) correct to 2 decimal places, giving the result of each iteration to 4 decimal places.
  2. Determine the exact value of \(\alpha\).
CAIE P2 2017 June Q5
5
  1. Express \(2 \cos \theta + ( \sqrt { } 5 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(2 \cos \theta + ( \sqrt { } 5 ) \sin \theta = 1\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2017 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-06_561_542_260_799} The diagram shows the curve \(y = \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\). The shaded region is bounded by the curve and the lines \(x = \frac { 1 } { 6 } \pi\) and \(y = 0\).
  1. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region, giving your answer correct to 3 significant figures.
  2. Find the exact volume of the solid formed when the shaded region is rotated completely about the \(x\)-axis.
CAIE P2 2017 June Q7
7 The parametric equations of a curve are $$x = t ^ { 3 } + 6 t + 1 , \quad y = t ^ { 4 } - 2 t ^ { 3 } + 4 t ^ { 2 } - 12 t + 5$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and use division to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be written in the form \(a t + b\), where \(a\) and \(b\) are constants to be found.
  2. The straight line \(x - 2 y + 9 = 0\) is the normal to the curve at the point \(P\). Find the coordinates of \(P\).
CAIE P2 2017 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-10_643_414_260_863} The diagram shows the curve with equation $$y = 3 x ^ { 2 } \ln \left( \frac { 1 } { 6 } x \right) .$$ The curve crosses the \(x\)-axis at the point \(P\) and has a minimum point \(M\).
  1. Find the gradient of the curve at the point \(P\).
  2. Find the exact coordinates of the point \(M\).
CAIE P2 2017 June Q1
1 Solve the equation \(| x + a | = | 2 x - 5 a |\), giving \(x\) in terms of the positive constant \(a\).
CAIE P2 2017 June Q2
2 Use logarithms to solve the equation \(3 ^ { x + 4 } = 5 ^ { 2 x }\), giving your answer correct to 3 significant figures.
CAIE P2 2017 June Q3
3
  1. By sketching a suitable pair of graphs, show that the equation $$x ^ { 3 } = 11 - 2 x$$ has exactly one real root.
  2. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 11 - 2 x _ { n } \right)$$ to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2017 June Q4
4 Find the equation of the tangent to the curve \(y = \frac { \mathrm { e } ^ { 4 x } } { 2 x + 3 }\) at the point on the curve for which \(x = 0\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
CAIE P2 2017 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{de2f8bf3-fd03-4199-9eb2-c9cbac4d4385-05_551_535_260_806} The variables \(x\) and \(y\) satisfy the equation \(y = \frac { K } { a ^ { 2 x } }\), where \(K\) and \(a\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.6,1.81 )\) and \(( 1.4,1.39 )\), as shown in the diagram. Find the values of \(K\) and \(a\) correct to 2 significant figures.
CAIE P2 2017 June Q6
6
  1. Use the factor theorem to show that ( \(x + 2\) ) is a factor of the expression $$6 x ^ { 3 } + 13 x ^ { 2 } - 33 x - 70$$ and hence factorise the expression completely.
  2. Deduce the roots of the equation $$6 + 13 y - 33 y ^ { 2 } - 70 y ^ { 3 } = 0$$
CAIE P2 2017 June Q7
7
  1. Find \(\int ( 2 \cos \theta - 3 ) ( \cos \theta + 1 ) \mathrm { d } \theta\).
    1. Find \(\int \left( \frac { 4 } { 2 x + 1 } + \frac { 1 } { 2 x } \right) \mathrm { d } x\).
    2. Hence find \(\int _ { 1 } ^ { 4 } \left( \frac { 4 } { 2 x + 1 } + \frac { 1 } { 2 x } \right) \mathrm { d } x\), giving your answer in the form \(\ln k\).
CAIE P2 2017 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{de2f8bf3-fd03-4199-9eb2-c9cbac4d4385-10_549_495_258_824} The diagram shows the curve with parametric equations $$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve \(( 1,4 )\) and \(( 3,3 )\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
  2. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
  3. Find the exact gradient of the normal to the curve at the point for which \(x = 2\).
CAIE P2 2017 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{bdc467f6-105e-4429-95c6-701eaa43deff-05_551_533_260_806} The variables \(x\) and \(y\) satisfy the equation \(y = \frac { K } { a ^ { 2 x } }\), where \(K\) and \(a\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \(( 0.6,1.81 )\) and \(( 1.4,1.39 )\), as shown in the diagram. Find the values of \(K\) and \(a\) correct to 2 significant figures.
CAIE P2 2017 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{bdc467f6-105e-4429-95c6-701eaa43deff-10_549_495_258_824} The diagram shows the curve with parametric equations $$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve are \(( 1,4 )\) and \(( 3,3 )\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
  2. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
  3. Find the exact gradient of the normal to the curve at the point for which \(x = 2\).
CAIE P2 2018 June Q1
1 Solve the equation \(3 \mathrm { e } ^ { 2 x } - 82 \mathrm { e } ^ { x } + 27 = 0\), giving your answers in the form \(k \ln 3\).
CAIE P2 2018 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{873a104f-e2e2-49bb-b943-583769728fbb-04_554_493_260_826} The variables \(x\) and \(y\) satisfy the equation \(y = A \times B ^ { \ln x }\), where \(A\) and \(B\) are constants. The graph of \(\ln y\) against \(\ln x\) is a straight line passing through the points (2.2, 4.908) and (5.9, 11.008), as shown in the diagram. Find the values of \(A\) and \(B\) correct to 2 significant figures.
CAIE P2 2018 June Q3
3 Without using a calculator, find the exact value of \(\int _ { 0 } ^ { 2 } 4 \mathrm { e } ^ { - x } \left( \mathrm { e } ^ { 3 x } + 1 \right) \mathrm { d } x\).
CAIE P2 2018 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{873a104f-e2e2-49bb-b943-583769728fbb-06_355_839_260_653} The diagram shows the curve with equation \(y = \frac { 5 \ln x } { 2 x + 1 }\). The curve crosses the \(x\)-axis at the point \(P\) and has a maximum point \(M\).
  1. Find the gradient of the curve at the point \(P\).
  2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \frac { x + 0.5 } { \ln x }\).
  3. Use an iterative formula based on the equation in part (ii) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Show the result of each iteration to 6 significant figures.
CAIE P2 2018 June Q5
5 The parametric equations of a curve are $$x = 2 \cos 2 \theta + 3 \sin \theta , \quad y = 3 \cos \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\).
  1. Find the gradient of the curve at the point for which \(\theta = 1\) radian.
  2. Find the value of \(\sin \theta\) at the point on the curve where the tangent is parallel to the \(y\)-axis.
CAIE P2 2018 June Q6
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + 14 x + a + 1$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).
  1. Use the factor theorem to find the value of \(a\) and hence factorise \(\mathrm { f } ( x )\) completely.
  2. Hence, without using a calculator, solve the equation \(\mathrm { f } ( 2 x ) = 3 \mathrm { f } ( x )\).
CAIE P2 2018 June Q7
7
  1. Express \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
  2. Using your answer from part (i), solve the equation $$5 \cot \theta - 4 \operatorname { cosec } \theta = 2$$ for \(0 < \theta < 2 \pi\).
  3. Find \(\int \frac { 1 } { \left( 5 \cos \frac { 1 } { 2 } x - 2 \sin \frac { 1 } { 2 } x \right) ^ { 2 } } \mathrm {~d} x\).
    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 P2 2018 June Q1
1 Solve the inequality \(| 3 x - 2 | < | x + 5 |\).
CAIE P2 2018 June Q2
2 A curve has equation \(y = 3 \ln ( 2 x + 9 ) - 2 \ln x\).
  1. Find the \(x\)-coordinate of the stationary point.
  2. Determine whether the stationary point is a maximum or minimum point.