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 2022 March Q2
2 A curve has equation \(y = 7 + 4 \ln ( 2 x + 5 )\).
Find the equation of the tangent to the curve at the point ( \(- 2,7\) ), giving your answer in the form \(y = m x + c\).
CAIE P2 2022 March Q3
3 The variables \(x\) and \(y\) satisfy the equation \(y = 3 ^ { 2 a } a ^ { x }\), where \(a\) is a constant. The graph of \(\ln y\) against \(x\) is a straight line with gradient 0.239 .
  1. Find the value of \(a\) correct to 3 significant figures.
  2. Hence find the value of \(x\) when \(y = 36\). Give your answer correct to 3 significant figures.
CAIE P2 2022 March Q4
4
  1. Show that \(\sin 2 \theta \cot \theta - \cos 2 \theta \equiv 1\).
  2. Hence find the exact value of \(\sin \frac { 1 } { 6 } \pi \cot \frac { 1 } { 12 } \pi\).
  3. Find the smallest positive value of \(\theta\) (in radians) satisfying the equation $$\sin 2 \theta \cot \theta - 3 \cos 2 \theta = 1 .$$
CAIE P2 2022 March Q5
5
  1. Given that \(y = \tan ^ { 2 } x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \tan x + 2 \tan ^ { 3 } x\).
  2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( \tan x + \tan ^ { 2 } x + \tan ^ { 3 } x \right) \mathrm { d } x\).
CAIE P2 2022 March Q6
6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + 16 x ^ { 2 } + 9 x - 15$$
  1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 2 x + 3 )\), and show that the remainder is - 6 .
  2. Find \(\int \frac { \mathrm { p } ( x ) } { 2 x + 3 } \mathrm {~d} x\).
  3. Factorise \(\mathrm { p } ( x ) + 6\) completely and hence solve the equation $$p ( \operatorname { cosec } 2 \theta ) + 6 = 0$$ for \(0 ^ { \circ } < \theta < 135 ^ { \circ }\).
CAIE P2 2022 March Q7
7 A curve has equation \(\mathrm { e } ^ { 2 x } y - \mathrm { e } ^ { y } = 100\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \mathrm { e } ^ { 2 x } y } { \mathrm { e } ^ { y } - \mathrm { e } ^ { 2 x } }\).
  2. Show that the curve has no stationary points.
    It is required to find the \(x\)-coordinate of \(P\), the point on the curve at which the tangent is parallel to the \(y\)-axis.
  3. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$x = \ln 10 - \frac { 1 } { 2 } \ln ( 2 x - 1 )$$
  4. Use an iterative formula, based on the equation in part (c), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.
    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 2023 March Q1
1 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 2 \tan ^ { 2 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
CAIE P2 2023 March Q2
2 Solve the equation \(\tan \left( \theta - 60 ^ { \circ } \right) = 3 \cot \theta\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P2 2023 March Q3
3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), and that the remainder is 35 when \(\mathrm { p } ( x )\) is divided by \(( x - 3 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\) and show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
CAIE P2 2023 March Q4
4
  1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 11 |\) and \(y = 3 x - 3\).
  2. Solve the inequality \(| 2 x - 11 | < 3 x - 3\).
  3. Find the smallest integer \(N\) satisfying the inequality \(| 2 \ln N - 11 | < 3 \ln N - 3\).
CAIE P2 2023 March Q5
5 It is given that \(\int _ { 1 } ^ { a } \left( \frac { 4 } { 1 + 2 x } + \frac { 3 } { x } \right) \mathrm { d } x = \ln 10\), where \(a\) is a constant greater than 1 .
  1. Show that \(a = \sqrt [ 3 ] { 90 ( 1 + 2 a ) ^ { - 2 } }\).
  2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures.
    \includegraphics[max width=\textwidth, alt={}, center]{ce0d5faa-9428-4afd-829d-7634c5bd150d-10_798_495_269_810} The diagram shows the curve with equation \(y = \frac { 4 \mathrm { e } ^ { 2 x } + 9 } { \mathrm { e } ^ { x } + 2 }\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).
  3. Find the exact value of the gradient of the curve at \(P\).
  4. Find the exact coordinates of \(M\).
CAIE P2 2023 March Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{ce0d5faa-9428-4afd-829d-7634c5bd150d-12_446_613_274_760} The diagram shows the curve with parametric equations $$x = k \tan t , \quad y = 3 \sin 2 t - 4 \sin t ,$$ for \(0 < t < \frac { 1 } { 2 } \pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).
  1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction.
  2. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(k\) and \(\cos t\).
  3. Given that the normal to the curve at \(P\) has gradient \(\frac { 9 } { 10 }\), find the value of \(k\), giving your answer as an exact fraction.
    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 2024 March Q1
4 marks
1 Use logarithms to solve the equation \(3 ^ { 4 x + 3 } = 5 ^ { 2 x + 7 }\). Give your answer correct to 3 significant figures. [4]
CAIE P2 2024 March Q2
2
  1. Sketch the graph of \(y = | 3 x - 7 |\), stating the coordinates of the points where the graph meets the axes.
  2. Hence find the set of values of the constant \(k\) for which the equation \(| 3 \mathrm { x } - 7 | = \mathrm { k } ( \mathrm { x } - 4 )\) has exactly two real roots.
CAIE P2 2024 March Q3
3 The polynomial \(\mathrm { p } ( x )\) is defined by $$p ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 3 x - 10$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\) completely.
  2. Solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
    \includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-06_442_706_278_667} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { 1 + \mathrm { e } ^ { 0.5 \mathrm { x } } }\). The shaded region is bounded by the curve and the straight lines \(x = 0 , x = 6\) and \(y = 0\).
  3. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
  4. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.
CAIE P2 2024 March Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-08_615_469_260_799} The diagram shows part of the curve with equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 3 } } { \mathrm { x } + 2 }\). At the point \(P\), the gradient of the curve is 6 .
  1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \sqrt [ 3 ] { 12 x + 12 }\).
  2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 3.8 and 4.0 .
  3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Show the result of each iteration to 5 significant figures.
CAIE P2 2024 March Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-10_629_620_278_717} The diagram shows the curve with parametric equations $$x = 1 + \sqrt { t } , \quad y = ( \ln t + 2 ) ( \ln t - 3 ) ,$$ for \(0 < t < 25\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\) and has a minimum point \(M\).
  1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 4 \mathrm { Int } - 2 } { \sqrt { \mathrm { t } } }\).
  2. Find the exact gradient of the curve at \(B\).
  3. Find the exact coordinates of \(M\).
CAIE P2 2024 March Q7
7
  1. Prove that $$\sin 2 \theta ( a \cot \theta + b \tan \theta ) \equiv a + b + ( a - b ) \cos 2 \theta$$ where \(a\) and \(b\) are constants.
  2. Find the exact value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } \sin 2 \theta ( 5 \cot \theta + 3 \tan \theta ) d \theta\).
  3. Solve the equation \(\sin \frac { 2 } { 3 } \alpha \left( 2 \cot \frac { 1 } { 3 } \alpha + 7 \tan \frac { 1 } { 3 } \alpha \right) = 11\) for \(- \pi < \alpha < \pi\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P2 2020 November Q1
1 Given that $$\ln ( 2 x + 1 ) - \ln ( x - 3 ) = 2$$ find \(x\) in terms of e.
CAIE P2 2020 November Q2
2 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 16$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 72 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\). Find the values of \(a\) and \(b\).
CAIE P2 2020 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{8beee722-7f86-454a-bc36-27e83f1483fd-04_684_455_260_845} The diagram shows the curve \(y = 2 + \mathrm { e } ^ { - 2 x }\). The curve crosses the \(y\)-axis at the point \(A\), and the point \(B\) on the curve has \(x\)-coordinate 1 . The shaded region is bounded by the curve and the line segment \(A B\). Find the exact area of the shaded region.
CAIE P2 2020 November Q4
4
  1. Solve the equation \(| 2 x - 5 | = | x + 6 |\).
  2. Hence find the value of \(y\) such that \(\left| 2 ^ { 1 - y } - 5 \right| = \left| 2 ^ { - y } + 6 \right|\). Give your answer correct to 3 significant figures.
CAIE P2 2020 November Q5
5 The sequence of values given by the iterative formula \(x _ { n + 1 } = \frac { 6 + 8 x _ { n } } { 8 + x _ { n } ^ { 2 } }\) with initial value \(x _ { 1 } = 2\) converges to \(\alpha\).
  1. Use the iterative formula to find the value of \(\alpha\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
  2. State an equation satisfied by \(\alpha\) and hence determine the exact value of \(\alpha\).
CAIE P2 2020 November Q6
6 It is given that \(3 \sin 2 \theta = \cos \theta\) where \(\theta\) is an angle such that \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
  1. Find the exact value of \(\sin \theta\).
  2. Find the exact value of \(\sec \theta\).
  3. Find the exact value of \(\cos 2 \theta\).
CAIE P2 2020 November Q7
7 A curve is defined by the parametric equations $$x = 3 t - 2 \sin t , \quad y = 5 t + 4 \cos t$$ where \(0 \leqslant t \leqslant 2 \pi\). At each of the points \(P\) and \(Q\) on the curve, the gradient of the curve is \(\frac { 5 } { 2 }\).
  1. Show that the values of \(t\) at \(P\) and \(Q\) satisfy the equation \(10 \cos t - 8 \sin t = 5\).
  2. Express \(10 \cos t - 8 \sin t\) in the form \(R \cos ( t + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
  3. Hence find the values of \(t\) at the points \(P\) and \(Q\).