Questions P2 (856 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 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 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 2018 June Q3
3
  1. Find the quotient when $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 13$$ is divided by \(x ^ { 2 } + 6\) and show that the remainder is 1 .
  2. Show that the equation $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 12 = 0$$ has no real roots.
CAIE P2 2018 June Q4
4
  1. Solve the equation \(2 \ln ( 2 x ) - \ln ( x + 3 ) = 4 \ln 2\).
  2. Hence solve the equation $$2 \ln \left( 2 ^ { u + 1 } \right) - \ln \left( 2 ^ { u } + 3 \right) = 4 \ln 2$$ giving the value of \(u\) correct to 4 significant figures.
CAIE P2 2018 June Q5
5 A curve has equation $$y ^ { 3 } \sin 2 x + 4 y = 8$$ Find the equation of the tangent to the curve at the point where it crosses the \(y\)-axis.
CAIE P2 2018 June Q6
6 It is given that \(\int _ { 0 } ^ { a } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 2 } x } \right) ^ { 2 } \mathrm {~d} x = 10\), where \(a\) is a positive constant.
  1. Show that \(a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)\).
  2. Use the equation in part (i) to show by calculation that \(1.5 < a < 1.6\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2018 June Q7
7
  1. Show that \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x\).
  2. Solve the equation \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21\) for \(0 < x < \pi\), giving your answers correct to 3 significant figures.
  3. Find \(\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y\).
    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 2019 June Q1
1 Show that \(\ln \left( x ^ { 3 } - 4 x \right) - \ln \left( x ^ { 2 } - 2 x \right) \equiv \ln ( x + 2 )\).
CAIE P2 2019 June Q2
2
  1. Solve the inequality \(| 3 x - 5 | < | x + 3 |\).
  2. Hence find the greatest integer \(n\) satisfying the inequality \(\left| 3 ^ { 0.1 n + 1 } - 5 \right| < \left| 3 ^ { 0.1 n } + 3 \right|\).
CAIE P2 2019 June Q3
3 Find the equation of the normal to the curve $$x ^ { 2 } \ln y + 2 x + 5 y = 11$$ at the point \(( 3,1 )\).
CAIE P2 2019 June Q4
4
  1. Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 3 x } + 4 } { \mathrm { e } ^ { x } } \mathrm {~d} x\). Show all necessary working.
CAIE P2 2019 June Q5
5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 5 x ^ { 3 } + a x ^ { 2 } + b x - 16$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 27 when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2019 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{6694ccc1-c8b1-42a7-8b21-829a89af74c9-08_732_807_258_667} The diagram shows the curve with equation \(y = \frac { 8 + x ^ { 3 } } { 2 - 5 x }\). The maximum point is denoted by \(M\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine the gradient of the curve at the point where the curve crosses the \(x\)-axis.
  2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \sqrt { } \left( 0.6 x + 4 x ^ { - 1 } \right)\).
  3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2019 June Q7
7
  1. Show that \(2 \operatorname { cosec } 2 \theta \cot \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
  2. Hence show that \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } \tan 15 ^ { \circ } = 4\).
  3. Solve the equation \(2 \operatorname { cosec } \phi \cot \frac { 1 } { 2 } \phi + \operatorname { cosec } \frac { 1 } { 2 } \phi = 12\) for \(- 360 ^ { \circ } < \phi < 360 ^ { \circ }\). Show all necessary working.
    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 2019 June Q1
1 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + ( k + 1 ) x ^ { 2 } - m x + 3 k$$ where \(k\) and \(m\) are constants. Given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\), express \(m\) in terms of \(k\).
CAIE P2 2019 June Q2
2
  1. Solve the equation \(| 4 + 2 x | = | 3 - 5 x |\).
  2. Hence solve the equation \(\left| 4 + 2 e ^ { 3 y } \right| = \left| 3 - 5 e ^ { 3 y } \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2019 June Q3
3 Find the exact coordinates of the stationary point of the curve with equation \(y = \frac { 3 x } { \ln x }\).
CAIE P2 2019 June Q4
4
  1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 4 \sin 2 x + 2 \cos ^ { 2 } x \right) \mathrm { d } x\). Show all necessary working.
  2. Use the trapezium rule with two intervals to find an approximation to \(\int _ { 2 } ^ { 8 } \sqrt { } ( \ln ( 1 + x ) ) \mathrm { d } x\)
CAIE P2 2019 June Q5
5
  1. Find the quotient and remainder when \(2 x ^ { 3 } + x ^ { 2 } - 8 x\) is divided by ( \(2 x + 1\) ).
  2. Hence find the exact value of \(\int _ { 0 } ^ { 3 } \frac { 2 x ^ { 3 } + x ^ { 2 } - 8 x } { 2 x + 1 } \mathrm {~d} x\), giving the answer in the form \(\ln \left( k \mathrm { e } ^ { a } \right)\) where \(k\) and \(a\) are constants.
CAIE P2 2019 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{f5e0b088-73db-405b-a832-aa01d9fcba64-08_396_716_260_712} The diagram shows the curve with parametric equations $$x = 3 t - 6 \mathrm { e } ^ { - 2 t } , \quad y = 4 t ^ { 2 } \mathrm { e } ^ { - t }$$ for \(0 \leqslant t \leqslant 2\). At the point \(P\) on the curve, the \(y\)-coordinate is 1 .
  1. Show that the value of \(t\) at the point \(P\) satisfies the equation \(t = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t }\).
  2. Use the iterative formula \(t _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t _ { n } }\) with \(t _ { 1 } = 0.7\) to find the value of \(t\) at \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
  3. Find the gradient of the curve at \(P\), giving the answer correct to 2 significant figures.
CAIE P2 2019 June Q7
7
    1. Express \(4 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
    2. Hence find the smallest positive value of \(\theta\) satisfying the equation \(4 \sin \theta + 4 \cos \theta = 5\).
  2. Solve the equation $$4 \cot 2 x = 5 + \tan x$$ for \(0 < x < \pi\), showing all necessary working and giving the answers correct to 2 decimal places.
    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 2019 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{0d15e5a1-d05f-48bc-8613-198804ff605c-08_396_716_260_712} The diagram shows the curve with parametric equations $$x = 3 t - 6 \mathrm { e } ^ { - 2 t } , \quad y = 4 t ^ { 2 } \mathrm { e } ^ { - t }$$ for \(0 \leqslant t \leqslant 2\). At the point \(P\) on the curve, the \(y\)-coordinate is 1 .
  1. Show that the value of \(t\) at the point \(P\) satisfies the equation \(t = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t }\).
  2. Use the iterative formula \(t _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t _ { n } }\) with \(t _ { 1 } = 0.7\) to find the value of \(t\) at \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
  3. Find the gradient of the curve at \(P\), giving the answer correct to 2 significant figures.
CAIE P2 2016 March Q1
1 Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 10\) is divided by \(( x + 2 )\).
CAIE P2 2016 March Q2
2 Solve the inequality \(| x - 5 | < | 2 x + 3 |\).
CAIE P2 2016 March Q3
3 It is given that \(k\) is a positive constant. Solve the equation \(2 \ln x = \ln ( 3 k + x ) + \ln ( 2 k - x )\), expressing \(x\) in terms of \(k\).
CAIE P2 2016 March Q4
4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } + 4 x _ { n } ^ { - 3 } \right)$$ with initial value \(x _ { 1 } = 1.5\), converges to \(\alpha\).
  1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).
CAIE P2 2016 March Q5
5 Given that \(\int _ { 0 } ^ { a } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x = 65\), find the value of \(a\) correct to 3 decimal places.