Questions (33218 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 P1 2024 March Q10
12 marks Standard +0.3
  1. Find the equation of the tangent to the circle at the point \(( - 6,9 )\).
  2. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  3. Find the value of \(\theta\) correct to 4 significant figures.
  4. Find the perimeter and area of the segment shaded in the diagram.
CAIE P1 2020 November Q10
10 marks Moderate -0.3
[diagram]
The diagram shows a sector \(CAB\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D\), \(E\), and \(F\), where \(COD\) is a straight line and angle \(ACD\) is \(\theta\) radians.
  1. Find \(C D\) in terms of \(r\) and \(\sin \theta\).
    It is now given that \(r = 4\) and \(\theta = \frac { 1 } { 6 } \pi\).
  2. Find the perimeter of sector \(C A B\) in terms of \(\pi\).
  3. Find the area of the shaded region in terms of \(\pi\) and \(\sqrt { 3 }\).
CAIE P1 2021 November Q6
7 marks Standard +0.3
  1. Find the perimeter of the plate, giving your answer in terms of \(\pi\).
  2. Find the area of the plate, giving your answer in terms of \(\pi\) and \(\sqrt { 3 }\).
CAIE P1 2022 November Q10
8 marks Standard +0.3
  1. Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
  2. Find the difference in area of the two triangles \(A O B\) and \(A P B\), giving your answer correct to 2 decimal places.
  3. Find the area of the cross-section RASB, giving your answer correct to 1 decimal place.
CAIE P1 2022 November Q10
10 marks Standard +0.3
  1. Find the coordinates of \(A\).
  2. Find the volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in the form \(\frac { \pi } { a } ( b \sqrt { c } - d )\), where \(a , b , c\) and \(d\) are integers.
  3. Find an exact expression for the perimeter of the shaded region.
CAIE P1 2017 June Q4
7 marks Moderate -0.8
  1. Express the perimeter of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 5\) and \(\theta = \frac { 1 } { 6 } \pi\), find the area of the shaded region.
CAIE P1 2018 June Q5
6 marks Moderate -0.5
  1. Express each of the vectors \(\overrightarrow { D A }\) and \(\overrightarrow { C A }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle \(C A D\).
CAIE P1 2017 March Q4
6 marks Standard +0.3
  1. Show that angle \(C B D = \frac { 9 } { 14 } \pi\) radians.
  2. Find the perimeter of the shaded region.
CAIE P1 2005 November Q5
7 marks Standard +0.3
  1. Express \(h\) in terms of \(r\) and hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 12 \pi r ^ { 2 } - 2 \pi r ^ { 3 }$$
  2. Given that \(r\) varies, find the stationary value of \(V\).
CAIE P1 2015 November Q10
9 marks Standard +0.3
  1. For the case where \(a = 2\), find the unit vector in the direction of \(\overrightarrow { P M }\).
  2. For the case where angle \(A T P = \cos ^ { - 1 } \left( \frac { 2 } { 7 } \right)\), find the value of \(a\).
CAIE P1 2016 November Q4
6 marks Easy -1.2
  1. Find the equation of the line \(C D\), giving your answer in the form \(y = m x + c\).
  2. Find the distance \(A D\).
CAIE P1 Specimen Q10
9 marks Standard +0.3
  1. For the case where \(a = 2\), find the unit vector in the direction of \(\overrightarrow { P M }\).
  2. For the case where angle \(A T P = \cos ^ { - 1 } \left( \frac { 2 } { 7 } \right)\), find the value of \(a\).
CAIE P2 2024 June Q6
9 marks Moderate -0.3
  1. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.
  2. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_65_1548_379_349} \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_67_1566_466_328} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_646_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_735_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_826_324} \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_74_1570_916_324} ........................................................................................................................................ . \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1572_1096_322} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_1187_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_67_1570_1279_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_70_1570_1367_324} \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_62_1570_1462_324} .......................................................................................................................................... ......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_72_1570_1724_324} .......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_71_1570_1905_324} \includegraphics[max width=\textwidth, alt={}, center]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_74_1570_1994_324} \includegraphics[max width=\textwidth, alt={}]{971a1d8d-a82e-4a3a-b72d-3c147e4f30bb-11_76_1570_2083_324} ......................................................................................................................................... . ........................................................................................................................................ ......................................................................................................................................... ........................................................................................................................................ . ......................................................................................................................................... . ........................................................................................................................................
CAIE P2 2024 November Q6
9 marks Moderate -0.3
  1. Use the trapezium rule with two intervals to find an approximation to the area of region \(A\). Give your answer correct to 3 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_2720_38_105_2010} \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-11_2716_29_107_22}
  2. Find the exact total area of regions \(A\) and \(B\). Give your answer in the form \(k \ln m\), where \(k\) and \(m\) are constants.
  3. Deduce an approximation to the area of region \(B\). Give your answer correct to 3 significant figures.
  4. State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region \(B\).
CAIE P2 2002 June Q5
8 marks Standard +0.3
  1. Find the exact coordinates of \(P\).
  2. Show that the \(x\)-coordinates of \(Q\) and \(R\) satisfy the equation $$x = \frac { 1 } { 4 } e ^ { x } .$$
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 4 } e ^ { x _ { n } }$$ with initial value \(x _ { 1 } = 0\), to find the \(x\)-coordinate of \(Q\) correct to 2 decimal places, showing the value of each approximation that you calculate.
CAIE P2 2010 June Q6
8 marks Standard +0.3
  1. By sketching a suitable pair of graphs, show that the equation $$\ln x = 2 - x ^ { 2 }$$ has only one root.
  2. Verify by calculation that this root lies between \(x = 1.3\) and \(x = 1.4\).
  3. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$ converges, then it converges to the root of the equation in part (i).
  4. Use the iterative formula \(x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P2 2014 June Q5
8 marks Standard +0.3
  1. Prove that \(\tan \theta + \cot \theta \equiv \frac { 2 } { \sin 2 \theta }\).
  2. Hence
    1. find the exact value of \(\tan \frac { 1 } { 8 } \pi + \cot \frac { 1 } { 8 } \pi\),
    2. evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 6 } { \tan \theta + \cot \theta } \mathrm { d } \theta\).
CAIE P2 2007 November Q7
8 marks Standard +0.3
  1. Prove the identity $$( \cos x + 3 \sin x ) ^ { 2 } \equiv 5 - 4 \cos 2 x + 3 \sin 2 x$$
  2. Using the identity, or otherwise, find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \cos x + 3 \sin x ) ^ { 2 } d x$$
CAIE P2 2017 November Q5
9 marks Standard +0.3
  1. Show that the \(x\)-coordinate of \(Q\) satisfies the equation \(x = \frac { 9 } { 8 } - \frac { 1 } { 2 } \mathrm { e } ^ { - 2 x }\).
  2. Use an iterative formula based on the equation in part (i) to find the \(x\)-coordinate of \(Q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P3 2020 June Q7
9 marks Challenging +1.2
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Using your answer to part (a), show that $$( f ( x ) ) ^ { 2 } = \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } - \frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$$
  3. Hence show that \(\int _ { 1 } ^ { 2 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x = \frac { 2 } { 5 } + \frac { 1 } { 2 } \ln \left( \frac { 5 } { 9 } \right)\).
CAIE P3 2021 June Q10
10 marks Standard +0.3
  1. Given that the sum of the areas of the shaded sectors is \(90 \%\) of the area of the trapezium, show that \(x\) satisfies the equation \(x = 0.9 ( 2 - \cos x ) \sin x\).
  2. Verify by calculation that \(x\) lies between 0.5 and 0.7 .
  3. Show that if a sequence of values in the interval \(0 < x < \frac { 1 } { 2 } \pi\) given by the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( 2 - \frac { x _ { n } } { 0.9 \sin x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (a).
  4. Use this iterative formula to determine \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 June Q10
10 marks Standard +0.3
  1. Find the exact coordinates of \(M\).
  2. Using the substitution \(u = 3 - 2 x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis. Give your answer in the form \(a \sqrt { 13 }\), where \(a\) is a rational number. [5]
CAIE P3 2021 November Q9
10 marks Standard +0.3
  1. Find the \(x\)-coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\).
  2. Using the substitution \(u = \sqrt { x }\), show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 3 } \ln 5\).
CAIE P3 2024 November Q6
8 marks Standard +0.3
  1. Given that the \(x\)-coordinate of \(M\) lies in the interval \(\frac { 1 } { 2 } \pi < x < \frac { 3 } { 4 } \pi\), find the exact coordinates of \(M\). \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-10_2718_35_107_2012} \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-11_2725_35_99_20}
  2. Find the exact area of the region \(R\).
CAIE P3 2012 June Q7
8 marks Standard +0.3
  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 the complex numbers \(u , 1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\) respectively.
  3. By considering the arguments of \(1 + 2 \mathrm { i }\) and \(1 - 3 \mathrm { i }\), show that $$\tan ^ { - 1 } 2 + \tan ^ { - 1 } 3 = \frac { 3 } { 4 } \pi$$