Questions — CAIE (7279 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 M1 2024 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-03_721_622_296_724} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a vertical wall. The particle is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), perpendicular to the string, with the string taut and making an angle of \(30 ^ { \circ }\) with the wall (see diagram). Find the tension in the string and the value of \(X\).
CAIE M1 2024 June Q3
3 A car travels along a straight road with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\), where \(a > 0\). The car passes through points \(A , B\) and \(C\) in that order. The speed of the car at \(A\) is \(u \mathrm {~ms} ^ { - 1 }\) in the direction \(A B\). The distance \(B C\) is twice the distance \(A B\). The car takes 8 seconds to travel from \(A\) to \(B\) and 10 seconds to travel from \(B\) to \(C\).
  1. Find \(u\) in terms of \(a\).
  2. Find the speed of the car at \(C\) in terms of \(a\).
CAIE M1 2024 June Q4
4 A particle travels in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving a point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = k t ^ { 2 } - 4 t + 3$$ The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that \(v > 0\) in the first 2s of its motion.
  1. Find the value of \(k\).
  2. Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.
CAIE M1 2024 June Q5
5 A van of mass 4500 kg is towing a trailer of mass 750 kg down a straight hill inclined at an angle of \(\theta\) to the horizontal where \(\sin \theta = 0.05\). The van and the trailer are connected by a light rigid tow-bar which is parallel to the road. There are constant resistance forces of 2500 N on the van and 300 N on the trailer.
  1. It is given that the tension in the tow-bar is 450 N . Find the acceleration of the trailer and the driving force of the van's engine.
    On another occasion, the van and trailer ascend a straight hill inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.09\). The driving force of the van's engine is now 9100 N , and the speed of the van at the bottom of the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances to motion are unchanged.
    1. Find the acceleration of the van and the tension in the tow-bar.
    2. Find the speed of the van when it has travelled a distance of 375 m up the hill.
CAIE M1 2024 June Q6
6 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg . There is a constant resistance force of magnitude 32 N to the cyclist's motion. At an instant when she is travelling at \(7 \mathrm {~ms} ^ { - 1 }\), her acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).
  1. Find the power output of the cyclist.
  2. Find the steady speed that the cyclist can maintain if her power output and the resistance force are both unchanged.
    \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-08_2718_35_141_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-09_2724_35_136_20} The cyclist later descends a straight hill of length 32.2 m , inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 20 } \right)\) to the horizontal. Her power output is now 120 W , and the resistance force now has variable magnitude such that the work done against this force in descending the hill is 1128 J . The time taken to descend the hill is 4 s .
  3. Given that the speed of the cyclist at the top of the hill is \(7.5 \mathrm {~ms} ^ { - 1 }\), find her speed at the bottom of the hill.
CAIE M1 2024 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_323_1308_292_376} The diagram shows a track \(A B C D\) which lies in a vertical plane. The section \(A B\) is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is smooth. The section \(B C\) is a horizontal straight line and is rough. The section CD is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is rough. The lengths \(A B , B C\) and \(C D\) are each 2 m . A particle is released from rest at \(A\). The coefficient of friction between the particle and both \(B C\) and \(C D\) is \(\mu\). There is no change in the speed of the particle when it passes through either of the points \(B\) or \(C\).
  1. It is given that \(\mu = 0.1\). Find the distance which the particle has moved up the section \(C D\) when its speed is \(1 \mathrm {~ms} ^ { - 1 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_2716_33_143_2014}
  2. It is given instead that with a different value of \(\mu\) the particle travels 1 m up the track from \(C\) before it comes instantaneously to rest. Find the value of \(\mu\) and the speed of the particle at the instant that it passes \(C\) for the second time.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P1 2021 March Q10
  1. For the case where angle \(B A C = \frac { 1 } { 6 } \pi\) radians, find \(k\) correct to 4 significant figures.
  2. For the general case in which angle \(B A C = \theta\) radians, where \(0 < \theta < \frac { 1 } { 2 } \pi\), it is given that \(\frac { \theta } { \sin \theta } > 1\). Find the set of possible values of \(k\).
CAIE P1 2022 March Q6
  1. Find, by calculation, the coordinates of \(A\) and \(B\).
  2. Find an equation of the circle which has its centre at \(C\) and for which the line with equation \(y = 3 x - 20\) is a tangent to the circle.
CAIE P1 2022 March Q10
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.
CAIE P1 2024 March Q10
  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
  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
  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
  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
  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
  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
  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
  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
  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 2014 November Q5
  1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos ^ { 2 } x - \cos x - 1 = 0$$
  2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). The equation of a curve is \(y = x ^ { 3 } + a x ^ { 2 } + b x\), where \(a\) and \(b\) are constants.
  3. In the case where the curve has no stationary point, show that \(a ^ { 2 } < 3 b\).
  4. In the case where \(a = - 6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\).
    \includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-3_634_711_952_717} The diagram shows a pyramid \(O A B C X\). The horizontal square base \(O A B C\) has side 8 units and the centre of the base is \(D\). The top of the pyramid, \(X\), is vertically above \(D\) and \(X D = 10\) units. The mid-point of \(O X\) is \(M\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(\overrightarrow { O A }\) and \(\overrightarrow { O C }\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.
  5. Express the vectors \(\overrightarrow { A M }\) and \(\overrightarrow { A C }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  6. Use a scalar product to find angle \(M A C\).
    (a) The sum, \(S _ { n }\), of the first \(n\) terms of an arithmetic progression is given by \(S _ { n } = 32 n - n ^ { 2 }\). Find the first term and the common difference.
    (b) A geometric progression in which all the terms are positive has sum to infinity 20 . The sum of the first two terms is 12.8 . Find the first term of the progression.
CAIE P1 2015 November Q10
  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
  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
  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
  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 June Q6
  1. Find an expression for \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
  2. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \frac { x + 3 } { \ln ( 2 x + 1 ) } - 0.5\).
  3. Show by calculation that the \(x\)-coordinate of \(M\) lies between 2.5 and 3.0 .
  4. Use an iterative formula based on the equation in part (b) to find the \(x\)-coordinate of \(M\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2024 November Q6
  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 \(B\).