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 P1 2010 June Q1
1
  1. Show that the equation $$3 ( 2 \sin x - \cos x ) = 2 ( \sin x - 3 \cos x )$$ can be written in the form \(\tan x = - \frac { 3 } { 4 }\).
  2. Solve the equation \(3 ( 2 \sin x - \cos x ) = 2 ( \sin x - 3 \cos x )\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{dcc8cfc5-7ed3-4e4d-9856-b12e38ac69ef-2_486_727_625_708} The diagram shows part of the curve \(y = \frac { a } { x }\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is \(24 \pi\), find the value of \(a\).
CAIE P1 2010 June Q3
3 The functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 4 x - 2 x ^ { 2 }
& \mathrm {~g} : x \mapsto 5 x + 3 \end{aligned}$$
  1. Find the range of f .
  2. Find the value of the constant \(k\) for which the equation \(\mathrm { gf } ( x ) = k\) has equal roots.
    \includegraphics[max width=\textwidth, alt={}, center]{dcc8cfc5-7ed3-4e4d-9856-b12e38ac69ef-2_607_780_1909_685} In the diagram, \(A\) is the point \(( - 1,3 )\) and \(B\) is the point \(( 3,1 )\). The line \(L _ { 1 }\) passes through \(A\) and is parallel to \(O B\). The line \(L _ { 2 }\) passes through \(B\) and is perpendicular to \(A B\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) meet at \(C\). Find the coordinates of \(C\).
CAIE P1 2010 June Q5
5 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 2
3
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 4
1
p \end{array} \right)$$
  1. Find the value of \(p\) for which \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O B }\).
  2. Find the values of \(p\) for which the magnitude of \(\overrightarrow { A B }\) is 7 .
CAIE P1 2010 June Q6
6
  1. Find the first 3 terms in the expansion of \(( 1 + a x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Given that there is no term in \(x\) in the expansion of \(( 1 - 2 x ) ( 1 + a x ) ^ { 5 }\), find the value of the constant \(a\).
  3. For this value of \(a\), find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - 2 x ) ( 1 + a x ) ^ { 5 }\).
CAIE P1 2010 June Q7
7
  1. Find the sum of all the multiples of 5 between 100 and 300 inclusive.
  2. A geometric progression has a common ratio of \(- \frac { 2 } { 3 }\) and the sum of the first 3 terms is 35 . Find
    1. the first term of the progression,
    2. the sum to infinity.
CAIE P1 2010 June Q8
8 A solid rectangular block has a square base of side \(x \mathrm {~cm}\). The height of the block is \(h \mathrm {~cm}\) and the total surface area of the block is \(96 \mathrm {~cm} ^ { 2 }\).
  1. Express \(h\) in terms of \(x\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the block is given by $$V = 24 x - \frac { 1 } { 2 } x ^ { 3 }$$ Given that \(x\) can vary,
  2. find the stationary value of \(V\),
  3. determine whether this stationary value is a maximum or a minimum.
CAIE P1 2010 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{dcc8cfc5-7ed3-4e4d-9856-b12e38ac69ef-4_723_919_248_612} The diagram shows the curve \(y = ( x - 2 ) ^ { 2 }\) and the line \(y + 2 x = 7\), which intersect at points \(A\) and \(B\). Find the area of the shaded region.
CAIE P1 2010 June Q10
10 The equation of a curve is \(y = \frac { 1 } { 6 } ( 2 x - 3 ) ^ { 3 } - 4 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the equation of the tangent to the curve at the point where the curve intersects the \(y\)-axis.
  3. Find the set of values of \(x\) for which \(\frac { 1 } { 6 } ( 2 x - 3 ) ^ { 3 } - 4 x\) is an increasing function of \(x\).
CAIE P1 2010 June Q11
11 The function f : \(x \mapsto 4 - 3 \sin x\) is defined for the domain \(0 \leqslant x \leqslant 2 \pi\).
  1. Solve the equation \(\mathrm { f } ( x ) = 2\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution. The function \(\mathrm { g } : x \mapsto 4 - 3 \sin x\) is defined for the domain \(\frac { 1 } { 2 } \pi \leqslant x \leqslant A\).
  4. State the largest value of \(A\) for which g has an inverse.
  5. For this value of \(A\), find the value of \(\mathrm { g } ^ { - 1 } ( 3 )\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P1 2010 June Q1
1 The first term of a geometric progression is 12 and the second term is - 6 . Find
  1. the tenth term of the progression,
  2. the sum to infinity.
CAIE P1 2010 June Q2
2
  1. Find the first three terms, in descending powers of \(x\), in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\).
  2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2010 June Q3
3 The function \(\mathrm { f } : x \mapsto a + b \cos x\) is defined for \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } ( 0 ) = 10\) and that \(\mathrm { f } \left( \frac { 2 } { 3 } \pi \right) = 1\), find
  1. the values of \(a\) and \(b\),
  2. the range of \(f\),
  3. the exact value of \(\mathrm { f } \left( \frac { 5 } { 6 } \pi \right)\).
CAIE P1 2010 June Q4
4
  1. Show that the equation \(2 \sin x \tan x + 3 = 0\) can be expressed as \(2 \cos ^ { 2 } x - 3 \cos x - 2 = 0\).
  2. Solve the equation \(2 \sin x \tan x + 3 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2010 June Q5
5 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 3 x - 2 ) }\). Given that the curve passes through the point \(P ( 2,11 )\), find
  1. the equation of the normal to the curve at \(P\),
  2. the equation of the curve.
CAIE P1 2010 June Q6
6 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 8 \mathbf { k } , \quad \overrightarrow { O C } = - \mathbf { i } - 2 \mathbf { j } + 10 \mathbf { k }$$
  1. Use a scalar product to find angle \(A B C\).
  2. Find the perimeter of triangle \(A B C\), giving your answer correct to 2 decimal places.
CAIE P1 2010 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_744_675_255_735} The diagram shows a metal plate \(A B C D E F\) which has been made by removing the two shaded regions from a circle of radius 10 cm and centre \(O\). The parallel edges \(A B\) and \(E D\) are both of length 12 cm .
  1. Show that angle \(D O E\) is 1.287 radians, correct to 4 significant figures.
  2. Find the perimeter of the metal plate.
  3. Find the area of the metal plate.
CAIE P1 2010 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_796_695_1539_726} The diagram shows a rhombus \(A B C D\) in which the point \(A\) is ( \(- 1,2\) ), the point \(C\) is ( 5,4 ) and the point \(B\) lies on the \(y\)-axis. Find
  1. the equation of the perpendicular bisector of \(A C\),
  2. the coordinates of \(B\) and \(D\),
  3. the area of the rhombus.
CAIE P1 2010 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-4_602_899_248_625} The diagram shows part of the curve \(y = x + \frac { 4 } { x }\) which has a minimum point at \(M\). The line \(y = 5\) intersects the curve at the points \(A\) and \(B\).
  1. Find the coordinates of \(A , B\) and \(M\).
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2010 June Q10
10 The function \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 14\) is defined for \(x \in \mathbb { R }\).
  1. Find the values of the constant \(k\) for which the line \(y + k x = 12\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
  3. Find the range of f . The function \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 8 x + 14\) is defined for \(x \geqslant A\).
  4. Find the smallest value of \(A\) for which g has an inverse.
  5. For this value of \(A\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P1 2011 June Q1
1 Find the coefficient of \(x\) in the expansion of \(\left( x + \frac { 2 } { x ^ { 2 } } \right) ^ { 7 }\).
CAIE P1 2011 June Q2
2 The volume of a spherical balloon is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the radius when the radius is 10 cm . [Volume of a sphere \(= \frac { 4 } { 3 } \pi r ^ { 3 }\).]
CAIE P1 2011 June Q3
3
  1. Sketch the curve \(y = ( x - 2 ) ^ { 2 }\).
  2. The region enclosed by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained, giving your answer in terms of \(\pi\).
CAIE P1 2011 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-2_750_855_902_646} The diagram shows a prism \(A B C D P Q R S\) with a horizontal square base \(A P S D\) with sides of length 6 cm . The cross-section \(A B C D\) is a trapezium and is such that the vertical edges \(A B\) and \(D C\) are of lengths 5 cm and 2 cm respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(A D , A P\) and \(A B\) respectively.
  1. Express each of the vectors \(\overrightarrow { C P }\) and \(\overrightarrow { C Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to calculate angle \(P C Q\).
CAIE P1 2011 June Q5
5
  1. Show that the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) can be written in the form $$2 \sin ^ { 4 } \theta + \sin ^ { 2 } \theta - 1 = 0 .$$
  2. Hence solve the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).