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 2021 June Q4
4 The coefficient of \(x\) in the expansion of \(\left( 4 x + \frac { 10 } { x } \right) ^ { 3 }\) is \(p\). The coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x + \frac { k } { x ^ { 2 } } \right) ^ { 5 }\) is \(q\). Given that \(p = 6 q\), find the possible values of \(k\).
CAIE P1 2021 June Q5
5 The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 3\) for \(x \geqslant 0\).
  1. Find and simplify an expression for \(\mathrm { ff } ( x )\).
  2. Solve the equation \(\mathrm { ff } ( x ) = 34 x ^ { 2 } + 19\).
CAIE P1 2021 June Q6
6 Points \(A\) and \(B\) have coordinates \(( 8,3 )\) and \(( p , q )\) respectively. The equation of the perpendicular bisector of \(A B\) is \(y = - 2 x + 4\). Find the values of \(p\) and \(q\).
CAIE P1 2021 June Q7
7 The point \(A\) has coordinates \(( 1,5 )\) and the line \(l\) has gradient \(- \frac { 2 } { 3 }\) and passes through \(A\). A circle has centre \(( 5,11 )\) and radius \(\sqrt { 52 }\).
  1. Show that \(l\) is the tangent to the circle at \(A\).
  2. Find the equation of the other circle of radius \(\sqrt { 52 }\) for which \(l\) is also the tangent at \(A\).
CAIE P1 2021 June Q8
8 The first, second and third terms of an arithmetic progression are \(a , \frac { 3 } { 2 } a\) and \(b\) respectively, where \(a\) and \(b\) are positive constants. The first, second and third terms of a geometric progression are \(a , 18\) and \(b + 3\) respectively.
  1. Find the values of \(a\) and \(b\).
  2. Find the sum of the first 20 terms of the arithmetic progression.
CAIE P1 2021 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-10_540_1113_260_516} The diagram shows part of the curve with equation \(y ^ { 2 } = x - 2\) and the lines \(x = 5\) and \(y = 1\). The shaded region enclosed by the curve and the lines is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained.
CAIE P1 2021 June Q10
10
  1. Prove the identity \(\frac { 1 + \sin x } { 1 - \sin x } - \frac { 1 - \sin x } { 1 + \sin x } \equiv \frac { 4 \tan x } { \cos x }\).
  2. Hence solve the equation \(\frac { 1 + \sin x } { 1 - \sin x } - \frac { 1 - \sin x } { 1 + \sin x } = 8 \tan x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
CAIE P1 2021 June Q11
11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 ( 3 x - 5 ) ^ { 3 } - k x ^ { 2 }\), where \(k\) is a constant. The curve has a stationary point at \(( 2 , - 3.5 )\).
  1. Find the value of \(k\).
    ................................................................................................................................................. . .
  2. Find the equation of the curve.
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  4. Determine the nature of the stationary point at \(( 2 , - 3.5 )\).
CAIE P1 2021 June Q12
12
\includegraphics[max width=\textwidth, alt={}, center]{5b8ddd32-c884-48a0-ad51-5582ef0d5128-16_598_609_264_769} The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm , held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are \(A , B , C , D\), \(E\) and \(F\). Points \(P\) and \(Q\) are situated where straight sections of the rope meet the pipe with centre \(A\).
  1. Show that angle \(P A Q = \frac { 1 } { 3 } \pi\) radians.
  2. Find the length of the rope.
  3. Find the area of the hexagon \(A B C D E F\), giving your answer in terms of \(\sqrt { 3 }\).
  4. Find the area of the complete region enclosed by the rope.
    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 P1 2021 June Q1
1 A curve with equation \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - \frac { 8 } { x ^ { 2 } }\). It is given that the curve passes through the point \(( 2,7 )\). Find \(\mathrm { f } ( x )\).
CAIE P1 2021 June Q2
2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 } ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 2 x\) for \(\frac { 1 } { 2 } < x < a\). It is given that f is a decreasing function. Find the maximum possible value of the constant \(a\).
CAIE P1 2021 June Q3
3 A line with equation \(y = m x - 6\) is a tangent to the curve with equation \(y = x ^ { 2 } - 4 x + 3\).
Find the possible values of the constant \(m\), and the corresponding coordinates of the points at which the line touches the curve.
CAIE P1 2021 June Q4
4
  1. Show that the equation $$\frac { \tan x + \sin x } { \tan x - \sin x } = k$$ where \(k\) is a constant, may be expressed as $$\frac { 1 + \cos x } { 1 - \cos x } = k$$
  2. Hence express \(\cos x\) in terms of \(k\).
  3. Hence solve the equation \(\frac { \tan x + \sin x } { \tan x - \sin x } = 4\) for \(- \pi < x < \pi\).
CAIE P1 2021 June Q5
5 The diagram shows a triangle \(A B C\), in which angle \(A B C = 90 ^ { \circ }\) and \(A B = 4 \mathrm {~cm}\). The sector \(A B D\) is part of a circle with centre \(A\). The area of the sector is \(10 \mathrm {~cm} ^ { 2 }\).
  1. Find angle \(B A D\) in radians.
  2. Find the perimeter of the shaded region.
CAIE P1 2021 June Q6
6 Functions f and g are both defined for \(x \in \mathbb { R }\) and are given by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 2 x + 5
& \mathrm {~g} ( x ) = x ^ { 2 } + 4 x + 13 \end{aligned}$$
  1. By first expressing each of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) in completed square form, express \(\mathrm { g } ( x )\) in the form \(\mathrm { f } ( x + p ) + q\), where \(p\) and \(q\) are constants.
  2. Describe fully the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).
CAIE P1 2021 June Q7
7
  1. Write down the first four terms of the expansion, in ascending powers of \(x\), of \(( a - x ) ^ { 6 }\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \frac { 2 } { a x } \right) ( a - x ) ^ { 6 }\) is - 20 , find in exact form the possible values of the constant \(a\).
CAIE P1 2021 June Q8
8 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x ^ { 2 } - 1 \text { for } x < 0
& \mathrm {~g} : x \mapsto \frac { 1 } { 2 x + 1 } \text { for } x < - \frac { 1 } { 2 } \end{aligned}$$
  1. Solve the equation \(\operatorname { fg } ( x ) = 3\).
  2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\).
CAIE P1 2021 June Q9
9
  1. A geometric progression is such that the second term is equal to \(24 \%\) of the sum to infinity. Find the possible values of the common ratio.
  2. An arithmetic progression \(P\) has first term \(a\) and common difference \(d\). An arithmetic progression \(Q\) has first term 2( \(a + 1\) ) and common difference ( \(d + 1\) ). It is given that $$\frac { 5 \text { th term of } P } { 12 \text { th term of } Q } = \frac { 1 } { 3 } \quad \text { and } \quad \frac { \text { Sum of first } 5 \text { terms of } P } { \text { Sum of first } 5 \text { terms of } Q } = \frac { 2 } { 3 } .$$ Find the value of \(a\) and the value of \(d\).
CAIE P1 2021 June Q10
10 Points \(A ( - 2,3 ) , B ( 3,0 )\) and \(C ( 6,5 )\) lie on the circumference of a circle with centre \(D\).
  1. Show that angle \(A B C = 90 ^ { \circ }\).
  2. Hence state the coordinates of \(D\).
  3. Find an equation of the circle.
    The point \(E\) lies on the circumference of the circle such that \(B E\) is a diameter.
  4. Find an equation of the tangent to the circle at \(E\).
CAIE P1 2021 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{aaba3158-b5be-464e-bea3-1a4c460f9637-16_622_1091_260_525} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } + k ^ { 2 } x ^ { - \frac { 1 } { 2 } }\), where \(k\) is a positive constant.
  1. Find the coordinates of the minimum point of the curve, giving your answer in terms of \(k\).
    The tangent at the point on the curve where \(x = 4 k ^ { 2 }\) intersects the \(y\)-axis at \(P\).
  2. Find the \(y\)-coordinate of \(P\) in terms of \(k\).
    The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = \frac { 9 } { 4 } k ^ { 2 }\) and \(x = 4 k ^ { 2 }\).
  3. Find the area of the shaded region in terms of \(k\).
    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 P1 2022 June Q1
1
  1. Express \(x ^ { 2 } - 8 x + 11\) in the form \(( x + p ) ^ { 2 } + q\) where \(p\) and \(q\) are constants.
  2. Hence find the exact solutions of the equation \(x ^ { 2 } - 8 x + 11 = 1\).
CAIE P1 2022 June Q2
2 The thirteenth term of an arithmetic progression is 12 and the sum of the first 30 terms is - 15 .
Find the sum of the first 50 terms of the progression.
CAIE P1 2022 June Q3
3 The coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 x ^ { 2 } + \frac { k ^ { 2 } } { x } \right) ^ { 5 }\) is \(a\). The coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 k x - 1 ) ^ { 4 }\) is \(b\).
  1. Find \(a\) and \(b\) in terms of the constant \(k\).
  2. Given that \(a + b = 216\), find the possible values of \(k\).
CAIE P1 2022 June Q4
4
  1. Prove the identity \(\frac { \sin ^ { 3 } \theta } { \sin \theta - 1 } - \frac { \sin ^ { 2 } \theta } { 1 + \sin \theta } \equiv - \tan ^ { 2 } \theta \left( 1 + \sin ^ { 2 } \theta \right)\).
  2. Hence solve the equation $$\frac { \sin ^ { 3 } \theta } { \sin \theta - 1 } - \frac { \sin ^ { 2 } \theta } { 1 + \sin \theta } = \tan ^ { 2 } \theta \left( 1 - \sin ^ { 2 } \theta \right)$$ for \(0 < \theta < 2 \pi\).
CAIE P1 2022 June Q5
4 marks
5
\includegraphics[max width=\textwidth, alt={}, center]{574b96b2-62f2-41b3-a178-8e68e16429ff-08_509_654_264_751} The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius \(r\). The line \(B D\) is perpendicular to \(A C\). Angle \(C A B\) is \(\theta\) radians.
  1. Given that \(\theta = \frac { 1 } { 6 } \pi\), find the exact area of \(B C D\) in terms of \(r\).
  2. Given instead that the length of \(B D\) is \(\frac { \sqrt { 3 } } { 2 } r\), find the exact perimeter of \(B C D\) in terms of \(r\). [4]