Questions — CAIE (7279 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 P1 2015 June Q6
2 marks
6
- 3
2 \end{array} \right)$$
  1. Find the cosine of angle \(A O B\). The position vector of \(C\) is given by \(\overrightarrow { O C } = \left( \begin{array} { c } k
    - 2 k
    2 k - 3 \end{array} \right)\).
  2. Given that \(A B\) and \(O C\) have the same length, find the possible values of \(k\). 5 A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r \mathrm {~cm}\).
  3. Show that the area of the sector, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 12 r - r ^ { 2 }\).
  4. Express \(A\) in the form \(a - ( r - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
  5. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2] 6 The line with gradient - 2 passing through the point \(P ( 3 t , 2 t )\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  6. Find the area of triangle \(A O B\) in terms of \(t\). The line through \(P\) perpendicular to \(A B\) intersects the \(x\)-axis at \(C\).
  7. Show that the mid-point of \(P C\) lies on the line \(y = x\).
CAIE P1 2015 June Q7
7
  1. The third and fourth terms of a geometric progression are \(\frac { 1 } { 3 }\) and \(\frac { 2 } { 9 }\) respectively. Find the sum to infinity of the progression.
  2. A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector.
CAIE P1 2015 June Q8
8 The function f : \(x \mapsto 5 + 3 \cos \left( \frac { 1 } { 2 } x \right)\) is defined for \(0 \leqslant x \leqslant 2 \pi\).
  1. Solve the equation \(\mathrm { f } ( x ) = 7\), giving your answer correct to 2 decimal places.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Explain why f has an inverse.
  4. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2015 June Q9
9 The equation of a curve is \(y = x ^ { 3 } + p x ^ { 2 }\), where \(p\) is a positive constant.
  1. Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of \(p\).
  2. Find the nature of each of the stationary points. Another curve has equation \(y = x ^ { 3 } + p x ^ { 2 } + p x\).
  3. Find the set of values of \(p\) for which this curve has no stationary points.
    [0pt] [Question 10 is printed on the next page.]
CAIE P1 2015 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{c8925c7a-cb3b-43b8-9d09-8adc800c6887-4_798_805_258_669} The diagram shows part of the curve \(y = \frac { 8 } { \sqrt { } ( 3 x + 4 ) }\). The curve intersects the \(y\)-axis at \(A ( 0,4 )\). The normal to the curve at \(A\) intersects the line \(x = 4\) at the point \(B\).
  1. Find the coordinates of \(B\).
  2. Show, with all necessary working, that the areas of the regions marked \(P\) and \(Q\) are equal.
CAIE P1 2015 June Q1
1 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 5 - 2 x ^ { 2 }\) and ( 3,5 ) is a point on the curve \(y = \mathrm { f } ( x )\). Find \(\mathrm { f } ( x )\).
CAIE P1 2015 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{5eb35648-db3f-4f29-b835-1aff4ef927b9-2_624_588_395_776} In the diagram, \(A Y B\) is a semicircle with \(A B\) as diameter and \(O A X B\) is a sector of a circle with centre \(O\) and radius \(r\). Angle \(A O B = 2 \theta\) radians. Find an expression, in terms of \(r\) and \(\theta\), for the area of the shaded region.
CAIE P1 2015 June Q3
3
  1. Find the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 2 - x ) ^ { 6 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 x + 1 ) ( 2 - x ) ^ { 6 }\).
CAIE P1 2015 June Q4
4 Variables \(u , x\) and \(y\) are such that \(u = 2 x ( y - x )\) and \(x + 3 y = 12\). Express \(u\) in terms of \(x\) and hence find the stationary value of \(u\).
  1. Prove the identity \(\frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } \equiv \frac { \tan \theta - 1 } { \tan \theta + 1 }\).
  2. Hence solve the equation \(\frac { \sin \theta - \cos \theta } { \sin \theta + \cos \theta } = \frac { \tan \theta } { 6 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2015 June Q6
6 A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, \(h \mathrm {~m}\), of a passenger above the ground is given by the formula \(h = 60 ( 1 - \cos k t )\). In this formula, \(k\) is a constant, \(t\) is the time in minutes that has elapsed since the passenger started the ride at ground level and \(k t\) is measured in radians.
  1. Find the greatest height of the passenger above the ground. One complete revolution of the wheel takes 30 minutes.
  2. Show that \(k = \frac { 1 } { 15 } \pi\).
  3. Find the time for which the passenger is above a height of 90 m .
CAIE P1 2015 June Q7
7 The point \(C\) lies on the perpendicular bisector of the line joining the points \(A ( 4,6 )\) and \(B ( 10,2 )\). \(C\) also lies on the line parallel to \(A B\) through \(( 3,11 )\).
  1. Find the equation of the perpendicular bisector of \(A B\).
  2. Calculate the coordinates of \(C\).
CAIE P1 2015 June Q8
8
  1. The first, second and last terms in an arithmetic progression are 56, 53 and -22 respectively. Find the sum of all the terms in the progression.
  2. The first, second and third terms of a geometric progression are \(2 k + 6,2 k\) and \(k + 2\) respectively, where \(k\) is a positive constant.
    1. Find the value of \(k\).
    2. Find the sum to infinity of the progression.
CAIE P1 2015 June Q9
9 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }$$
  1. Use a vector method to find angle \(A O B\). The point \(C\) is such that \(\overrightarrow { A B } = \overrightarrow { B C }\).
  2. Find the unit vector in the direction of \(\overrightarrow { O C }\).
  3. Show that triangle \(O A C\) is isosceles.
CAIE P1 2015 June Q10
10 The equation of a curve is \(y = \frac { 4 } { 2 x - 1 }\).
  1. Find, showing all necessary working, the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  2. Given that the line \(2 y = x + c\) is a normal to the curve, find the possible values of the constant \(c\).
CAIE P1 2015 June Q11
11 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(x \in \mathbb { R }\).
  1. Find the set of values of \(p\) for which the equation \(\mathrm { f } ( x ) = p\) has no real roots. The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(0 \leqslant x \leqslant 4\).
  2. Express \(\mathrm { g } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
  3. Find the range of g . The function h is defined by \(\mathrm { h } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.
  4. State the smallest value of \(k\) for which h has an inverse.
  5. For this value of \(k\), find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
CAIE P1 2015 June Q1
1 Express \(2 x ^ { 2 } - 12 x + 7\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
CAIE P1 2015 June Q2
2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) ^ { \frac { 1 } { 2 } }\) and the point (4,7) lies on the curve. Find the equation of the curve.
CAIE P1 2015 June Q3
3
  1. Write down the first 4 terms, in ascending powers of \(x\), of the expansion of \(( a - x ) ^ { 5 }\).
  2. The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - a x ) ( a - x ) ^ { 5 }\) is - 200 . Find the possible values of the constant \(a\).
CAIE P1 2015 June Q4
4
  1. Express the equation \(3 \sin \theta = \cos \theta\) in the form \(\tan \theta = k\) and solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
  2. Solve the equation \(3 \sin ^ { 2 } 2 x = \cos ^ { 2 } 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P1 2015 June Q5
5 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3
2
- 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 5
- 1
- 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l }
CAIE P1 2015 June Q6
6
1
2 \end{array} \right)$$
  1. Show that angle \(A B C\) is \(90 ^ { \circ }\).
  2. Find the area of triangle \(A B C\), giving your answer correct to 1 decimal place. 6
    \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-2_519_670_1640_735} The diagram shows the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), where \(\mathrm { f } ^ { - 1 }\) is defined by \(\mathrm { f } ^ { - 1 } ( x ) = \frac { 1 - 5 x } { 2 x }\) for \(0 < x \leqslant 2\).
  3. Find an expression for \(\mathrm { f } ( x )\) and state the domain of f .
  4. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x }\) for \(x \geqslant 1\). Find an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\), giving your answer in the form \(a x + b\), where \(a\) and \(b\) are constants to be found.
CAIE P1 2015 June Q7
7 The point \(A\) has coordinates \(( p , 1 )\) and the point \(B\) has coordinates \(( 9,3 p + 1 )\), where \(p\) is a constant.
  1. For the case where the distance \(A B\) is 13 units, find the possible values of \(p\).
  2. For the case in which the line with equation \(2 x + 3 y = 9\) is perpendicular to \(A B\), find the value of \(p\).
CAIE P1 2015 June Q8
8 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x > - 1\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. State, with a reason, whether f is an increasing function, a decreasing function or neither. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x < - 1\).
  3. Find the coordinates of the stationary point on the curve \(y = \mathrm { g } ( x )\).
CAIE P1 2015 June Q9
9
  1. The first term of an arithmetic progression is - 2222 and the common difference is 17 . Find the value of the first positive term.
  2. The first term of a geometric progression is \(\sqrt { } 3\) and the second term is \(2 \cos \theta\), where \(0 < \theta < \pi\). Find the set of values of \(\theta\) for which the progression is convergent.
CAIE P1 2015 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-3_682_1319_1525_413} Points \(A ( 2,9 )\) and \(B ( 3,0 )\) lie on the curve \(y = 9 + 6 x - 3 x ^ { 2 }\), as shown in the diagram. The tangent at \(A\) intersects the \(x\)-axis at \(C\). Showing all necessary working,
  1. find the equation of the tangent \(A C\) and hence find the \(x\)-coordinate of \(C\),
  2. find the area of the shaded region \(A B C\).
    [0pt] [Question 11 is printed on the next page.]