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 2018 June Q3
Moderate -0.8
3 A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by \(2 \%\) of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained 8000 kg of salt.
  1. Find the amount of salt obtained in the 12th week after the change.
  2. Find the total amount of salt obtained in the first 12 weeks after the change.
CAIE P1 2018 June Q4
Moderate -0.8
4 The function f is such that \(\mathrm { f } ( x ) = a + b \cos x\) for \(0 \leqslant x \leqslant 2 \pi\). It is given that \(\mathrm { f } \left( \frac { 1 } { 3 } \pi \right) = 5\) and \(\mathrm { f } ( \pi ) = 11\).
  1. Find the values of the constants \(a\) and \(b\).
    \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-05_63_1566_397_328}
  2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution.
    \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-06_622_878_260_632} The diagram shows a three-dimensional shape. The base \(O A B\) is a horizontal triangle in which angle \(A O B\) is \(90 ^ { \circ }\). The side \(O B C D\) is a rectangle and the side \(O A D\) lies in a vertical plane. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O B\) respectively and the unit vector \(\mathbf { k }\) is vertical. The position vectors of \(A , B\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i } , \overrightarrow { O B } = 5 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }\).
CAIE P1 2018 June Q6
Standard +0.8
6
\includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-08_454_684_255_726} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor \(\operatorname { arc } A B\) and the lines \(A T\) and \(B T\). Angle \(A O B\) is \(2 \theta\) radians.
  1. In the case where the area of the sector \(A O B\) is the same as the area of the shaded region, show that \(\tan \theta = 2 \theta\).
  2. In the case where \(r = 8 \mathrm {~cm}\) and the length of the minor \(\operatorname { arc } A B\) is 19.2 cm , find the area of the shaded region.
CAIE P1 2018 June Q7
Moderate -0.3
7 The function f is defined by \(\mathrm { f } : x \mapsto 7 - 2 x ^ { 2 } - 12 x\) for \(x \in \mathbb { R }\).
  1. Express \(7 - 2 x ^ { 2 } - 12 x\) in the form \(a - 2 ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
  2. State the coordinates of the stationary point on the curve \(y = \mathrm { f } ( x )\).
    The function g is defined by \(\mathrm { g } : x \mapsto 7 - 2 x ^ { 2 } - 12 x\) for \(x \geqslant k\).
  3. State the smallest value of \(k\) for which g has an inverse.
  4. For this value of \(k\), find \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2018 June Q8
Moderate -0.3
8 Points \(A\) and \(B\) have coordinates \(( h , h )\) and \(( 4 h + 6,5 h )\) respectively. The equation of the perpendicular bisector of \(A B\) is \(3 x + 2 y = k\). Find the values of the constants \(h\) and \(k\).
CAIE P1 2018 June Q9
Standard +0.3
9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 4 x + 1 )\) and \(( 2,5 )\) is a point on the curve.
  1. Find the equation of the curve.
  2. A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \(( 2,5 )\).
  3. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \times \frac { \mathrm { d } y } { \mathrm {~d} x }\) is constant.
CAIE P1 2018 June Q10
Moderate -0.3
10
  1. Solve the equation \(2 \cos x + 3 \sin x = 0\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Sketch, on the same diagram, the graphs of \(y = 2 \cos x\) and \(y = - 3 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  3. Use your answers to parts (i) and (ii) to find the set of values of \(x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) for which \(2 \cos x + 3 \sin x > 0\).
CAIE P1 2018 June Q11
Standard +0.3
11
\includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-18_643_969_258_587} The diagram shows part of the curve \(y = \frac { x } { 2 } + \frac { 6 } { x }\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).
  1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\).
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in terms of \(\pi\).
    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 2018 June Q1
Easy -1.2
1 Express \(3 x ^ { 2 } - 12 x + 7\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
CAIE P1 2018 June Q2
Moderate -0.8
2 Find the coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 5 }\).
CAIE P1 2018 June Q3
Standard +0.3
3 The common ratio of a geometric progression is 0.99 . Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures.
CAIE P1 2018 June Q4
Moderate -0.3
4 A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(A ( 3,1 )\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\). Find the \(y\)-coordinate of \(B\).
CAIE P1 2018 June Q5
Moderate -0.5
5
\includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-06_323_775_260_685} The diagram shows a triangle \(O A B\) in which angle \(O A B = 90 ^ { \circ }\) and \(O A = 5 \mathrm {~cm}\). The arc \(A C\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(O B\) at \(C\). Find the area of the shaded region.
CAIE P1 2018 June Q6
Moderate -0.3
6 The coordinates of points \(A\) and \(B\) are \(( - 3 k - 1 , k + 3 )\) and \(( k + 3,3 k + 5 )\) respectively, where \(k\) is a constant ( \(k \neq - 1\) ).
  1. Find and simplify the gradient of \(A B\), showing that it is independent of \(k\).
  2. Find and simplify the equation of the perpendicular bisector of \(A B\).
CAIE P1 2018 June Q7
3 marks Moderate -0.3
7
    1. Express \(\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 }\) in the form \(a \sin ^ { 2 } \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
    2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 } = \frac { 1 } { 4 }$$ for \(- 90 ^ { \circ } \leqslant \theta \leqslant 0 ^ { \circ }\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-11_549_796_267_717} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
    1. Find the \(x\)-coordinate of \(A\).
    2. Find the \(y\)-coordinate of \(B\).
CAIE P1 2018 June Q8
Moderate -0.3
8
  1. The tangent to the curve \(y = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12\) at a point \(A\) is parallel to the line \(y = 2 - 3 x\). Find the equation of the tangent at \(A\).
  2. The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12\) for \(x > k\), where \(k\) is a constant. Find the smallest value of \(k\) for f to be an increasing function.
CAIE P1 2018 June Q9
Standard +0.3
9
\includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-14_670_857_260_644} The diagram shows a pyramid \(O A B C D\) with a horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(O B\) is such that \(O E = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O C\) and \(E D\) respectively.
  1. Show that \(\overrightarrow { O E } = 1.6 \mathbf { i } + 1.2 \mathbf { j }\).
  2. Use a scalar product to find angle \(B D O\).
CAIE P1 2018 June Q10
Moderate -0.8
10 The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } + 2\) for \(x \geqslant c\), where \(c\) is a constant.
  1. State the smallest possible value of \(c\).
    In parts (ii) and (iii) the value of \(c\) is 4 .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Solve the equation \(\mathrm { ff } ( x ) = 51\), giving your answer in the form \(a + \sqrt { } b\).
CAIE P1 2018 June Q11
Standard +0.3
11
\includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-18_645_723_258_573} The diagram shows part of the curve \(y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.
  1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2 ( x + 1 ) ^ { 3 } = 1\) and find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
    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 2019 June Q1
Moderate -0.8
1 The term independent of \(x\) in the expansion of \(\left( 2 x + \frac { k } { x } \right) ^ { 6 }\), where \(k\) is a constant, is 540.
  1. Find the value of \(k\).
  2. For this value of \(k\), find the coefficient of \(x ^ { 2 }\) in the expansion.
CAIE P1 2019 June Q2
Moderate -0.3
2 The line \(4 y = x + c\), where \(c\) is a constant, is a tangent to the curve \(y ^ { 2 } = x + 3\) at the point \(P\) on the curve.
  1. Find the value of \(c\).
  2. Find the coordinates of \(P\).
CAIE P1 2019 June Q3
Moderate -0.8
3 A sector of a circle of radius \(r \mathrm {~cm}\) has an area of \(A \mathrm {~cm} ^ { 2 }\). Express the perimeter of the sector in terms of \(r\) and \(A\).
CAIE P1 2019 June Q4
Standard +0.3
4
\includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-06_625_750_260_699} The diagram shows a trapezium \(A B C D\) in which the coordinates of \(A , B\) and \(C\) are (4, 0), (0, 2) and \(( h , 3 h )\) respectively. The lines \(B C\) and \(A D\) are parallel, angle \(A B C = 90 ^ { \circ }\) and \(C D\) is parallel to the \(x\)-axis.
  1. Find, by calculation, the value of \(h\).
  2. Hence find the coordinates of \(D\).
CAIE P1 2019 June Q5
Moderate -0.8
5 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } + 12 x - 3\) for \(x \in \mathbb { R }\).
  1. Express \(- 2 x ^ { 2 } + 12 x - 3\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. State the greatest value of \(\mathrm { f } ( x )\).
    The function g is defined by \(\mathrm { g } ( x ) = 2 x + 5\) for \(x \in \mathbb { R }\).
  3. Find the values of \(x\) for which \(\operatorname { gf } ( x ) + 1 = 0\).
CAIE P1 2019 June Q6
Standard +0.3
6
  1. Prove the identity \(\left( \frac { 1 } { \cos x } - \tan x \right) ^ { 2 } \equiv \frac { 1 - \sin x } { 1 + \sin x }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \cos 2 x } - \tan 2 x \right) ^ { 2 } = \frac { 1 } { 3 }\) for \(0 \leqslant x \leqslant \pi\).