Questions — CAIE P1 (1228 questions)

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CAIE P1 2017 June Q6
7 marks Standard +0.8
6
[diagram]
The diagram shows the straight line \(x + y = 5\) intersecting the curve \(y = \frac { 4 } { x }\) at the points \(A ( 1,4 )\) and \(B ( 4,1 )\). Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2017 June Q7
8 marks Standard +0.3
7
  1. The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum to exceed 20000.
  2. A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric progression is formed by squaring each of the terms of the original progression. Find the sum to infinity of the new progression.
CAIE P1 2017 June Q8
8 marks Standard +0.3
8 Relative to an origin \(O\), the position vectors of three points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = 3 \mathbf { i } + p \mathbf { j } - 2 p \mathbf { k } , \quad \overrightarrow { O B } = 6 \mathbf { i } + ( p + 4 ) \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = ( p - 1 ) \mathbf { i } + 2 \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.
  1. In the case where \(p = 2\), use a scalar product to find angle \(A O B\).
  2. In the case where \(\overrightarrow { A B }\) is parallel to \(\overrightarrow { O C }\), find the values of \(p\) and \(q\).
CAIE P1 2017 June Q9
9 marks Moderate -0.3
9 The equation of a curve is \(y = 8 \sqrt { } x - 2 x\).
  1. Find the coordinates of the stationary point of the curve.
  2. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence, or otherwise, determine the nature of the stationary point.
  3. Find the values of \(x\) at which the line \(y = 6\) meets the curve.
  4. State the set of values of \(k\) for which the line \(y = k\) does not meet the curve.
CAIE P1 2017 June Q10
11 marks Standard +0.2
10 The function f is defined by \(\mathrm { f } ( x ) = 3 \tan \left( \frac { 1 } { 2 } x \right) - 2\), for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Solve the equation \(\mathrm { f } ( x ) + 4 = 0\), giving your answer correct to 1 decimal place.
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2018 June Q1
5 marks Moderate -0.8
1
  1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 5 }\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + a x + 2 x ^ { 2 } \right) ( 1 - 2 x ) ^ { 5 }\) is 12 , find the value of the constant \(a\).
CAIE P1 2018 June Q2
4 marks Moderate -0.5
2 A point is moving along the curve \(y = 2 x + \frac { 5 } { x }\) in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.02 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 1\).
CAIE P1 2018 June Q3
6 marks Moderate -0.3
3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }\). The point \(( 1,1 )\) lies on the curve. Find the coordinates of the point at which the curve intersects the \(x\)-axis.
CAIE P1 2018 June Q4
6 marks Standard +0.3
4
  1. Prove the identity \(( \sin \theta + \cos \theta ) ( 1 - \sin \theta \cos \theta ) \equiv \sin ^ { 3 } \theta + \cos ^ { 3 } \theta\).
  2. Hence solve the equation \(( \sin \theta + \cos \theta ) ( 1 - \sin \theta \cos \theta ) = 3 \cos ^ { 3 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2018 June Q5
7 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-08_558_785_258_680} The diagram shows a kite \(O A B C\) in which \(A C\) is the line of symmetry. The coordinates of \(A\) and \(C\) are \(( 0,4 )\) and \(( 8,0 )\) respectively and \(O\) is the origin.
  1. Find the equations of \(A C\) and \(O B\).
  2. Find, by calculation, the coordinates of \(B\).
CAIE P1 2018 June Q6
7 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-10_499_922_262_607} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circle and \(A T\) is a tangent to the circle. Angle \(A O B = \theta\) radians and \(O B T\) is a straight line.
  1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 3\) and \(\theta = 1.2\), find the perimeter of the shaded region.
CAIE P1 2018 June Q7
8 marks Moderate -0.8
7 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 1 \\ 3 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right)$$
  1. Find \(\overrightarrow { A C }\).
  2. The point \(M\) is the mid-point of \(A C\). Find the unit vector in the direction of \(\overrightarrow { O M }\).
  3. Evaluate \(\overrightarrow { A B } \cdot \overrightarrow { A C }\) and hence find angle \(B A C\).
CAIE P1 2018 June Q8
9 marks Moderate -0.3
8
  1. A geometric progression has a second term of 12 and a sum to infinity of 54 . Find the possible values of the first term of the progression.
  2. The \(n\)th term of a progression is \(p + q n\), where \(p\) and \(q\) are constants, and \(S _ { n }\) is the sum of the first \(n\) terms.
    1. Find an expression, in terms of \(p , q\) and \(n\), for \(S _ { n }\).
    2. Given that \(S _ { 4 } = 40\) and \(S _ { 6 } = 72\), find the values of \(p\) and \(q\).
CAIE P1 2018 June Q9
11 marks Moderate -0.3
9 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 1 } { 2 } x - 2 \\ & \mathrm {~g} : x \mapsto 4 + x - \frac { 1 } { 2 } x ^ { 2 } \end{aligned}$$
  1. Find the points of intersection of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\).
  2. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { g } ( x )\).
  3. Find an expression for \(\mathrm { fg } ( x )\) and deduce the range of fg .
    The function h is defined by \(\mathrm { h } : x \mapsto 4 + x - \frac { 1 } { 2 } x ^ { 2 }\) for \(x \geqslant k\).
  4. Find the smallest value of \(k\) for which h has an inverse.
CAIE P1 2018 June Q10
12 marks Standard +0.3
10 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 5 x\) passes through the origin.
  1. Show that the curve has no stationary points.
  2. Denoting the gradient of the curve by \(m\), find the stationary value of \(m\) and determine its nature.
  3. Showing all necessary working, find the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = 6\).
    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
5 marks Moderate -0.3
1 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 + \frac { x } { 2 } \right) ^ { 6 } + ( a + x ) ^ { 5 }\) is 330 . Find the value of the constant \(a\).
CAIE P1 2018 June Q2
5 marks Moderate -0.5
2 The equation of a curve is \(y = x ^ { 2 } - 6 x + k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis.
  2. Find the value of \(k\) for which the line \(y + 2 x = 7\) is a tangent to the curve.
CAIE P1 2018 June Q3
5 marks 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
6 marks 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
6 marks 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
7 marks 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
7 marks 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
8 marks 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
8 marks 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
12 marks 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.