Questions — CAIE P3 (1110 questions)

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CAIE P3 2022 March Q10
10 marks Standard +0.3
10 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) respectively. The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\).
  1. Find a vector equation for the line through \(A\) and \(B\).
  2. Find the acute angle between the directions of \(A B\) and \(l\), giving your answer in degrees.
  3. Show that the line through \(A\) and \(B\) does not intersect the line \(l\).
CAIE P3 2022 March Q11
11 marks Standard +0.8
11 \includegraphics[max width=\textwidth, alt={}, center]{7cdf4db7-7217-4ef1-becf-359a70cfeb62-16_556_698_274_712} The diagram shows the curve \(y = \sin x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 significant figures.
  2. Using the substitution \(u = \cos x\), find the area of the shaded region enclosed by the curve and the \(x\)-axis in the first quadrant, giving your answer in a simplified exact form.
    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 P3 2023 March Q1
3 marks Moderate -0.5
1 It is given that \(x = \ln ( 2 y - 3 ) - \ln ( y + 4 )\).
Express \(y\) in terms of \(x\).
CAIE P3 2023 March Q2
5 marks Standard +0.8
2
  1. On an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(- \frac { 1 } { 3 } \pi \leqslant \arg ( z - 1 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 3 } \pi\) and \(\operatorname { Re } z \leqslant 3\).
  2. Calculate the least value of \(\arg z\) for points in the region from (a). Give your answer in radians correct to 3 decimal places.
CAIE P3 2023 March Q3
5 marks Standard +0.3
3 The polynomial \(2 x ^ { 4 } + a x ^ { 3 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - x + 1\) the remainder is \(3 x + 2\). Find the values of \(a\) and \(b\).
CAIE P3 2023 March Q4
5 marks Standard +0.8
4 Solve the equation $$\frac { 5 z } { 1 + 2 \mathrm { i } } - z z ^ { * } + 30 + 10 \mathrm { i } = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2023 March Q5
6 marks Moderate -0.3
5 The parametric equations of a curve are $$x = t \mathrm { e } ^ { 2 t } , \quad y = t ^ { 2 } + t + 3$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 t }\).
  2. Hence show that the normal to the curve, where \(t = - 1\), passes through the point \(\left( 0,3 - \frac { 1 } { \mathrm { e } ^ { 4 } } \right)\).
CAIE P3 2023 March Q6
7 marks Standard +0.3
6
  1. Express \(5 \sin \theta + 12 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  2. Hence solve the equation \(5 \sin 2 x + 12 \cos 2 x = 6\) for \(0 \leqslant x \leqslant \pi\).
CAIE P3 2023 March Q7
9 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-10_627_611_255_767} The diagram shows a circle with centre \(O\) and radius \(r\). The angle of the minor sector \(A O B\) of the circle is \(x\) radians. The area of the major sector of the circle is 3 times the area of the shaded region.
  1. Show that \(x = \frac { 3 } { 4 } \sin x + \frac { 1 } { 2 } \pi\).
  2. Show by calculation that the root of the equation in (a) lies between 2 and 2.5.
  3. Use an iterative formula based on the equation in (a) to calculate this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 March Q8
9 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{8c26235b-c78c-40d8-9e8e-213dc1311186-12_437_686_274_719} The diagram shows the curve \(y = x ^ { 3 } \ln x\), for \(x > 0\), and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { 1 } { 2 }\). [5]
CAIE P3 2023 March Q9
7 marks Standard +0.3
9 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 3 y } \sin ^ { 2 } 2 x$$ It is given that \(y = 0\) when \(x = 0\).
Solve the differential equation and find the value of \(y\) when \(x = \frac { 1 } { 2 }\).
CAIE P3 2023 March Q10
9 marks Standard +0.8
10 With respect to the origin \(O\), the points \(A , B , C\) and \(D\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 1 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 2 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 5 \\ - 6 \\ 11 \end{array} \right)$$
  1. Find the obtuse angle between the vectors \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) The line \(l\) passes through the points \(A\) and \(B\).
  2. Find a vector equation for the line \(l\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
  3. Find the position vector of the point of intersection of the line \(l\) and the line passing through \(C\) and \(D\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
CAIE P3 2023 March Q11
10 marks Challenging +1.2
11 Let \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } + x + 11 } { \left( 4 + x ^ { 2 } \right) ( 1 + x ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
  2. Hence show that \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln 54 - \frac { 1 } { 8 } \pi\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
CAIE P3 2024 March Q1
3 marks Moderate -0.5
1 Find the quotient and remainder when \(x ^ { 4 } - 3 x ^ { 3 } + 9 x ^ { 2 } - 12 x + 27\) is divided by \(x ^ { 2 } + 5\).
CAIE P3 2024 March Q2
5 marks Moderate -0.3
2
  1. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 x - 5 ) \sqrt { 4 - x }\).
  2. State the set of values of \(x\) for which the expansion in part (a) is valid.
CAIE P3 2024 March Q3
6 marks Standard +0.3
3 It is given that \(z = - \sqrt { 3 } + \mathrm { i }\).
  1. Express \(z ^ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  2. The complex number \(\omega\) is such that \(z ^ { 2 } \omega\) is real and \(\left| \frac { z ^ { 2 } } { \omega } \right| = 12\). Find the two possible values of \(\omega\), giving your answers in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\) and \(- \pi < \alpha \leqslant \pi\).
CAIE P3 2024 March Q4
4 marks Moderate -0.5
4 The positive numbers \(p\) and \(q\) are such that $$\ln \left( \frac { p } { q } \right) = a \text { and } \ln \left( q ^ { 2 } p \right) = b .$$ Express \(\ln \left( p ^ { 7 } q \right)\) in terms of \(a\) and \(b\).
CAIE P3 2024 March Q5
6 marks Standard +0.8
5
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - 4 - 2 i | \leqslant 3\) and \(| z | \geqslant | 10 - z |\).
  2. Find the greatest value of \(\arg z\) for points in this region.
CAIE P3 2024 March Q6
7 marks Standard +0.3
6 The equation of a curve is \(2 y ^ { 2 } + 3 x y + x = x ^ { 2 }\).
  1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 2 \mathrm { x } - 3 \mathrm { y } - 1 } { 4 \mathrm { y } + 3 \mathrm { x } }\).
  2. Hence show that the curve does not have a tangent that is parallel to the \(x\)-axis.
CAIE P3 2024 March Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-10_620_517_260_774} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { 2 \mathrm { x } } - 5 \mathrm { x }\) and its minimum point \(M\), where \(x = \alpha\).
  1. Show that \(\alpha\) satisfies the equation \(\alpha = \frac { 1 } { 2 } \ln \left( \frac { 5 } { 1 + 2 \alpha } \right)\).
  2. Verify by calculation that \(\alpha\) lies between 0.4 and 0.5.
  3. Use an iterative formula based on the equation in part (a) to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2024 March Q8
9 marks Challenging +1.2
8
  1. Express \(3 \sin x + 2 \sqrt { 2 } \cos \left( x + \frac { 1 } { 4 } \pi \right)\) in the form \(\mathrm { R } \sin ( \mathrm { x } + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
  2. Hence solve the equation $$6 \sin \frac { 1 } { 2 } \theta + 4 \sqrt { 2 } \cos \left( \frac { 1 } { 2 } \theta + \frac { 1 } { 4 } \pi \right) = 3$$ for \(- 4 \pi < \theta < 4 \pi\).
CAIE P3 2024 March Q9
8 marks Standard +0.3
9 Relative to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { \mathrm { OA } } = 5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { \mathrm { OB } } = 8 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } \quad \text { and } \quad \overrightarrow { \mathrm { OC } } = 3 \mathbf { i } + 4 \mathbf { j } - 7 \mathbf { k }$$
  1. Show that \(O A B C\) is a rectangle. \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_67_1573_557_324} \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_68_1575_648_322} \includegraphics[max width=\textwidth, alt={}]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_70_1573_737_324} ....................................................................................................................................... .........................................................................................................................................
  2. Use a scalar product to find the acute angle between the diagonals of \(O A B C\).
CAIE P3 2024 March Q10
10 marks Challenging +1.2
10 Let \(f ( x ) = \frac { 36 a ^ { 2 } } { ( 2 a + x ) ( 2 a - x ) ( 5 a - 2 x ) }\), where \(a\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find the exact value of \(\int _ { - a } ^ { a } f ( x ) d x\), giving your answer in the form plnq+rlns where \(p\) and \(r\) are integers and \(q\) and \(s\) are prime numbers.
CAIE P3 2024 March Q11
9 marks Standard +0.3
11 The variables \(y\) and \(\theta\) satisfy the differential equation $$( 1 + y ) ( 1 + \cos 2 \theta ) \frac { d y } { d \theta } = e ^ { 3 y }$$ It is given that \(y = 0\) when \(\theta = \frac { 1 } { 4 } \pi\).
Solve the differential equation and find the exact value of \(\tan \theta\) when \(y = 1\).
If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2020 November Q1
4 marks Standard +0.3
1 Solve the inequality \(2 - 5 x > 2 | x - 3 |\).