Questions P3 (1203 questions)

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CAIE P3 2023 June Q4
4 The parametric equations of a curve are $$x = \frac { \cos \theta } { 2 - \sin \theta } , \quad y = \theta + 2 \cos \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 - \sin \theta ) ^ { 2 }\).
CAIE P3 2023 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{72042f09-3495-42e9-bee9-96ec5ac0bf0c-06_352_643_274_744} The diagram shows the part of the curve \(y = x ^ { 2 } \cos 3 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\), and its maximum point \(M\), where \(x = a\).
  1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { 2 } { 3 a } \right)\).
  2. Use an iterative formula based on the equation in (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2023 June Q6
6
  1. Express \(3 \cos x + 2 \cos \left( x - 60 ^ { \circ } \right)\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$3 \cos 2 \theta + 2 \cos \left( 2 \theta - 60 ^ { \circ } \right) = 2.5$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2023 June Q7
7
  1. Use the substitution \(u = \cos x\) to show that $$\int _ { 0 } ^ { \pi } \sin 2 x \mathrm { e } ^ { 2 \cos x } \mathrm {~d} x = \int _ { - 1 } ^ { 1 } 2 u \mathrm { e } ^ { 2 u } \mathrm {~d} u$$
  2. Hence find the exact value of \(\int _ { 0 } ^ { \pi } \sin 2 x \mathrm { e } ^ { 2 \cos x } \mathrm {~d} x\).
CAIE P3 2023 June Q8
8 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } + 4 } { x ( y + 4 ) }$$ for \(x > 0\). It is given that \(x = 4\) when \(y = 2 \sqrt { 3 }\).
Solve the differential equation to obtain the value of \(x\) when \(y = 2\).
CAIE P3 2023 June Q9
4 marks
9 The lines \(l\) and \(m\) have equations $$\begin{aligned} l : & \mathbf { r } = a \mathbf { i } + 3 \mathbf { j } + b \mathbf { k } + \lambda ( c \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } )
m : & \mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ Relative to the origin \(O\), the position vector of the point \(P\) is \(4 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\).
  1. Given that \(l\) is perpendicular to \(m\) and that \(P\) lies on \(l\), find the values of the constants \(a , b\) and \(c\). [4]
  2. The perpendicular from \(P\) meets line \(m\) at \(Q\). The point \(R\) lies on \(P Q\) extended, with \(P Q : Q R = 2 : 3\). Find the position vector of \(R\).
CAIE P3 2023 June Q10
10 Let \(\mathrm { f } ( x ) = \frac { 21 - 8 x - 2 x ^ { 2 } } { ( 1 + 2 x ) ( 3 - x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2023 June Q11
11 The complex number \(z\) is defined by \(z = \frac { 5 a - 2 \mathrm { i } } { 3 + a \mathrm { i } }\), where \(a\) is an integer. It is given that \(\arg z = - \frac { 1 } { 4 } \pi\).
  1. Find the value of \(a\) and hence express \(z\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Express \(z ^ { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the simplified exact values of \(r\) and \(\theta\).
    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 2024 June Q1
1 Expand \(( 3 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2024 June Q2
2 Solve the equation \(\ln ( x - 5 ) = 7 - \ln x\). Give your answer correct to 2 decimal places.
CAIE P3 2024 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{37f00894-e6b1-4961-bd3c-4852e43173d0-04_597_921_260_573} The variables \(x\) and \(y\) satisfy the equation \(\mathrm { a } ^ { \mathrm { y } } = \mathrm { bx }\), where \(a\) and \(b\) are constants. The graph of \(y\) against \(\ln x\) is a straight line passing through the points ( \(0.336,1.00\) ) and ( \(1.31,1.50\) ), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to the nearest integer.
CAIE P3 2024 June Q4
4 The complex number \(u\) is given by \(u = - 1 - \mathrm { i } \sqrt { 3 }\).
  1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
    The complex number \(v\) is given by \(v = 5 \left( \cos \frac { 1 } { 6 } \pi + \mathrm { i } \sin \frac { 1 } { 6 } \pi \right)\).
  2. Express the complex number \(\frac { \mathrm { v } } { \mathrm { u } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
CAIE P3 2024 June Q5
5 The equation of a curve is \(y = \frac { e ^ { \sin x } } { \cos ^ { 2 } x }\) for \(0 \leqslant x \leqslant 2 \pi\).
Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) and hence find the \(x\)-coordinates of the stationary points of the curve.
CAIE P3 2024 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } \frac { 1 } { 2 } x = \mathrm { e } ^ { x } - 3\) has exactly one root, denoted by \(\alpha\), in the interval \(0 < x < \pi\).
  2. Verify by calculation that \(\alpha\) lies between 1 and 2 .
  3. Show that if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula $$x _ { n + 1 } = \ln \left( \operatorname { cosec } \frac { 1 } { 2 } x _ { n } + 3 \right)$$ converges, then it converges to \(\alpha\).
  4. Use this iterative formula with an initial value of 1.4 to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
  5. State the minimum number of calculated iterations needed with this initial value to determine \(\alpha\) correct to 2 decimal places.
CAIE P3 2024 June Q7
7
  1. On a single Argand diagram sketch the loci given by the equations \(| z - 3 + 2 i | = 2\) and \(| w - 3 + 2 \mathrm { i } | = | w + 3 - 4 \mathrm { i } |\) where z and \(w\) are complex numbers.
  2. Hence find the least value of \(| \mathbf { z } - \mathbf { w } |\) for points on these loci. Give your answer in an exact form.
CAIE P3 2024 June Q8
8 Use the substitution \(\mathrm { u } = 1 - \sin \mathrm { x }\) to find the exact value of $$\int _ { \pi } ^ { \frac { 3 } { 2 } \pi } \frac { \sin 2 x } { \sqrt { 1 - \sin x } } d x$$ Give your answer in the form \(\mathrm { a } + \mathrm { b } \sqrt { 2 }\) where \(a\) and \(b\) are rational numbers to be determined.
CAIE P3 2024 June Q9
9 The equations of two straight lines \(l _ { 1 }\) and \(l _ { 2 }\) are $$l _ { 1 } : \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \quad \mathbf { r } = - \mathbf { i } - \mathbf { j } - \mathbf { k } + \mu ( 3 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) ,$$ where \(a\) is a constant.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
  1. Show that \(a = 4\).
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) also intersect.
  2. Find the position vector of the point of intersection.
    The point \(A\) has position vector \(- 5 \mathbf { i } + \mathbf { j } - 9 \mathbf { k }\).
  3. Show that \(A\) lies on \(l _ { 1 }\).
    The point \(B\) is the image of \(A\) after a reflection in the line \(l _ { 2 }\).
  4. Find the position vector of \(B\).
CAIE P3 2024 June Q10
10
  1. Given that \(2 x = \tan y\), show that \(\frac { d y } { d x } = \frac { 2 } { 1 + 4 x ^ { 2 } }\).
  2. Hence find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { \frac { \sqrt { 3 } } { 2 } } x \tan ^ { - 1 } ( 2 x ) \mathrm { d } x\).
CAIE P3 2024 June Q11
11 In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time \(t\) after the first plant is infected there are \(x\) infected plants. The rate of change of \(x\) is proportional to the product of the number of plants infected and the number of plants that are not yet infected. The variables \(x\) and \(t\) are treated as continuous, and it is given that \(\frac { \mathrm { dx } } { \mathrm { dt } } = 0.2\) and \(x = 1\) when \(t = 0\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$1495 \frac { \mathrm { dx } } { \mathrm { dt } } = x ( 300 - x )$$
  2. Using partial fractions, solve the differential equation and obtain an expression for \(t\) in terms of a single logarithm involving \(x\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2024 June Q1
1
  1. Sketch the graph of \(\mathrm { y } = | \mathrm { x } - 2 \mathrm { a } |\), where \(a\) is a positive constant.
  2. Solve the inequality \(2 \mathrm { x } - 3 \mathrm { a } < | \mathrm { x } - 2 \mathrm { a } |\).
CAIE P3 2024 June Q2
2 Express \(\frac { 6 x ^ { 2 } - 9 x - 16 } { 2 x ^ { 2 } - 5 x - 12 }\) in partial fractions.
CAIE P3 2024 June Q3
3 The variables \(x\) and \(y\) satisfy the equation \(\mathrm { a } ^ { 2 \mathrm { y } - 1 } = \mathrm { b } ^ { \mathrm { x } - \mathrm { y } }\), where \(a\) and \(b\) are constants.
  1. Show that the graph of \(y\) against \(x\) is a straight line.
  2. Given that \(\mathrm { a } = \mathrm { b } ^ { 3 }\), state the equation of the straight line in the form \(\mathrm { y } = \mathrm { px } + \mathrm { q }\), where \(p\) and \(q\) are rational numbers in their simplest form.
CAIE P3 2024 June Q4
4 The equation of a curve is \(\mathrm { ye } ^ { 2 \mathrm { x } } + \mathrm { y } ^ { 2 } \mathrm { e } ^ { \mathrm { x } } = 6\).
Find the gradient of the curve at the point where \(y = 1\).
CAIE P3 2024 June Q5
5
  1. It is given that the equation \(\mathrm { e } ^ { 2 x } = 5 + \cos 3 x\) has only one root.
    Show by calculation that this root lies in the interval \(0.7 < x < 0.8\).
  2. Show that if a sequence of values in the interval \(0.7 < x < 0.8\) given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 5 + \cos 3 x _ { n } \right)$$ converges then it converges to the root of the equation in part (a).
  3. Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2024 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-08_351_1031_264_516} The diagram shows the curve \(\mathrm { y } = \mathrm { xe } ^ { - \mathrm { ax } }\), where \(a\) is a positive constant, and its maximum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 2 } { a } } x e ^ { - a x } d x\).