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CAIE P3 2016 June Q9
11 marks Standard +0.3
9 With respect to the origin \(O\), the points \(A , B , C , D\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } , \quad \overrightarrow { O C } = 2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { O D } = - 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$
  1. Find the equation of the plane containing \(A , B\) and \(C\), giving your answer in the form \(a x + b y + c z = d\).
  2. The line through \(D\) parallel to \(O A\) meets the plane with equation \(x + 2 y - z = 7\) at the point \(P\). Find the position vector of \(P\) and show that the length of \(D P\) is \(2 \sqrt { } ( 14 )\).
CAIE P3 2016 June Q10
11 marks Standard +0.3
10
  1. Showing all your working and without the use of a calculator, find the square roots of the complex number \(7 - ( 6 \sqrt { } 2 ) \mathrm { i }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    1. On an Argand diagram, sketch the loci of points representing complex numbers \(w\) and \(z\) such that \(| w - 1 - 2 \mathrm { i } | = 1\) and \(\arg ( z - 1 ) = \frac { 3 } { 4 } \pi\).
    2. Calculate the least value of \(| w - z |\) for points on these loci.
CAIE P3 2016 June Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(4 ^ { 3 x - 1 } = 3 \left( 5 ^ { x } \right)\), giving your answer correct to 3 decimal places.
CAIE P3 2016 June Q2
4 marks Moderate -0.3
2 Expand \(\frac { 1 } { \sqrt { ( 1 - 2 x ) } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2016 June Q3
5 marks Standard +0.3
3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { 2 } \sin 2 x \mathrm {~d} x\).
CAIE P3 2016 June Q4
6 marks Standard +0.3
4 The curve with equation \(y = \frac { ( \ln x ) ^ { 2 } } { x }\) has two stationary points. Find the exact values of the coordinates of these points.
CAIE P3 2016 June Q5
8 marks Standard +0.3
5
  1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta \equiv 8 \sin ^ { 4 } \theta - 3\).
  2. Hence solve the equation $$\cos 4 \theta = 4 \cos 2 \theta + 3$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2016 June Q6
8 marks Standard +0.3
6 The variables \(x\) and \(\theta\) satisfy the differential equation $$( 3 + \cos 2 \theta ) \frac { \mathrm { d } x } { \mathrm {~d} \theta } = x \sin 2 \theta$$ and it is given that \(x = 3\) when \(\theta = \frac { 1 } { 4 } \pi\).
  1. Solve the differential equation and obtain an expression for \(x\) in terms of \(\theta\).
  2. State the least value taken by \(x\).
CAIE P3 2016 June Q7
10 marks Standard +0.3
7 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 7 x + 4 } { ( 2 x + 1 ) ( x + 2 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = 8 - \ln 3\).
CAIE P3 2016 June Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{9d3af34a-670f-425e-8156-0ad4d08fbdc0-3_499_552_258_792} The diagram shows the curve \(y = \operatorname { cosec } x\) for \(0 < x < \pi\) and part of the curve \(y = \mathrm { e } ^ { - x }\). When \(x = a\), the tangents to the curves are parallel.
  1. By differentiating \(\frac { 1 } { \sin x }\), show that if \(y = \operatorname { cosec } x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x \cot x\).
  2. By equating the gradients of the curves at \(x = a\), show that $$a = \tan ^ { - 1 } \left( \frac { \mathrm { e } ^ { a } } { \sin a } \right)$$
  3. Verify by calculation that \(a\) lies between 1 and 1.5.
  4. Use an iterative formula based on the equation in part (ii) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2016 June Q9
10 marks Standard +0.3
9 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\), \(\overrightarrow { O B } = 4 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }\). A fourth point \(D\) is such that the quadrilateral \(A B C D\) is a parallelogram.
  1. Find the position vector of \(D\) and verify that the parallelogram is a rhombus.
  2. The plane \(p\) is parallel to \(O A\) and the line \(B C\) lies in \(p\). Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2016 June Q10
10 marks Standard +0.3
10
  1. Showing all necessary working, solve the equation \(\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
    1. On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation \(| z | = | z - 4 - 3 \mathrm { i } |\).
    2. Find the complex number represented by the point on the locus where \(| z |\) is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.
CAIE P3 2016 June Q1
4 marks Standard +0.3
1 Solve the inequality \(2 | x - 2 | > | 3 x + 1 |\).
CAIE P3 2016 June Q2
5 marks Moderate -0.8
2 The variables \(x\) and \(y\) satisfy the relation \(3 ^ { y } = 4 ^ { 2 - x }\).
  1. By taking logarithms, show that the graph of \(y\) against \(x\) is a straight line. State the exact value of the gradient of this line.
  2. Calculate the exact \(x\)-coordinate of the point of intersection of this line with the line with equation \(y = 2 x\), simplifying your answer.
CAIE P3 2016 June Q3
6 marks Standard +0.3
3
  1. Express \(( \sqrt { } 5 ) \cos x + 2 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$( \sqrt { } 5 ) \cos \frac { 1 } { 2 } x + 2 \sin \frac { 1 } { 2 } x = 1.2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P3 2016 June Q4
8 marks Moderate -0.3
4 The parametric equations of a curve are $$x = t + \cos t , \quad y = \ln ( 1 + \sin t )$$ where \(- \frac { 1 } { 2 } \pi < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec t\).
  2. Hence find the \(x\)-coordinates of the points on the curve at which the gradient is equal to 3 . Give your answers correct to 3 significant figures.
CAIE P3 2016 June Q5
8 marks Moderate -0.3
5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 y } \tan ^ { 2 } x$$ for \(0 \leqslant x < \frac { 1 } { 2 } \pi\), and it is given that \(y = 0\) when \(x = 0\). Solve the differential equation and calculate the value of \(y\) when \(x = \frac { 1 } { 4 } \pi\).
CAIE P3 2016 June Q6
8 marks Standard +0.3
6 The curve with equation \(y = x ^ { 2 } \cos \frac { 1 } { 2 } x\) has a stationary point at \(x = p\) in the interval \(0 < x < \pi\).
  1. Show that \(p\) satisfies the equation \(\tan \frac { 1 } { 2 } p = \frac { 4 } { p }\).
  2. Verify by calculation that \(p\) lies between 2 and 2.5.
  3. Use the iterative formula \(p _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 4 } { p _ { n } } \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2016 June Q7
8 marks Standard +0.3
7 Let \(I = \int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \left( 1 + x ^ { 2 } \right) ^ { 3 } } \mathrm {~d} x\).
  1. Using the substitution \(u = 1 + x ^ { 2 }\), show that \(I = \int _ { 1 } ^ { 2 } \frac { ( u - 1 ) ^ { 2 } } { 2 u ^ { 3 } } \mathrm {~d} u\).
  2. Hence find the exact value of \(I\).
CAIE P3 2016 June Q8
9 marks Standard +0.3
8 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 3 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).
  1. Show that the line passing through \(A\) and \(B\) does not intersect \(l\).
  2. Show that the length of the perpendicular from \(A\) to \(l\) is \(\frac { 1 } { \sqrt { 2 } }\).
CAIE P3 2016 June Q10
10 marks Standard +0.3
10 Let \(\mathrm { f } ( x ) = \frac { 10 x - 2 x ^ { 2 } } { ( x + 3 ) ( x - 1 ) ^ { 2 } }\).
  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 2017 June Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x + 1 | < 3 | x - 2 |\).
CAIE P3 2017 June Q2
4 marks Moderate -0.3
2 Expand \(\frac { 1 } { \sqrt [ 3 ] { } ( 1 + 6 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2017 June Q3
6 marks Standard +0.3
3 It is given that \(x = \ln ( 1 - y ) - \ln y\), where \(0 < y < 1\).
  1. Show that \(y = \frac { \mathrm { e } ^ { - x } } { 1 + \mathrm { e } ^ { - x } }\).
  2. Hence show that \(\int _ { 0 } ^ { 1 } y \mathrm {~d} x = \ln \left( \frac { 2 \mathrm { e } } { \mathrm { e } + 1 } \right)\).
CAIE P3 2017 June Q4
8 marks Standard +0.3
4 The parametric equations of a curve are $$x = \ln \cos \theta , \quad y = 3 \theta - \tan \theta ,$$ where \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan \theta\).
  2. Find the exact \(y\)-coordinate of the point on the curve at which the gradient of the normal is equal to 1 . \includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-08_378_689_260_726} The diagram shows a semicircle with centre \(O\), radius \(r\) and diameter \(A B\). The point \(P\) on its circumference is such that the area of the minor segment on \(A P\) is equal to half the area of the minor segment on \(B P\). The angle \(A O P\) is \(x\) radians.