Questions P3 (1203 questions)

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CAIE P3 2015 June Q10
10 Let \(\mathrm { f } ( x ) = \frac { 11 x + 7 } { ( 2 x - 1 ) ( x + 2 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 4 } + \ln \left( \frac { 9 } { 4 } \right)\).
CAIE P3 2016 June Q1
1
  1. Solve the equation \(2 | x - 1 | = 3 | x |\).
  2. Hence solve the equation \(2 \left| 5 ^ { x } - 1 \right| = 3 \left| 5 ^ { x } \right|\), giving your answer correct to 3 significant figures.
CAIE P3 2016 June Q2
2 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
CAIE P3 2016 June Q3
3 By expressing the equation \(\operatorname { cosec } \theta = 3 \sin \theta + \cot \theta\) in terms of \(\cos \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2016 June Q4
4 The variables \(x\) and \(y\) satisfy the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = y \left( 1 - 2 x ^ { 2 } \right)$$ and it is given that \(y = 2\) when \(x = 1\). Solve the differential equation and obtain an expression for \(y\) in terms of \(x\) in a form not involving logarithms.
CAIE P3 2016 June Q5
5 The curve with equation \(y = \sin x \cos 2 x\) has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.
CAIE P3 2016 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation $$5 \mathrm { e } ^ { - x } = \sqrt { } x$$ has one root.
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 25 } { x _ { n } } \right)$$ converges, then it converges to the root of the equation in part (i).
  3. Use this iterative formula, with initial value \(x _ { 1 } = 1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2016 June Q7
7 The equation of a curve is \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 3 } = 3\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - 2 x y } { x ^ { 2 } - y ^ { 2 } }\).
  2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.
CAIE P3 2016 June Q8
8 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 12 } { ( x + 1 ) ( x - 3 ) ^ { 2 } }\).
  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 2016 June Q9
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
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
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
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
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
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
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
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
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
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
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
  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
1 Solve the inequality \(2 | x - 2 | > | 3 x + 1 |\).
CAIE P3 2016 June Q2
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
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
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.