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CAIE P3 2015 June Q4
6 marks Standard +0.3
4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 + \mathrm { e } ^ { 3 x } }\) has one stationary point. Find the exact values of the coordinates of this point.
CAIE P3 2015 June Q5
8 marks Standard +0.8
5 The parametric equations of a curve are $$x = a \cos ^ { 4 } t , \quad y = a \sin ^ { 4 } t$$ where \(a\) is a positive constant.
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Show that the equation of the tangent to the curve at the point with parameter \(t\) is $$x \sin ^ { 2 } t + y \cos ^ { 2 } t = a \sin ^ { 2 } t \cos ^ { 2 } t$$
  3. Hence show that if the tangent meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\), then $$O P + O Q = a$$ where \(O\) is the origin.
CAIE P3 2015 June Q6
9 marks Standard +0.3
6 It is given that \(\int _ { 0 } ^ { a } x \cos x \mathrm {~d} x = 0.5\), where \(0 < a < \frac { 1 } { 2 } \pi\).
  1. Show that \(a\) satisfies the equation \(\sin a = \frac { 1.5 - \cos a } { a }\).
  2. Verify by calculation that \(a\) is greater than 1 .
  3. Use the iterative formula $$a _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 1.5 - \cos a _ { n } } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 4 decimal places, giving the result of each iteration to 6 decimal places.
CAIE P3 2015 June Q7
9 marks Standard +0.3
7 The number of micro-organisms in a population at time \(t\) is denoted by \(M\). At any time the variation in \(M\) is assumed to satisfy the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = k ( \sqrt { } M ) \cos ( 0.02 t )$$ where \(k\) is a constant and \(M\) is taken to be a continuous variable. It is given that when \(t = 0 , M = 100\).
  1. Solve the differential equation, obtaining a relation between \(M , k\) and \(t\).
  2. Given also that \(M = 196\) when \(t = 50\), find the value of \(k\).
  3. Obtain an expression for \(M\) in terms of \(t\) and find the least possible number of micro-organisms.
CAIE P3 2015 June Q8
9 marks Standard +0.3
8 The complex number 1 - i is denoted by \(u\).
  1. Showing your working and without using a calculator, express $$\frac { \mathrm { i } } { u }$$ in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(| z - u | = | z |\) and \(| z - \mathrm { i } | = 2\).
  3. Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).
CAIE P3 2015 June Q9
10 marks Standard +0.3
9 Two planes have equations \(x + 3 y - 2 z = 4\) and \(2 x + y + 3 z = 5\). The planes intersect in the straight line \(l\).
  1. Calculate the acute angle between the two planes.
  2. Find a vector equation for the line \(l\).
CAIE P3 2015 June Q10
10 marks Standard +0.3
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
5 marks Moderate -0.3
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
5 marks Moderate -0.8
2 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
CAIE P3 2016 June Q3
5 marks Standard +0.3
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
6 marks Moderate -0.3
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
6 marks Standard +0.3
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
7 marks Standard +0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
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.