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Pre-U Pre-U 9794/2 2014 June Q4
7 marks Standard +0.3
4 The points \(A , B , C\) and \(D\) have coordinates \(( 2 , - 1,0 ) , ( 3,2,5 ) , ( 4,2,3 )\) and \(( - 1 , a , b )\) respectively, where \(a\) and \(b\) are constants.
  1. Find the angle \(A B C\).
  2. Given that the lines \(A B\) and \(C D\) are parallel, find the values of \(a\) and \(b\).
Pre-U Pre-U 9794/2 2014 June Q5
3 marks Easy -1.3
5 An arithmetic progression has first term 5 and common difference 7.
  1. Find the value of the 10th term.
  2. Find the sum of the first 15 terms. The terms of the progression are given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\).
  3. Evaluate \(\sum _ { n = 1 } ^ { 15 } \left( 2 x _ { n } + 1 \right)\).
Pre-U Pre-U 9794/2 2014 June Q6
5 marks Moderate -0.8
6 Given that the angle \(\theta\) is acute and \(\cos \theta = \frac { 3 } { 4 }\) find, without using a calculator, the exact value of \(\sin 2 \theta\) and of \(\cot \theta\).
Pre-U Pre-U 9794/2 2014 June Q7
2 marks Moderate -0.8
7
  1. Express \(z ^ { 4 } + 3 z ^ { 2 } - 4\) in the form \(\left( z ^ { 2 } + a \right) \left( z ^ { 2 } + b \right)\) where \(a\) and \(b\) are real constants to be found.
  2. Hence draw an Argand diagram showing the points that represent the roots of the equation \(z ^ { 4 } + 3 z ^ { 2 } - 4 = 0\).
Pre-U Pre-U 9794/2 2014 June Q8
6 marks Standard +0.3
8 Show that the graph of \(y = x ^ { 2 } - \ln x\) has only one stationary point and give the coordinates of that point in exact form.
Pre-U Pre-U 9794/2 2014 June Q9
7 marks Challenging +1.2
9 A new lake is stocked with fish. Let \(P _ { t }\) be the population of fish in the lake after \(t\) years. Two models using recurrence relations are proposed for \(P _ { t }\), with \(P _ { 0 } = 550\). $$\begin{aligned} & \text { Model } 1 : P _ { t } = 2 P _ { t - 1 } \mathrm { e } ^ { - 0.001 P _ { t - 1 } } \\ & \text { Model } 2 : P _ { t } = \frac { 1 } { 2 } P _ { t - 1 } \left( 7 - \frac { 1 } { 160 } P _ { t - 1 } \right) \end{aligned}$$
  1. Evaluate the population predicted by each model when \(t = 3\).
  2. Identify, with evidence, which one of the models predicts a stable population in the long term.
  3. Describe the long term behaviour of the population for the other model.
Pre-U Pre-U 9794/2 2014 June Q10
11 marks Standard +0.3
10 Let \(\mathrm { f } ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 10 x ^ { 2 } + 28 x - 15\).
  1. Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\).
  2. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(x - 5\).
  3. Factorise \(\mathrm { f } ( x )\) fully and hence sketch the graph of \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9794/2 2014 June Q11
12 marks Challenging +1.2
11 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } + 4 x - 7 = 0\) has a single root \(\alpha\), close to 1.9 , which can be found using an iteration of the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\). Three possible functions that can be used for such an iteration are $$\mathrm { F } _ { 1 } ( x ) = \frac { 7 } { 4 } + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 } , \quad \mathrm {~F} _ { 2 } ( x ) = \sqrt [ 3 ] { 2 x ^ { 2 } - 4 x + 7 } , \quad \mathrm {~F} _ { 3 } ( x ) = \frac { 7 - 4 x } { x ^ { 2 } - 2 x }$$
  1. Differentiate each of these functions with respect to \(x\).
  2. Without performing any iterations, and using \(x = 1.9\), show that an iterative process based on only two of the given functions will converge. Determine which one will do so more rapidly. The sequence of errors, \(e _ { n }\), is such that \(e _ { n + 1 } \approx \mathrm {~F} ^ { \prime } ( \alpha ) e _ { n }\).
  3. Using the iteration from part (ii) with the most rapid convergence, estimate the number of iterations required to reduce the magnitude of the error from \(\left| e _ { 1 } \right|\) in the first term to less than \(10 ^ { - 10 } \left| e _ { 1 } \right|\).
Pre-U Pre-U 9794/2 2014 June Q12
9 marks Standard +0.8
12 A curve \(C\) is defined parametrically by $$x = \cos t ( 1 - 2 \sin t ) , \quad y = \sin t ( 1 - 3 \sin t ) , \quad 0 \leqslant t < 2 \pi$$
  1. Show that \(C\) intersects the \(y\)-axis at exactly three points, and state the values of \(t\) and \(y\) at these points.
  2. Find the range of values of \(t\) for which \(C\) lies above the \(x\)-axis.
Pre-U Pre-U 9795/1 2014 June Q1
4 marks Standard +0.8
1 The series \(S\) is given by \(S = \sum _ { r = 0 } ^ { N } ( N + r ) ^ { 2 }\).
  1. Write out the first three terms and the last three terms of the series for \(S\).
  2. Use the standard result \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } N ( N + 1 ) ( a N + 1 )\) for some positive integer \(a\) to be determined.
Pre-U Pre-U 9795/1 2014 June Q2
8 marks Standard +0.3
2
  1. Show that there is a value of \(t\) for which \(\mathbf { A B }\) is an integer multiple of the \(3 \times 3\) identity matrix \(\mathbf { I }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 2 & 1 \\ t & 1 & - t \\ 3 & 2 & 1 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { c r r } t - 2 & 0 & 5 \\ 12 & - 2 & - 6 \\ 3 t & 4 & 7 \end{array} \right) .$$
  2. Express the system of equations $$\begin{aligned} - 5 x + 5 z & = 8 \\ 12 x - 2 y - 6 z & = 12 \\ - 9 x + 4 y + 7 z & = 22 \end{aligned}$$ in the form \(\mathbf { C x } = \mathbf { u }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, and \(\mathbf { x }\) and \(\mathbf { u }\) are suitable column vectors.
  3. Use the result of part (i) to solve the system of equations given in part (ii).
Pre-U Pre-U 9795/1 2014 June Q3
5 marks Standard +0.3
3
  1. On a single copy of an Argand diagram, sketch the loci defined by $$| z + 2 | = 3 \quad \text { and } \quad \arg ( z - \mathrm { i } ) = - \frac { 1 } { 4 } \pi$$
  2. State the complex number \(z\) which corresponds to the point of intersection of these two loci.
Pre-U Pre-U 9795/1 2014 June Q4
5 marks Challenging +1.2
4 Let \(I _ { n } = \int _ { 0 } ^ { 4 } x ^ { n } \sqrt { 2 x + 1 } \mathrm {~d} x\) for \(n \geqslant 0\). Show that, for \(n \geqslant 1\), $$( 2 n + 3 ) I _ { n } = 27 \times 4 ^ { n } - n I _ { n - 1 }$$
Pre-U Pre-U 9795/1 2014 June Q5
6 marks Standard +0.3
5 The curve \(C\) has equation \(y = \frac { 12 ( x + 1 ) } { ( x - 2 ) ^ { 2 } }\).
  1. Determine the coordinates of any stationary points of \(C\).
  2. Sketch \(C\).
Pre-U Pre-U 9795/1 2014 June Q6
8 marks Standard +0.3
6 Solve the first-order differential equation \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 \ln x\) given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9795/1 2014 June Q7
6 marks Standard +0.3
7 Let \(\mathrm { f } ( n ) = 11 ^ { 2 n - 1 } + 7 \times 4 ^ { n }\). Prove by induction that \(\mathrm { f } ( n )\) is divisible by 13 for all positive integers \(n\).
Pre-U Pre-U 9795/1 2014 June Q8
6 marks Standard +0.3
8
  1. Show that the line \(l\) with vector equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 5 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 5 \\ - 2 \\ 3 \end{array} \right)\) lies in the plane \(\Pi\) with cartesian equation \(x + 4 y + z + 11 = 0\).
  2. The plane \(\Pi\) is horizontal, and the point \(P ( 1,2 , k )\) is above it. Given that the point in \(\Pi\) which is directly beneath \(P\) is on the line \(l\), determine the value of \(k\).
Pre-U Pre-U 9795/1 2014 June Q9
2 marks Hard +2.3
9
  1. Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2).
  2. Prove that any group in which every non-identity element is self-inverse is abelian.
  3. Simon believes that if \(x\) and \(y\) are two distinct self-inverse elements of a group, then the element \(x y\) is also self-inverse. By considering the group of the six permutations of \(\left( \begin{array} { l l } 1 & 2 \end{array} \right)\), produce a counter-example to prove him wrong.
  4. A group \(G\) has order \(4 n + 2\), for some positive integer \(n\), and \(i\) is the identity element of \(G\). Let \(x\) and \(y\) be two distinct self-inverse elements of \(G\). By considering the set \(H = \{ i , x , y , x y \}\), prove by contradiction that \(G\) cannot contain all self-inverse elements.
Pre-U Pre-U 9795/1 2014 June Q10
13 marks Challenging +1.8
10
  1. Use de Moivre's theorem to show that \(2 \cos 6 \theta \equiv 64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 2\).
  2. Hence find, in exact trigonometric form, the six roots of the equation $$x ^ { 6 } - 6 x ^ { 4 } + 9 x ^ { 2 } - 3 = 0$$
  3. By considering the product of these six roots, determine the exact value of $$\cos \left( \frac { 1 } { 18 } \pi \right) \cos \left( \frac { 5 } { 18 } \pi \right) \cos \left( \frac { 7 } { 18 } \pi \right) .$$
Pre-U Pre-U 9795/1 2014 June Q11
9 marks Challenging +1.2
11 A curve has polar equation \(r = \mathrm { e } ^ { \sin \theta }\) for \(- \pi < \theta \leqslant \pi\).
  1. State the polar coordinates of the point where the curve crosses the initial line.
  2. State also the polar coordinates of the points where \(r\) takes its least and greatest values.
  3. Sketch the curve.
  4. By deriving a suitable Maclaurin series up to and including the term in \(\theta ^ { 2 }\), find an approximation, to 3 decimal places, for the area of the region enclosed by the curve, the initial line and the line \(\theta = 0.3\).
Pre-U Pre-U 9795/1 2014 June Q12
10 marks Challenging +1.8
12
  1. (a) Show that \(\tanh x = \frac { \mathrm { e } ^ { 2 x } - 1 } { \mathrm { e } ^ { 2 x } + 1 }\).
    (b) Hence, or otherwise, show that, if \(\tanh x = \frac { 1 } { k }\) for \(k > 1\), then \(x = \frac { 1 } { 2 } \ln \left( \frac { k + 1 } { k - 1 } \right)\) and find an expression in terms of \(k\) for \(\sinh 2 x\).
  2. A curve has equation \(y = \frac { 1 } { 2 } \ln ( \tanh x )\) for \(\alpha \leqslant x \leqslant \beta\), where \(\tanh \alpha = \frac { 1 } { 3 }\) and \(\tanh \beta = \frac { 1 } { 2 }\). Find, in its simplest exact form, the arc length of this curve.
Pre-U Pre-U 9795/1 2014 June Q13
8 marks Challenging +1.8
13 The complex number \(w\) has modulus 1. It is given that $$w ^ { 2 } - \frac { 2 } { w } + k \mathrm { i } = 0$$ where \(k\) is a positive real constant.
  1. Show that \(k = ( 3 - \sqrt { 3 } ) \sqrt { \frac { 1 } { 2 } \sqrt { 3 } }\).
  2. Prove that at least one of the remaining two roots of the equation \(z ^ { 2 } - \frac { 2 } { z } + k i = 0\) has modulus greater than 1 .
Pre-U Pre-U 9794/1 2014 June Q4
2 marks Moderate -0.8
4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-2_949_1127_1041_507} Draw the graphs of
  1. \(\mathrm { f } ( x + 2 ) + 1\),
  2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2014 June Q6
7 marks Moderate -0.3
6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-3_648_684_342_731}
  1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
  3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.
Pre-U Pre-U 9794/1 2014 June Q4
2 marks Moderate -0.3
4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-2_949_1127_1041_507} Draw the graphs of
  1. \(\mathrm { f } ( x + 2 ) + 1\),
  2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).