Questions Pre-U 9795/1 (179 questions)

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Pre-U Pre-U 9795/1 2013 June Q1
4 marks Standard +0.3
1 By completing the square, or otherwise, find the exact value of \(\int _ { 2 } ^ { 6 } \frac { 1 } { x ^ { 2 } - 6 x + 12 } \mathrm {~d} x\).
Pre-U Pre-U 9795/1 2013 June Q2
4 marks Challenging +1.2
2 Use the standard Maclaurin series expansions given in the List of Formulae MF20 to show that $$\frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \equiv \tanh ^ { - 1 } x \text { for } - 1 < x < 1$$
Pre-U Pre-U 9795/1 2013 June Q3
2 marks Standard +0.3
3 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } - 4 }\).
  1. Show that the gradient of \(C\) is always negative.
  2. Sketch \(C\), showing all significant features.
Pre-U Pre-U 9795/1 2013 June Q4
2 marks Standard +0.8
4
  1. Find a vector which is perpendicular to both of the vectors $$\mathbf { d } _ { 1 } = \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \mathbf { d } _ { 2 } = 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } .$$
  2. Determine the shortest distance between the skew lines with equations $$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \mathbf { i } + \mathbf { j } + 10 \mathbf { k } + \mu ( 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) .$$
Pre-U Pre-U 9795/1 2013 June Q5
5 marks Standard +0.8
5 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).
  1. Prove the result \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta\).
  2. Use this result to express \(\sin ^ { 5 } \theta\) in the form \(A \sin 5 \theta + B \sin 3 \theta + C \sin \theta\), for constants \(A , B\) and \(C\) to be determined.
Pre-U Pre-U 9795/1 2013 June Q6
8 marks Standard +0.8
6 The curve \(P\) has polar equation \(r = \frac { 1 } { 1 - \sin \theta }\) for \(0 \leqslant \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi\).
  1. Determine, in the form \(y = \mathrm { f } ( x )\), the cartesian equation of \(P\).
  2. Sketch \(P\).
  3. Evaluate \(\int _ { \pi } ^ { 2 \pi } \frac { 1 } { ( 1 - \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).
Pre-U Pre-U 9795/1 2013 June Q7
7 marks Standard +0.8
7
  1. Express \(x ^ { 3 } + y ^ { 3 }\) in terms of \(( x + y )\) and \(x y\).
  2. The equation \(t ^ { 2 } - 3 t + \frac { 8 } { 9 } = 0\) has roots \(\alpha\) and \(\beta\).
    1. Determine the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\).
    2. Hence express 19 as the sum of the cubes of two positive rational numbers.
Pre-U Pre-U 9795/1 2013 June Q8
8 marks Challenging +1.8
8 Let \(G = \left\{ g _ { 1 } , g _ { 2 } , g _ { 3 } , \ldots , g _ { n } \right\}\) be a finite abelian group of order \(n\) under a multiplicative binary operation, where \(g _ { 1 } = e\) is the identity of \(G\).
  1. Let \(x \in G\). Justify the following statements:
    1. \(x g _ { i } = x g _ { j } \Leftrightarrow g _ { i } = g _ { j }\);
    2. \(\left\{ x g _ { 1 } , x g _ { 2 } , x g _ { 3 } , \ldots , x g _ { n } \right\} = G\).
    3. By considering the product of all \(G\) 's elements, and using the result of part (i)(b), prove that \(x ^ { n } = e\) for each \(x \in G\).
    4. Explain why
      (a) this does not imply that all elements of \(G\) have order \(n\),
      (b) this argument cannot be used to justify the same result for non-abelian groups.
Pre-U Pre-U 9795/1 2013 June Q9
8 marks Challenging +1.8
9 The plane transformation \(T\) is the composition (in this order) of
  • a reflection in the line \(y = x \tan \frac { 1 } { 8 } \pi\); followed by
  • a shear parallel to the \(y\)-axis, mapping \(( 1,0 )\) to \(( 1,2 )\); followed by
  • a clockwise rotation through \(\frac { 1 } { 4 } \pi\) radians about the origin; followed by
  • a shear parallel to the \(x\)-axis, mapping \(( 0,1 )\) to \(( - 2,1 )\).
Determine the matrix \(\mathbf { M }\) which represents \(T\), and hence give a full geometrical description of \(T\) as a single plane transformation.
Pre-U Pre-U 9795/1 2013 June Q10
18 marks Challenging +1.3
10
  1. Given that \(y = k x \cos x\) is a particular integral for the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 4 \sin x$$ determine the value of \(k\) and find the general solution of this differential equation.
  2. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 5 x - 19$$
    1. Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 1\), find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 1\).
    2. Deduce the Taylor series expansion for \(y\) in ascending powers of \(( x - 1 )\), up to and including the term in \(( x - 1 ) ^ { 3 }\), and use this series to find an approximation correct to 3 decimal places for the value of \(y\) when \(x = 1.1\).
Pre-U Pre-U 9795/1 2013 June Q11
13 marks Standard +0.3
11
  1. Determine \(p\) and \(q\) given that \(( p + \mathrm { i } q ) ^ { 2 } = 63 - 16 \mathrm { i }\) and that \(p\) and \(q\) are real.
  2. Let \(\mathrm { f } ( z ) = z ^ { 3 } - A z ^ { 2 } + B z - C\) for complex numbers \(A , B\) and \(C\).
    1. Given that the cubic equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha = - 7 \mathrm { i } , \beta = 3 \mathrm { i }\) and \(\gamma = 4\), determine each of \(A , B\) and \(C\).
    2. Find the roots of the equation \(\mathrm { f } ^ { \prime } ( z ) = 0\).
Pre-U Pre-U 9795/1 2013 June Q12
11 marks Challenging +1.2
12 Given \(y = x \mathrm { e } ^ { 2 x }\),
  1. find the first four derivatives of \(y\) with respect to \(x\),
  2. conjecture an expression for \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) in the form \(( a x + b ) \mathrm { e } ^ { 2 x }\), where \(a\) and \(b\) are functions of \(n\),
  3. prove by induction that your result holds for all positive integers \(n\).
Pre-U Pre-U 9795/1 2013 June Q13
4 marks Challenging +1.8
13
  1. Use the definitions \(\tanh \theta = \frac { \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }\) and \(\operatorname { sech } \theta = \frac { 2 } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }\) to prove the results
    1. \(\tanh ^ { 2 } \theta \equiv 1 - \operatorname { sech } ^ { 2 } \theta\),
    2. \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tanh \theta ) = \operatorname { sech } ^ { 2 } \theta\).
    3. Let \(I _ { n } = \int _ { 0 } ^ { \alpha } \tanh ^ { 2 n } \theta \mathrm {~d} \theta\) for \(n \geqslant 0\), where \(\alpha > 0\).
      (a) Show that \(I _ { n - 1 } - I _ { n } = \frac { \tanh ^ { 2 n - 1 } \alpha } { 2 n - 1 }\) for \(n \geqslant 1\). Given that \(\alpha = \frac { 1 } { 2 } \ln 3\),
      (b) evaluate \(I _ { 0 }\),
    4. use the method of differences to show that \(I _ { n } = \frac { 1 } { 2 } \ln 3 - \sum _ { r = 1 } ^ { n } \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 r - 1 } } { 2 r - 1 }\) and deduce the sum of the infinite series \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } }\).
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.