Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) 2013 June

1 By completing the square, or otherwise, find the exact value of \(\int _ { 2 } ^ { 6 } \frac { 1 } { x ^ { 2 } - 6 x + 12 } \mathrm {~d} x\).

2 Use the standard Maclaurin series expansions given in the List of Formulae MF20 to show that $$\frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \equiv \tanh ^ { - 1 } x \text { for } - 1 < x < 1$$

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.

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 $$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \mathbf { i } + \mathbf { j } + 10 \mathbf { k } + \mu ( 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) .$$

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.

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 \(\int _ { \pi } ^ { 2 \pi } \frac { 1 } { ( 1 - \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).

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.

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.

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.

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 \(x = 1.1\).

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 \(\mathrm { f } ^ { \prime } ( z ) = 0\).

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 \(n\).

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 \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } }\).