Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) 2017 June

1 Without using a calculator, determine the possible values of \(a\) and \(b\) for which \(( a + \mathrm { i } b ) ^ { 2 } = 21 - 20 \mathrm { i }\).

2 The equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Evaluate \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and use your answer to comment on the nature of these roots.

3

1. Sketch the curve with polar equation \(r = \frac { 1 } { 1 + \theta } , 0 \leqslant \theta \leqslant 2 \pi\).
2. Find, in terms of \(\pi\), the area of the region enclosed by the curve and the part of the initial line between the endpoints of the curve.

4 The curve \(C\) has parametric equations \(x = \frac { 1 } { 2 } t ^ { 2 } - \ln t , y = 2 t\), for \(1 \leqslant t \leqslant 4\). When \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed of surface area \(S\). Determine the exact value of \(S\).

5

1. Use the definition \(\tanh y = \frac { \mathrm { e } ^ { 2 y } - 1 } { \mathrm { e } ^ { 2 y } + 1 }\) to show that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\) for \(| x | < 1\).
2. Solve the equation \(\tanh x + \operatorname { coth } x = 4\), giving your answer in the form \(p \ln m\), where \(p\) is a positive rational number and \(m\) is a positive integer.

6 The curve \(S\) has equation \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ^ { 2 } }\).

1. Write down the equations of the asymptotes of \(S\).
2. Determine \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any turning points of \(S\).
3. Sketch \(S\).

7

1. Find the value of the constant \(k\) for which \(y = k x \sin 2 x\) is a particular integral of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 \cos 2 x\).
2. Solve \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 \cos 2 x\), given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\).

8 The line \(l\) has equation \(\mathbf { r } = \lambda \mathbf { d }\) and the plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } . \mathbf { n } = 35\), where $$\mathbf { d } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) \quad \text { and } \quad \mathbf { n } = \left( \begin{array} { r } 6 \\ - 2 \\ 3 \end{array} \right) .$$

1. (a) Determine the exact value of \(\cos \theta\), where \(\theta\) is the angle between \(\mathbf { d }\) and \(\mathbf { n }\).
   (b) Determine the position vector of the point of intersection of \(l\) and \(\Pi _ { 1 }\).
   (c) Determine the shortest distance from \(O\) to \(\Pi _ { 1 }\).
2. The plane \(\Pi _ { 2 }\) has cartesian equation \(12 x - 4 y + 6 z + 21 = 0\). Determine the distance between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

9

1. Given that \(x \geqslant 1\), use the substitution \(x = \cosh \theta\) to show that $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { \sqrt { x ^ { 2 } - 1 } } { x } + C$$ where \(C\) is an arbitrary constant.
2. By differentiating sec \(y = x\) implicitly, show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { x \sqrt { x ^ { 2 } - 1 } }\) for \(x \geqslant 1\).
3. Use integration by parts to determine \(\int \frac { \sec ^ { - 1 } x } { x ^ { 2 } } \mathrm {~d} x\) for \(x \geqslant 1\).

10

1. Express \(\frac { 1 } { ( k - 1 ) k ( k + 1 ) }\) in partial fractions.
2. Let \(S _ { n } = \sum _ { k = 3 } ^ { n } \frac { 1 } { ( k - 1 ) k ( k + 1 ) }\) for \(n \geqslant 3\). Use the method of differences to show that $$S _ { n } = \frac { 1 } { 12 } - \frac { 1 } { 2 n ( n + 1 ) }$$ and write down the limit of \(S _ { n }\) as \(n \rightarrow \infty\).
3. Given that \(k\) is a positive integer greater than 1 , explain why \(\frac { 1 } { k ^ { 3 } } < \frac { 1 } { ( k - 1 ) k ( k + 1 ) }\).
4. Show that \(\frac { 27 } { 24 } < \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ^ { 3 } } < \frac { 29 } { 24 }\).

11

1. (a) Given \(\mathbf { A } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } e & f \\ g & h \end{array} \right)\), work out the matrix \(\mathbf { A B }\) and write down expressions for \(\operatorname { det } \mathbf { A }\) and \(\operatorname { det } \mathbf { B }\).
   (b) Verify, by direct calculation, that \(\operatorname { det } ( \mathbf { A B } ) = \operatorname { det } \mathbf { A } \times \operatorname { det } \mathbf { B }\). Let \(S\) be the set of all \(2 \times 2\) matrices with determinant equal to 1 .
2. Show that \(\left( S , \times _ { \mathrm { M } } \right)\) forms a group, \(G\), where \(\times _ { \mathrm { M } }\) is the operation of matrix multiplication. [You may assume that \(\mathrm { X } _ { \mathrm { M } }\) is associative.]
3. (a) Show that \(\mathbf { K } = \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right)\) is an element of \(G\). Let \(H\) be the smallest subgroup of \(G\) that contains \(\mathbf { K }\) and let \(n\) be the order of \(H\).
   (b) Determine the value of \(n\).
   (c) Give a second subgroup of \(G\), also of order \(n\), which is isomorphic to \(H\).

12 For each positive integer \(n\), the function \(\mathrm { F } _ { n }\) is defined for all real angles \(\theta\) by $$\mathrm { F } _ { n } ( \theta ) = c ^ { 2 n } + s ^ { 2 n }$$ where \(c = \cos \theta\) and \(s = \sin \theta\).

1. Prove the identity $$\mathrm { F } _ { n + 2 } ( \theta ) - \frac { 1 } { 4 } \sin ^ { 2 } 2 \theta \times \mathrm { F } _ { n + 1 } ( \theta ) \equiv \mathrm { F } _ { n + 3 } ( \theta )$$ Let \(z\) denote the complex number \(c + \mathrm { i } s\).
2. Using de Moivre's theorem,
   1. express \(z + z ^ { - 1 }\) and \(z - z ^ { - 1 }\) in terms of \(c\) and \(s\) respectively,
   2. prove the identity \(8 \left( c ^ { 6 } + s ^ { 6 } \right) \equiv 3 \cos 4 \theta + 5\) and deduce that $$c ^ { 6 } + s ^ { 6 } \equiv \cos ^ { 2 } 2 \theta + \frac { 1 } { 4 } \sin ^ { 2 } 2 \theta$$
   3. Prove by induction that, for all positive integers \(n\), $$c ^ { 2 n + 4 } + s ^ { 2 n + 4 } \leqslant \cos ^ { 2 } 2 \theta + \frac { 1 } { 2 ^ { n + 1 } } \sin ^ { 2 } 2 \theta$$ [You are given that the range of the function \(\mathrm { F } _ { n }\) is \(\frac { 1 } { 2 ^ { n - 1 } } \leqslant \mathrm {~F} _ { n } ( \theta ) \leqslant 1\).] {www.cie.org.uk} after the live examination series. }