Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) 2016 June

1 Using standard summation results, show that \(\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } + r \right) \equiv \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 ) ^ { 2 }\).

2 Find a vector which is perpendicular to both of the lines $$\mathbf { r } = \left( \begin{array} { r } 11 \\ 5 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { l } 6 \\ 2 \\ 5 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 1 \\ 7 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { r } - 6 \\ 1 \\ 4 \end{array} \right)$$ and hence find the shortest distance between them.

3 A curve has equation \(y = \frac { 2 x ^ { 2 } - x - 1 } { 2 x - 3 }\).

1. Show that the curve meets the line \(y = k\) when \(2 x ^ { 2 } - ( 2 k + 1 ) x + ( 3 k - 1 ) = 0\), and hence show that no part of the curve exists in the interval \(\frac { 1 } { 2 } < y < \frac { 9 } { 2 }\).
2. Deduce the coordinates of the turning points of this curve.

4 A \(3 \times 3\) system of equations is given by the matrix equation \(\left( \begin{array} { r r r } - 1 & 3 & 1 \\ 5 & - 1 & 2 \\ - 1 & 1 & 0 \end{array} \right) \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 1 \\ 16 \\ - 2 \end{array} \right)\).

1. Show that this system of equations does not have a unique solution.
2. Solve this system of equations and describe the geometrical significance of the solution.

5 Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 24 \mathrm { e } ^ { 2 x }\).

6 The equation \(\sinh x + \sin x = 3 x\) has one positive root \(\alpha\).

1. Show that \(2.5 < \alpha < 3\).
2. By using the first two non-zero terms in the Maclaurin series for \(\sinh x + \sin x\), show that \(\alpha \approx \sqrt [ 4 ] { 60 }\).
3. By taking the third non-zero term in this series, find a second approximation to \(\alpha\), giving your answer correct to 4 decimal places.

7

1. Find all values of \(z\) for which \(z ^ { 3 } = 2 + 2 \mathrm { i }\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(\theta\) is an exact multiple of \(\pi\) in the interval \(0 < \theta < 2 \pi\).
2. The vertices of a triangle in the Argand diagram correspond to the three roots of the equation \(z ^ { 3 } = 2 + 2 \mathrm { i }\). Sketch the triangle and determine its area.

8

1. \(S\) is the set \(\{ 1,2,4,8,16,32 \}\) and \(\times _ { 63 }\) is the operation of multiplication modulo 63 .
   1. Construct the multiplication table for \(\left( S , \times _ { 63 } \right)\).
   2. Show that \(\left( S , \times _ { 63 } \right)\) forms a group, \(G\). (You may assume that \(\times _ { 63 }\) is associative.)
   3. The group \(H\), also of order 6, has identity element \(e\) and contains two further elements \(x\) and \(y\) with the properties $$x ^ { 2 } = y ^ { 3 } = e \quad \text { and } \quad x y x = y ^ { 2 } .$$ (a) Construct the group table of \(H\).
      (b) List all the proper subgroups of \(H\).
   4. State, with justification, whether \(G\) and \(H\) are isomorphic.

9 The cubic equation \(x ^ { 3 } - a x ^ { 2 } + b x - c = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. State, in terms of \(a , b\) and \(c\), the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. Find, in terms of \(a , b\) and \(c\), the values of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
3. Show that \(( \alpha - 2 \beta \gamma ) ( \beta - 2 \gamma \alpha ) ( \gamma - 2 \alpha \beta ) = c ( 2 a + 1 ) ^ { 2 } - 2 ( b + 2 c ) ^ { 2 }\).
4. Deduce that one root of the equation \(x ^ { 3 } - a x ^ { 2 } + b x - c = 0\) is twice the product of the other two roots if and only if \(c ( 2 a + 1 ) ^ { 2 } = 2 ( b + 2 c ) ^ { 2 }\).

10

1. Sketch the curve with polar equation \(r = \left| \frac { 1 } { 2 } + \sin \theta \right|\), for \(0 \leqslant \theta < 2 \pi\).
2. Find in an exact form the total area enclosed by the curve.

11

1. The sequence of Fibonacci Numbers \(\left\{ F _ { n } \right\}\) is given by $$F _ { 1 } = 1 , \quad F _ { 2 } = 1 \quad \text { and } \quad F _ { n + 1 } = F _ { n } + F _ { n - 1 } \text { for } n \geqslant 2 .$$ Write down the values of \(F _ { 3 }\) to \(F _ { 6 }\).
2. The sequence of functions \(\left\{ \mathrm { p } _ { n } ( x ) \right\}\) is given by $$\mathrm { p } _ { 1 } ( x ) = x + 1 \quad \text { and } \quad \mathrm { p } _ { n + 1 } ( x ) = 1 + \frac { 1 } { \mathrm { p } _ { n } ( x ) } \text { for } n \geqslant 1$$
   1. Find \(\mathrm { p } _ { 2 } ( x )\) and \(\mathrm { p } _ { 3 } ( x )\), giving each answer as a single algebraic fraction, and show that \(\mathrm { p } _ { 4 } ( x ) = \frac { 3 x + 5 } { 2 x + 3 }\).
   2. Conjecture an expression for \(\mathrm { p } _ { n } ( x )\) as a single algebraic fraction involving Fibonacci numbers, and prove it by induction for all integers \(n \geqslant 2\).

12 The curve \(C\) has equation \(y = \ln \left( \tanh \frac { 1 } { 2 } x \right)\), for \(x > 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x\).
2. For positive integers \(n\), the length of the arc of \(C\) between \(x = n\) and \(x = 2 n\) is \(L _ { n }\).
   1. Show by calculus that, when \(n\) is large, \(L _ { n } \approx n\).
   2. Explain how this result corresponds to the shape of \(C\).

13

1. (a) Given that \(x \geqslant 1\), show that \(\sec ^ { - 1 } x = \cos ^ { - 1 } \left( \frac { 1 } { x } \right)\), and deduce that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { x \sqrt { x ^ { 2 } - 1 } }\).
   (b) Use integration by parts to determine \(\int \sec ^ { - 1 } x \mathrm {~d} x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{5d526fd9-72f8-42b1-b156-fd4a0c764c82-4_670_1029_1073_596} The diagram shows the curve \(S\) with equation \(y = \sec ^ { - 1 } x\) for \(x \geqslant 1\). The line \(L\), with gradient \(\frac { 1 } { \sqrt { 2 } }\), is the tangent to \(S\) at the point \(P\) and cuts the \(x\)-axis at the point \(Q\). The point \(I\) has coordinates \(( 1,0 )\).
   (a) Determine the exact coordinates of \(P\) and \(Q\).
   (b) The region \(R\), shaded on the diagram, is bounded by the line segments \(P Q\) and \(Q I\) and the \(\operatorname { arc } I P\) of \(S\). Show that \(R\) has area $$\ln ( 1 + \sqrt { 2 } ) - \frac { \pi ( 8 - \pi ) \sqrt { 2 } } { 32 } .$$ {www.cie.org.uk} after the live examination series. }