Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) 2010 June

1 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Use the substitution \(y = 1 + x ^ { 2 }\) to find an equation, with integer coefficients, whose roots are \(1 + \alpha ^ { 2 } , 1 + \beta ^ { 2 }\) and \(1 + \gamma ^ { 2 }\).

2 Use the method of differences to express \(\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }\) in terms of \(n\), and hence deduce the sum of the infinite series $$\frac { 1 } { 3 } + \frac { 1 } { 15 } + \frac { 1 } { 35 } + \ldots + \frac { 1 } { 4 n ^ { 2 } - 1 } + \ldots$$

3 The points \(A ( 1,3 ) , B ( 4,36 )\) and \(C ( 9,151 )\) lie on the curve with equation \(y = p + q x + r x ^ { 2 }\).

1. Using this information, write down three simultaneous equations in \(p , q\) and \(r\).
2. Re-write this system of equations in the matrix form \(\mathbf { C x } = \mathbf { a }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, \(\mathbf { x }\) is an unknown vector, and \(\mathbf { a }\) is a fixed vector.
3. By finding \(\mathbf { C } ^ { - 1 }\), determine the values of \(p , q\) and \(r\).

4

1. Using the definitions of sinh and cosh in terms of exponentials, prove that $$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$
2. Solve the equation \(5 \cosh x + 3 \sinh x = 12\), giving your answers in the form \(\ln ( p \pm q \sqrt { 2 } )\) for rational numbers \(p\) and \(q\) to be determined.

5 A curve has equation \(y = \frac { x ^ { 2 } + 5 x - 6 } { x + 3 }\) for \(x \neq - 3\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points on the curve.
2. Sketch the curve, justifying all significant features.

6

1. The set \(S\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n \in \mathbb { Z }\).
   1. Show that \(S\), under the operation of matrix multiplication, forms a group \(G\). [You may assume that matrix multiplication is associative.]
   2. State, giving a reason, whether \(G\) is abelian.
   3. The group \(H\) is the set \(\mathbb { Z }\) together with the operation of addition. Explain why \(G\) is isomorphic to \(H\).
   4. The plane transformation \(T\) is given by the matrix \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n\) is a non-zero integer. Describe \(T\) fully.

7 A curve \(C\) has polar equation \(r = 2 + \cos \theta\) for \(- \pi < \theta \leqslant \pi\).

1. The point \(P\) on \(C\) corresponds to \(\theta = \alpha\), and the point \(Q\) on \(C\) is such that \(P O Q\) is a straight line, where \(O\) is the pole. Show that the length \(P Q\) is independent of \(\alpha\).
2. Find, in an exact form, the area of the region enclosed by \(C\).
3. Show that \(\left( x ^ { 2 } + y ^ { 2 } - x \right) ^ { 2 } = 4 \left( x ^ { 2 } + y ^ { 2 } \right)\) is a cartesian equation for \(C\). Identify the coordinates of the point which is included in this cartesian equation but is not on \(C\).

8 For the differential equation \(t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 6 - 4 t ^ { 2 } \right) x = 0\), use the substitution \(x = t ^ { 2 } u\) to find a differential equation involving \(t\) and \(u\) only. Hence solve the above differential equation, given that \(x = \mathrm { e } - 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 4 \mathrm { e }\) when \(t = 1\).

9 Three non-collinear points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to the origin \(O\). The plane through \(A , B\) and \(C\) is denoted by \(\Pi\).

1. (a) Prove that the area of triangle \(A B C\) is \(\frac { 1 } { 2 } | \mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a } |\).
   (b) Describe the significance of the vector \(\mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a }\) in relation to \(\Pi\).
2. (a) In the case when \(\mathbf { a } = a \mathbf { i } , \mathbf { b } = b \mathbf { j }\) and \(\mathbf { c } = c \mathbf { k }\), where \(a , b\) and \(c\) are positive scalar constants, determine the equation of \(\Pi\) in the form r.n \(= d\), where the components of \(\mathbf { n }\) and the value of the scalar constant \(d\) are to be given in terms of \(a , b\) and \(c\).
   (b) Deduce the shortest distance from the origin \(O\) to \(\Pi\) in this case.

10 One root of the equation \(z ^ { 5 } - 1 = 0\) is the complex number \(\omega = \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { i } }\).

1. Show that
   1. \(\quad \omega ^ { 5 } = 1\),
   2. \(\quad \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\),
   3. \(\quad \omega + \omega ^ { 4 } = 2 \cos \frac { 2 } { 5 } \pi\), and write down a similar expression for \(\omega ^ { 2 } + \omega ^ { 3 }\).
   4. Using these results, find the values of \(\cos \frac { 2 } { 5 } \pi + \cos \frac { 4 } { 5 } \pi\) and \(\cos \frac { 2 } { 5 } \pi \times \cos \frac { 4 } { 5 } \pi\), and deduce a quadratic equation, with integer coefficients, which has roots $$\cos \frac { 2 } { 5 } \pi \quad \text { and } \quad \cos \frac { 4 } { 5 } \pi$$

11

1. At all points \(( x , y )\) on the curve \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 0\).
   1. Prove by induction that, for all integers \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n + 1 } y } { \mathrm {~d} x ^ { n + 1 } } + x \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } + n \frac { \mathrm {~d} ^ { n - 1 } y } { \mathrm {~d} x ^ { n - 1 } } = 0$$ where \(\frac { \mathrm { d } ^ { 0 } y } { \mathrm {~d} x ^ { 0 } } = y\).
   2. Given that \(y = 1\) when \(x = 0\), determine the Maclaurin expansion of \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 6 }\).
   3. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = 0\) given that \(y = 1\) when \(x = 0\).
   4. Given that \(Z \sim \mathrm {~N} ( 0,1 )\), use your answers to parts (i) and (ii) to find an approximation, to 4 decimal places, to the probability \(\mathrm { P } ( Z \leqslant 1 )\).
      [0pt] [Note that the probability density function of the standard normal distribution is \(\mathrm { f } ( z ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - \frac { 1 } { 2 } z ^ { 2 } }\).]

12

1. Let \(I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } } \mathrm {~d} x\), for integers \(n \geqslant 0\).
   By writing \(\frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } }\) as \(x ^ { n - 1 } \times \frac { x } { \sqrt { x ^ { 2 } + 1 } }\), or otherwise, show that, for \(n \geqslant 2\), $$n I _ { n } = x ^ { n - 1 } \sqrt { x ^ { 2 } + 1 } - ( n - 1 ) I _ { n - 2 } .$$
2. The diagram shows a sketch of the hyperbola \(H\) with equation \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1\). \includegraphics[max width=\textwidth, alt={}, center]{32ed7cc8-3456-4cf0-952a-ee04eada1298-6_593_666_776_776}
   1. Find the coordinates of the points where \(H\) crosses the \(x\)-axis.
   2. The curve \(J\) has parametric equations \(x = 2 \cosh \theta , y = 4 \sinh \theta\), for \(\theta \geqslant 0\). Show that these parametric equations satisfy the cartesian equation of \(H\), and indicate on a copy of the above diagram which part of \(H\) is \(J\).
   3. The arc of the curve \(J\) between the points where \(x = 2\) and \(x = 34\) is rotated once completely about the \(x\)-axis to form a surface of revolution with area \(S\). Show that $$S = 16 \pi \int _ { \alpha } ^ { \beta } \sinh \theta \sqrt { 5 \cosh ^ { 2 } \theta - 1 } \mathrm {~d} \theta$$ for suitable constants \(\alpha\) and \(\beta\).
   4. Use the substitution \(u ^ { 2 } = 5 \cosh ^ { 2 } \theta - 1\) to show that $$S = \frac { 8 \pi } { \sqrt { 5 } } ( 644 \sqrt { 5 } - \ln ( 9 + 4 \sqrt { 5 } ) )$$