Pre-U Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) Specimen

2

1. Verify that, for all positive values of \(n\), $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find
2. the sum to \(N\) terms,
3. the sum to infinity.

3 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where \(x = 0\) to the point where \(x = 1\) is denoted by \(s\).

1. Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { 1 + x ^ { 4 } } \mathrm {~d} x$$ The surface area generated when this arc is rotated through one complete revolution about the \(x\)-axis is denoted by \(S\).
2. Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { 2 } - 1 )$$ [Do not attempt to evaluate \(s\) or \(S\).]

4

1. Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
2. On the same diagram draw the line \(\theta = \alpha\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The region bounded by \(C\) and the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(R\).
3. Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.

5 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where \(n \geqslant 0\).

1. Show that, for all \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
2. Hence prove by induction that, for all positive integers \(n\), $$I _ { n } < n ! .$$

6

1. Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73 .$$
2. Show that, whatever the initial conditions, \(\frac { y } { x ^ { 2 } } \rightarrow 1\) as \(x \rightarrow \infty\).

7 The multiplicative group \(G\) has eight elements \(e , a , b , c , a b , a c , b c , a b c\), where \(e\) is the identity. The group is commutative, and the order of each of the elements \(a , b , c\) is 2 .

1. Find four subgroups of \(G\) of order 4.
2. Give a reason why no group of order 8 can have a subgroup of order 3 . The group \(H\) has elements \(0,1,2 , \ldots , 7\) with group operation addition modulo 8 .
3. Find the order of each element of \(H\).
4. Determine whether \(G\) and \(H\) are isomorphic and justify your conclusion.

8

1. Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 5 a \\ 3 x + 8 y + a z & = b \end{aligned}$$ has a unique solution.
2. By use of the inverse matrix of a suitable \(3 \times 3\) matrix, find the unique solution in the case \(a = 1\) and \(b = 2\).
3. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.

9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant.

1. Obtain the equation of each of the asymptotes of \(C\).
2. Find the coordinates of the turning points of \(C\).
3. In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\).

10 The line \(l _ { 1 }\) is parallel to the vector \(4 \mathbf { j } - \mathbf { k }\) and passes through the point \(A\) whose position vector is \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\). The variable line \(l _ { 2 }\) is parallel to the vector \(\mathbf { i } - ( 2 \sin t ) \mathbf { j }\), where \(0 \leqslant t < 2 \pi\), and passes through the point \(B\) whose position vector is \(\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\). The points \(P\) and \(Q\) are on \(l _ { 1 }\) and \(l _ { 2 }\) respectively, and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length of \(P Q\) in terms of \(t\).
2. Hence find the values of \(t\) for which \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. For the case \(t = \frac { 1 } { 4 } \pi\), find the perpendicular distance from \(A\) to the plane \(B P Q\), giving your answer correct to 3 decimal places.

11 The complex number \(z\) is defined as \(z = \cos \theta + \mathrm { i } \sin \theta\).

1. Show that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
2. By expanding \(\left( z + z ^ { - 1 } \right) ^ { 5 }\), show that \(16 \cos ^ { 5 } \theta = \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta\).
3. Hence find \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 5 } \theta \mathrm {~d} \theta\).
4. Sketch the graphs of \(\mathrm { f } ( \theta ) = \sin ^ { 5 } \theta\) and \(\mathrm { f } ( \theta ) = \cos ^ { 5 } \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), and hence give the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 5 } \theta \mathrm {~d} \theta$$

12 The curve \(C\) is defined parametrically by $$x = t + \ln ( \cosh t ) , \quad y = \sinh t$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - t } \cosh ^ { 2 } t\).
2. Hence show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \cosh ^ { 2 } t ( 2 \sinh t - \cosh t )\).
3. Find the exact value of \(t\) at the point on \(C\) where \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).