Pre-U Pre-U 9795 (Pre-U Further Mathematics) Specimen

1 The \(n\)th triangular number, \(T _ { n }\), is given by the formula \(T _ { n } = \frac { 1 } { 2 } n ( n + 1 )\).

1. Express \(\frac { 1 } { T _ { n } }\) in terms of partial fractions.
2. Hence, using the method of differences, show that \(\sum _ { n = 1 } ^ { N } \left( \frac { 1 } { T _ { n } } \right) = \frac { 2 N } { N + 1 }\).

2

1. On a single Argand diagram, sketch and clearly label each of the following loci:
   1. \(| z | = 4\),
   2. \(\quad \arg ( z + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
   3. On the same Argand diagram, shade the region \(R\) defined by the inequalities $$| z | \leqslant 4 \quad \text { and } \quad 0 \leqslant \arg ( z + 4 i ) \leqslant \frac { 1 } { 4 } \pi$$

3 Solve the equation $$5 \cosh x - \sinh x = 7$$ giving your answers in an exact logarithmic form.

4 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).

1. Use de Moivre's theorem to prove that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
2. Deduce the identity \(\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )\).

5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 72 \mathrm { e } ^ { 3 x }$$

6

1. Given that \(y = \cos ( \ln ( 1 + x ) )\), prove that
   1. \(\quad ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = - \sin ( \ln ( 1 + x ) )\),
   2. \(( 1 + x ) ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } + y = 0\).
   3. Obtain an equation relating \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   4. Hence find the Maclaurin series for \(y\), up to and including the term in \(x ^ { 3 }\).

7 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x - 3 } { x + 2 } .$$

1. Find the equations of the asymptotes of \(C\).
2. Sketch \(C\), indicating clearly the asymptotes and any points where \(C\) meets the coordinate axes.

8 The equation \(8 x ^ { 3 } + 12 x ^ { 2 } + 4 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. By considering a suitable substitution, or otherwise, show that the equation whose roots are \(2 \alpha + 1,2 \beta + 1,2 \gamma + 1\) can be written in the form $$y ^ { 3 } - y - 1 = 0 .$$
2. The sum \(( 2 \alpha + 1 ) ^ { n } + ( 2 \beta + 1 ) ^ { n } + ( 2 \gamma + 1 ) ^ { n }\) is denoted by \(S _ { n }\). Evaluate \(S _ { 3 }\) and \(S _ { - 2 }\).

9

1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 1 & 3 & 4 \\ 2 & 5 & - 1 \\ 3 & 8 & 2 \end{array} \right)\), and hence solve the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 10 \\ 3 x + 8 y + 2 z & = 8 \end{aligned}$$
2. Find the value of \(k\) for which the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 15 \\ 3 x + 8 y + 3 z & = k \end{aligned}$$ is consistent. Find the solution in this case and interpret it geometrically.

10 A group \(G\) has distinct elements \(e , a , b , c , \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation.

1. Prove that if \(a \circ a = b\) and \(b \circ b = a\), then the set of elements \(\{ e , a , b \}\) forms a subgroup of \(G\).
2. Prove that if \(a \circ a = b , b \circ b = c\) and \(c \circ c = a\), then the set of elements \(\{ e , a , b , c \}\) does not form a subgroup of \(G\).

11 With respect to an origin \(O\), the points \(A , B , C\) and \(D\) have position vectors $$\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad \mathbf { b } = \mathbf { i } - 2 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { d } = - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ,$$ respectively. Find

1. a vector perpendicular to the plane \(O A B\),
2. the acute angle between the planes \(O A B\) and \(O C D\), correct to the nearest \(0.1 ^ { \circ }\),
3. the shortest distance between the line \(A B\) and the line \(C D\),
4. the perpendicular distance from the point \(A\) to the line \(C D\).

12 \includegraphics[max width=\textwidth, alt={}, center]{0f5edc87-cb14-4583-a54d-badec47741d1-08_414_659_804_744} The diagram shows a sketch of the curve \(C\) with polar equation \(r = 4 \cos ^ { 2 } \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Explain briefly how you can tell from this form of the equation that \(C\) is symmetrical about the line \(\theta = 0\) and that the tangent to \(C\) at the pole \(O\) is perpendicular to the line \(\theta = 0\).
2. The equation of \(C\) may be expressed in the form \(r = k ( 1 + \cos 2 \theta )\). State the value of \(k\) and use this form to show that the area of the region enclosed by \(C\) is given by $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } ( 3 + 4 \cos 2 \theta + \cos 4 \theta ) d \theta ,$$ and hence find this area.
3. The length of \(C\) is denoted by \(L\). Show that $$L = 8 \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos \theta \sqrt { 1 + 3 \sin ^ { 2 } \theta } \mathrm {~d} \theta$$ and use the substitution \(\sinh x = \sqrt { 3 } \sin \theta\) to determine \(L\) in an exact form.

13 Let \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\), where \(n\) is a positive integer.

1. By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ( \ln x ) ^ { n } \right)\), or otherwise, show that \(I _ { n } = \mathrm { e } - n I _ { n - 1 }\).
2. Let \(J _ { n } = \frac { I _ { n } } { n ! }\). Prove by induction that $$\sum _ { r = 2 } ^ { n } \frac { ( - 1 ) ^ { r } } { r ! } = \frac { 1 } { \mathrm { e } } \left( 1 + ( - 1 ) ^ { n } J _ { n } \right)$$ for all positive integers \(n \geqslant 2\).