7
- 1
\end{array} \right) + \mu \left( \begin{array} { c }
1
- 2
4
\end{array} \right) .$$
(ii) Find the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
(iii) Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
2 In this question you must show detailed reasoning.
(i) Find \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } 2 \tan x \mathrm {~d} x\) giving your answer in the form \(\ln p\).
(ii) Show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 2 \tan x \mathrm {~d} x\) is undefined explaining your reasoning.
3 The equation of a plane, \(\Pi\), is
$$\Pi : \quad \mathbf { r } = \left( \begin{array} { c }
2
- 3
5
\end{array} \right) + \lambda \left( \begin{array} { l }
1
1
3
\end{array} \right) + \mu \left( \begin{array} { c }
- 1
2
1
\end{array} \right) .$$
(i) Find a vector which is perpendicular to \(\Pi\).
(ii) Hence find an equation for \(\Pi\) in the form r.n \(= p\).
(iii) Find in the form \(\sqrt { q }\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number.
4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c r c } a & 2 & 3
4 & 4 & 6
- 2 & 2 & 9 \end{array} \right)\) where \(a\) is a constant. It is given that if \(\mathbf { A }\) is not singular then
$$\mathbf { A } ^ { - 1 } = \frac { 1 } { 24 a - 48 } \left( \begin{array} { c c c }
24 & - 12 & 0
- 48 & 9 a + 6 & 12 - 6 a
16 & - 2 a - 4 & 4 a - 8
\end{array} \right)$$
(i) Use \(\mathbf { A } ^ { - 1 }\) to solve the simultaneous equations below, giving your answer in terms of \(k\).
$$\begin{array} { r }
x + 2 y + 3 z = 6
4 x + 4 y + 6 z = 8
- 2 x + 2 y + 9 z = k
\end{array}$$
(ii) Consider the equations below where \(a\) takes the value which makes \(\mathbf { A }\) singular.
$$\begin{aligned}
a x + 2 y + 3 z & = b
4 x + 4 y + 6 z & = 10
- 2 x + 2 y + 9 z & = - 13
\end{aligned}$$
\(b\) takes the value for which the equations have an infinite number of solutions.
- Determine the value of \(b\).
- Find the solutions for \(y\) and \(z\) in terms of \(x\).
(iii) For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations.
5 The region \(R\) between the \(x\)-axis, the curve \(y = \frac { 1 } { \sqrt { p + x ^ { 2 } } }\) and the lines \(x = \sqrt { p }\) and \(x = \sqrt { 3 p }\), where \(p\) is a positive parameter, is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
(i) Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\).
(ii) Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt { 48 }\) find in exact form
- the greatest possible value of the volume of \(S\)
- the least possible value of the volume of \(S\).
6 (i) By considering \(\sum _ { r = 1 } ^ { n } \left( ( r + 1 ) ^ { 5 } - r ^ { 5 } \right)\) show that \(\sum _ { r = 1 } ^ { n } r ^ { 4 } = \frac { 1 } { 30 } n ( n + 1 ) ( 2 n + 1 ) \left( 3 n ^ { 2 } + 3 n - 1 \right)\).
(ii) Use the formula given in part (i) to find \(50 ^ { 4 } + 51 ^ { 4 } + \ldots + 80 ^ { 4 }\).
7 The roots of the equation \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are positive integers, are \(\alpha\) and \(\beta\).
(i) Find a quadratic equation with integer coefficients whose roots are \(\alpha + \beta\) and \(\alpha \beta\).
(ii) Show that it is not possible for the original equation and the equation found in part (i) both to have repeated roots.
(iii) Show that the discriminant of the equation found in part (i) is always positive.