- (a) Show that \(\left( 1 \frac { 1 } { 24 } \right) ^ { - \frac { 1 } { 2 } } = k \sqrt { 6 }\), where \(k\) is rational.
(b) Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { - \frac { 1 } { 2 } } , | x | < 2\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(c) Use your answer to part (b) with \(x = \frac { 1 } { 12 }\) to find an approximate value for \(\sqrt { 6 }\), giving your answer to 5 decimal places. - continued
- Relative to a fixed origin, two lines have the equations
$$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$
and
$$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$
where \(s\) and \(t\) are scalar parameters.
(a) Show that the two lines intersect and find the position vector of the point where they meet.
(b) Find, in degrees to 1 decimal place, the acute angle between the lines.