OCR MEI FP3 (Further Pure Mathematics 3) 2010 June

1 Four points have coordinates $$\mathrm { A } ( 3,8,27 ) , \quad \mathrm { B } ( 5,9,25 ) , \quad \mathrm { C } ( 8,0,1 ) \quad \text { and } \quad \mathrm { D } ( 11 , p , p ) ,$$ where \(p\) is a constant.

1. Find the perpendicular distance from C to the line AB .
2. Find \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\) in terms of \(p\), and show that the shortest distance between the lines AB and CD is $$\frac { 21 | p - 5 | } { \sqrt { 17 p ^ { 2 } - 2 p + 26 } }$$
3. Find, in terms of \(p\), the volume of the tetrahedron ABCD .
4. State the value of \(p\) for which the lines AB and CD intersect, and find the coordinates of the point of intersection in this case. Option 2: Multi-variable calculus

2 In this question, \(L\) is the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2
1
- 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
2
1 \end{array} \right)\), and \(\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Show that the normal to the surface \(\mathrm { g } ( x , y , z ) = 3\) at the point \(( 2,1 , - 1 )\) is the line \(L\). On the line \(L\), there are two points at which \(\mathrm { g } ( x , y , z ) = 0\).
3. Show that one of these points is \(\mathrm { P } ( 0,3,0 )\), and find the coordinates of the other point Q .
4. Show that, if \(x = - 2 \mu , y = 3 + 2 \mu , z = \mu\), and \(\mu\) is small, then $$\mathrm { g } ( x , y , z ) \approx - 6 \mu \mathrm { e } ^ { - 6 }$$ You are given that \(h\) is a small number.
5. There is a point on \(L\), close to P , at which \(\mathrm { g } ( x , y , z ) = h\). Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
6. Find the approximate coordinates of the point on \(L\), close to Q , at which \(\mathrm { g } ( x , y , z ) = h\).

3 A curve \(C\) has equation \(y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } }\), for \(x \geqslant 0\).

1. Show that the arc of \(C\) for which \(0 \leqslant x \leqslant a\) has length \(a ^ { \frac { 1 } { 2 } } + \frac { 1 } { 3 } a ^ { \frac { 3 } { 2 } }\).
2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Find the coordinates of the centre of curvature corresponding to the point \(\left( 4 , - \frac { 2 } { 3 } \right)\) on \(C\). The curve \(C\) is one member of the family of curves defined by $$y = p ^ { 2 } x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } p ^ { 3 } x ^ { \frac { 3 } { 2 } } \quad ( \text { for } x \geqslant 0 )$$ where \(p\) is a parameter (and \(p > 0\) ).
4. Find the equation of the envelope of this family of curves.

4 The group \(F = \{ \mathrm { p } , \mathrm { q } , \mathrm { r } , \mathrm { s } , \mathrm { t } , \mathrm { u } \}\) consists of the six functions defined by $$\mathrm { p } ( x ) = x \quad \mathrm { q } ( x ) = 1 - x \quad \mathrm { r } ( x ) = \frac { 1 } { x } \quad \mathrm {~s} ( x ) = \frac { x - 1 } { x } \quad \mathrm { t } ( x ) = \frac { x } { x - 1 } \quad \mathrm { u } ( x ) = \frac { 1 } { 1 - x } ,$$ the binary operation being composition of functions.

1. Show that st \(= \mathrm { r }\) and find ts.
2. Copy and complete the following composition table for \(F\).

   	p	q	r	s	t	u
   p	p	q	r	s	t	u
   q	q	p	s	r	u	t
   r	r	u	p	t	s	q
   s	s	t	q	u	r	p
   t	t	s	u
   u	u	r	t

3. Give the inverse of each element of \(F\).
4. List all the subgroups of \(F\). The group \(M\) consists of \(\left\{ 1 , - 1 , e ^ { \frac { \pi } { 3 } \mathrm { j } } , e ^ { - \frac { \pi } { 3 } \mathrm { j } } , e ^ { \frac { 2 \pi } { 3 } \mathrm { j } } , e ^ { - \frac { 2 \pi } { 3 } \mathrm { j } } \right\}\) with multiplication of complex numbers as its binary operation.
5. Find the order of each element of \(M\). The group \(G\) consists of the positive integers between 1 and 18 inclusive, under multiplication modulo 19.
6. Show that \(G\) is a cyclic group which can be generated by the element 2 .
7. Explain why \(G\) has no subgroup which is isomorphic to \(F\).
8. Find a subgroup of \(G\) which is isomorphic to \(M\).