OCR MEI FP3 (Further Pure Mathematics 3) 2007 June

1 Three planes \(P , Q\) and \(R\) have the following equations. $$\begin{array} { l l } \text { Plane } P : & 8 x - y - 14 z = 20
\text { Plane } Q : & 6 x + 2 y - 5 z = 26
\text { Plane } R : & 2 x + y - z = 40 \end{array}$$ The line of intersection of the planes \(P\) and \(Q\) is \(K\).
The line of intersection of the planes \(P\) and \(R\) is \(L\).

1. Show that \(K\) and \(L\) are parallel lines, and find the shortest distance between them.
2. Show that the shortest distance between the line \(K\) and the plane \(R\) is \(5 \sqrt { 6 }\). The line \(M\) has equation \(\mathbf { r } = ( \mathbf { i } - 4 \mathbf { j } ) + \lambda ( 5 \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } )\).
3. Show that the lines \(K\) and \(M\) intersect, and find the coordinates of the point of intersection.
4. Find the shortest distance between the lines \(L\) and \(M\).

2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.

3 The curve \(C\) has equation \(y = \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } \ln x\), and \(a\) is a constant with \(a \geqslant 1\).

1. Show that the length of the arc of \(C\) for which \(1 \leqslant x \leqslant a\) is \(\frac { 1 } { 2 } a ^ { 2 } + \frac { 1 } { 4 } \ln a - \frac { 1 } { 2 }\).
2. Find the area of the surface generated when the arc of \(C\) for which \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(\boldsymbol { y }\)-axis.
3. Show that the radius of curvature of \(C\) at the point where \(x = a\) is \(a \left( a + \frac { 1 } { 4 a } \right) ^ { 2 }\).
4. Find the centre of curvature corresponding to the point \(\left( 1 , \frac { 1 } { 2 } \right)\) on \(C\).
   \(C\) is one member of the family of curves defined by \(y = p x ^ { 2 } - p ^ { 2 } \ln x\), where \(p\) is a parameter.
5. Find the envelope of this family of curves.

4

1. Prove that, for a group of order 10, every proper subgroup must be cyclic. The set \(M = \{ 1,2,3,4,5,6,7,8,9,10 \}\) is a group under the binary operation of multiplication modulo 11.
2. Show that \(M\) is cyclic.
3. List all the proper subgroups of \(M\). The group \(P\) of symmetries of a regular pentagon consists of 10 transformations $$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$ and the binary operation is composition of transformations. The composition table for \(P\) is given below.

   	A	B	C	D	E	F	G	H	I	J
   A	C	J	G	H	A	B	I	F	E	D
   B	F	E	H	G	B	A	D	C	J	I
   C	G	D	I	F	C	J	E	B	A	H
   D	J	C	B	E	D	G	F	I	H	A
   E	A	B	C	D	E	F	G	H	I	J
   F	H	I	D	C	F	E	J	A	B	G
   G	I	H	E	B	G	D	A	J	C	F
   H	D	G	J	A	H	I	B	E	F	C
   I	E	F	A	J	I	H	C	D	G	B
   J	B	A	F	I	J	C	H	G	D	E

   One of these transformations is the identity transformation, some are rotations and the rest are reflections.
4. Identify which transformation is the identity, which are rotations and which are reflections.
5. State, giving a reason, whether \(P\) is isomorphic to \(M\).
6. Find the order of each element of \(P\).
7. List all the proper subgroups of \(P\).

5 A computer is programmed to generate a sequence of letters. The process is represented by a Markov chain with four states, as follows. The first letter is \(A , B , C\) or \(D\), with probabilities \(0.4,0.3,0.2\) and 0.1 respectively.
After \(A\), the next letter is either \(C\) or \(D\), with probabilities 0.8 and 0.2 respectively.
After \(B\), the next letter is either \(C\) or \(D\), with probabilities 0.1 and 0.9 respectively.
After \(C\), the next letter is either \(A\) or \(B\), with probabilities 0.4 and 0.6 respectively.
After \(D\), the next letter is either \(A\) or \(B\), with probabilities 0.3 and 0.7 respectively.

1. Write down the transition matrix \(\mathbf { P }\).
2. Use your calculator to find \(\mathbf { P } ^ { 4 }\) and \(\mathbf { P } ^ { 7 }\). (Give elements correct to 4 decimal places.)
3. Find the probability that the 8th letter is \(C\).
4. Find the probability that the 12th letter is the same as the 8th letter.
5. By investigating the behaviour of \(\mathbf { P } ^ { n }\) when \(n\) is large, find the probability that the ( \(n + 1\) )th letter is \(A\) when
   (A) \(n\) is a large even number,
   (B) \(n\) is a large odd number. The program is now changed. The initial probabilities and the transition probabilities are the same as before, except for the following. After \(D\), the next letter is \(A , B\) or \(D\), with probabilities \(0.3,0.6\) and 0.1 respectively.
6. Write down the new transition matrix \(\mathbf { Q }\).
7. Verify that \(\mathbf { Q } ^ { n }\) approaches a limit as \(n\) becomes large, and hence write down the equilibrium probabilities for \(A , B , C\) and \(D\).
8. When \(n\) is large, find the probability that the \(( n + 1 )\) th, \(( n + 2 )\) th and \(( n + 3 )\) th letters are DDD.