OCR MEI FP3 (Further Pure Mathematics 3) 2014 June

1 Three points have coordinates \(\mathrm { A } ( - 3,12 , - 7 ) , \mathrm { B } ( - 2,6,9 ) , \mathrm { C } ( 6,0 , - 10 )\). The plane \(P\) passes through the points \(\mathrm { A } , \mathrm { B }\) and C .

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\). Hence or otherwise find an equation for the plane \(P\) in the form \(a x + b y + c z = d\). The plane \(Q\) has equation \(6 x + 3 y + 2 z = 32\). The perpendicular from A to the plane \(Q\) meets \(Q\) at the point D. The planes \(P\) and \(Q\) intersect in the line \(L\).
2. Find the distance AD .
3. Find an equation for the line \(L\).
4. Find the shortest distance from A to the line \(L\).
5. Find the volume of the tetrahedron ABCD .

2 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 2 z ^ { 2 } + 2 y z + 6 x z - 4 x y - 24\). \(\mathrm { P } ( 2,6 , - 2 )\) is a point on the surface \(S\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Find the equation of the normal line to the surface \(S\) at the point P .
3. The point Q is on this normal line and close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find, in terms of \(h\), the approximate perpendicular distance from Q to the surface \(S\).
4. Find the coordinates of the two points on the surface at which the normal line is parallel to the \(y\)-axis.
5. Given that \(10 x - y + 2 z = 6\) is the equation of a tangent plane to the surface \(S\), find the coordinates of the point of contact.

3

1. A curve has intrinsic equation \(s = 2 \ln \left( \frac { \pi } { \pi - 3 \psi } \right)\) for \(0 \leqslant \psi < \frac { 1 } { 3 } \pi\), where \(s\) is the arc length measured from a fixed point P and \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x } . \mathrm { P }\) is in the third quadrant. The curve passes through the origin O , at which point \(\psi = \frac { 1 } { 6 } \pi . \mathrm { Q }\) is the point on the curve at which \(\psi = \frac { 3 } { 10 } \pi\).
   1. Express \(\psi\) in terms of \(s\), and sketch the curve, indicating the points \(\mathrm { O } , \mathrm { P }\) and Q .
   2. Find the arc length OQ .
   3. Find the radius of curvature at the point O .
   4. Find the coordinates of the centre of curvature corresponding to the point O .
2. 1. Find the surface area of revolution formed when the curve \(y = \frac { 1 } { 3 } \sqrt { x } ( x - 3 )\) for \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
   2. The curve in part (b)(i) is one member of the family \(y = \frac { 1 } { 9 } \lambda \sqrt { x } ( x - \lambda )\), where \(\lambda\) is a positive parameter. Find the equation of the envelope of this family of curves.

4 The twelve distinct elements of an abelian multiplicative group \(G\) are $$e , a , a ^ { 2 } , a ^ { 3 } , a ^ { 4 } , a ^ { 5 } , b , a b , a ^ { 2 } b , a ^ { 3 } b , a ^ { 4 } b , a ^ { 5 } b$$ where \(e\) is the identity element, \(a ^ { 6 } = e\) and \(b ^ { 2 } = e\).

1. Show that the element \(a ^ { 2 } b\) has order 6 .
2. Show that \(\left\{ e , a ^ { 3 } , b , a ^ { 3 } b \right\}\) is a subgroup of \(G\).
3. List all the cyclic subgroups of \(G\). You are given that the set $$H = \{ 1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89 \}$$ with binary operation multiplication modulo 90 is a group.
4. Determine the order of each of the elements 11, 17 and 19 .
5. Give a cyclic subgroup of \(H\) with order 4.
6. By identifying possible values for the elements \(a\) and \(b\) above, or otherwise, give one example of each of the following:
   (A) a non-cyclic subgroup of \(H\) with order 12,
   (B) a non-cyclic subgroup of \(H\) with order 4.

5 In this question, give probabilities correct to 4 decimal places.
The speeds of vehicles are measured on a busy stretch of road and are categorised as A (not more than 30 mph ), B (more than 30 mph but not more than 40 mph ) or C (more than 40 mph ).

• Following a vehicle in category A , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.9,0.07,0.03\) respectively.
• Following a vehicle in category B , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.3,0.6,0.1\) respectively.
• Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.

This is modelled as a Markov chain with three states corresponding to the categories A, B, C. The speed of the first vehicle is measured as 28 mph .

1. Write down the transition matrix \(\mathbf { P }\).
2. Find the probabilities that the 10th vehicle is in each of the three categories.
3. Find the probability that the 12th and 13th vehicles are in the same category.
4. Find the smallest value of \(n\) for which the probability that the \(n\)th and \(( n + 1 )\) th vehicles are in the same category is less than 0.8, and give the value of this probability.
5. Find the expected number of vehicles (including the first vehicle) in category A before a vehicle in a different category.
6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity, and hence write down the equilibrium probabilities for the three categories.
7. Find the probability that, after many vehicles have passed by, the next three vehicles are all in category A. On a new stretch of road, the same categories are used but some of the transition probabilities are different.
   • Following a vehicle in category A , the probability that the next vehicle is in category B is equal to the probability that it is in category C .
   • Following a vehicle in category B , the probability that the next vehicle is in category A is equal to the probability that it is in category C .
   • Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.
   In the long run, the proportions of vehicles in categories A, B, C are 50\%, 40\%, 10\% respectively.
8. Find the transition matrix for the new stretch of road.