Questions - OCR MEI FP3 - A-Level Maths

Find the vector product $\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }$.
For the case when AB is parallel to CD ,
(A) state the value of $k$,
(B) find the shortest distance between the parallel lines AB and CD ,
(C) find, in the form $a x + b y + c z + d = 0$, the equation of the plane containing AB and CD .
When AB is not parallel to CD , find the shortest distance between the lines AB and CD , in terms of $k$.
Find the value of $k$ for which the line AB intersects the line CD , and find the coordinates of the point of intersection in this case.

OCR MEI FP3 2006 June Q2

2 A surface has equation $x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0$.

Find a normal vector at the point $( x , y , z )$ on the surface.
Find the equation of the tangent plane to the surface at the point $\mathrm { Q } ( 17,4,1 )$.
The point $( 17 + h , 4 + p , 1 - h )$, where $h$ and $p$ are small, is on the surface and is close to Q . Find an approximate expression for $p$ in terms of $h$.
Show that there is no point on the surface where the normal line is parallel to the $z$-axis.
Find the two values of $k$ for which $5 x - 6 y + 2 z = k$ is a tangent plane to the surface.

OCR MEI FP3 2006 June Q3

3 The curve $C$ has parametric equations $x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }$.

Find the length of the arc of $C$ for which $0 \leqslant t \leqslant 1$.
Find the area of the surface generated when the arc of $C$ for which $0 \leqslant t \leqslant 1$ is rotated through $2 \pi$ radians about the $x$-axis.
Show that the equation of the normal to $C$ at the point with parameter $t$ is $$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$
Find the cartesian equation of the envelope of the normals to $C$.
The point $\mathrm { P } ( 64 , a )$ is the centre of curvature corresponding to a point on $C$. Find $a$.

OCR MEI FP3 2006 June Q4

$\mathbf { 4 }$ The group $G$ consists of the 8 complex matrices $\{ \mathbf { I } , \mathbf { J } , \mathbf { K } , \mathbf { L } , - \mathbf { I } , - \mathbf { J } , - \mathbf { K } , - \mathbf { L } \}$ under matrix multiplication, where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right) , \quad \mathbf { J } = \left( \begin{array} { r r } \mathrm { j } & 0
0 & - \mathrm { j } \end{array} \right) , \quad \mathbf { K } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right) , \quad \mathbf { L } = \left( \begin{array} { c c } 0 & \mathrm { j }
\mathrm { j } & 0 \end{array} \right)$$

Copy and complete the following composition table for $G$.
I J K L -I -J -K $- \mathbf { L }$
I I J K L -I -J -K -L
J J -I L -K -J I -L K
K K -L -I
L L K
-I -I -J
-J -J I
-K -K L
-L -L -K
(Note that $\mathbf { J K } = \mathbf { L }$ and $\mathbf { K J } = - \mathbf { L }$.)
State the inverse of each element of $G$.
Find the order of each element of $G$.
Explain why, if $G$ has a subgroup of order 4, that subgroup must be cyclic.
Find all the proper subgroups of $G$.
Show that $G$ is not isomorphic to the group of symmetries of a square.

OCR MEI FP3 2006 June Q5

5 A local hockey league has three divisions. Each team in the league plays in a division for a year. In the following year a team might play in the same division again, or it might move up or down one division. This question is about the progress of one particular team in the league. In 2007 this team will be playing in either Division 1 or Division 2. Because of its present position, the probability that it will be playing in Division 1 is 0.6 , and the probability that it will be playing in Division 2 is 0.4 . The following transition probabilities apply to this team from 2007 onwards.

When the team is playing in Division 1, the probability that it will play in Division 2 in the following year is 0.2 .
When the team is playing in Division 2, the probability that it will play in Division 1 in the following year is 0.1 , and the probability that it will play in Division 3 in the following year is 0.3 .
When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 .

This process is modelled as a Markov chain with three states corresponding to the three divisions.

Write down the transition matrix.
Determine in which division the team is most likely to be playing in 2014.
Find the equilibrium probabilities for each division for this team. In 2015 the rules of the league are changed. A team playing in Division 3 might now be dropped from the league in the following year. Once dropped, a team does not play in the league again.
-The transition probabilities from Divisions 1 and 2 remain the same as before.
- When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 , and the probability that it will be dropped from the league is 0.1 .
The team plays in Division 2 in 2015.
The new situation is modelled as a Markov chain with four states: 'Division1', 'Division 2', 'Division 3' and 'Out of league'.
Write down the transition matrix which applies from 2015.
Find the probability that the team is still playing in the league in 2020.
Find the first year for which the probability that the team is out of the league is greater than 0.5 .

OCR MEI FP3 2008 June Q1

1 A tetrahedron ABCD has vertices $\mathrm { A } ( - 3,5,2 ) , \mathrm { B } ( 3,13,7 ) , \mathrm { C } ( 7,0,3 )$ and $\mathrm { D } ( 5,4,8 )$.

Find the vector product $\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }$, and hence find the equation of the plane ABC .
Find the shortest distance from $D$ to the plane $A B C$.
Find the shortest distance between the lines AB and CD .
Find the volume of the tetrahedron ABCD . The plane $P$ with equation $3 x - 2 z + 5 = 0$ contains the point B , and meets the lines AC and AD at E and F respectively.
Find $\lambda$ and $\mu$ such that $\overrightarrow { \mathrm { AE } } = \lambda \overrightarrow { \mathrm { AC } }$ and $\overrightarrow { \mathrm { AF } } = \mu \overrightarrow { \mathrm { AD } }$. Deduce that E is between A and C , and that F is between A and D.
Hence, or otherwise, show that $P$ divides the tetrahedron ABCD into two parts having volumes in the ratio 4 to 17.

OCR MEI FP3 2008 June Q2

2 You are given $\mathrm { g } ( x , y , z ) = 6 x z - ( x + 2 y + 3 z ) ^ { 2 }$.

Find $\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }$ and $\frac { \partial \mathrm { g } } { \partial z }$. A surface $S$ has equation $\mathrm { g } ( x , y , z ) = 125$.
Find the equation of the normal line to $S$ at the point $\mathrm { P } ( 7 , - 7.5,3 )$.
The point Q is on this normal line and is close to P . At $\mathrm { Q } , \mathrm { g } ( x , y , z ) = 125 + h$, where $h$ is small. Find the vector $\mathbf { n }$ such that $\overrightarrow { \mathrm { PQ } } = h \mathbf { n }$ approximately.
Show that there is no point on $S$ at which the normal line is parallel to the $z$-axis.
Find the two points on $S$ at which the tangent plane is parallel to $x + 5 y = 0$.

OCR MEI FP3 2008 June Q3

3 The curve $C$ has parametric equations $x = 8 t ^ { 3 } , y = 9 t ^ { 2 } - 2 t ^ { 4 }$, for $t \geqslant 0$.

Show that $\dot { x } ^ { 2 } + \dot { y } ^ { 2 } = \left( 18 t + 8 t ^ { 3 } \right) ^ { 2 }$. Find the length of the arc of $C$ for which $0 \leqslant t \leqslant 2$.
Find the area of the surface generated when the arc of $C$ for which $0 \leqslant t \leqslant 2$ is rotated through $2 \pi$ radians about the $x$-axis.
Show that the curvature at a general point on $C$ is $\frac { - 6 } { t \left( 4 t ^ { 2 } + 9 \right) ^ { 2 } }$.
Find the coordinates of the centre of curvature corresponding to the point on $C$ where $t = 1$.

OCR MEI FP3 2008 June Q4

4 A binary operation * is defined on real numbers $x$ and $y$ by $$x * y = 2 x y + x + y$$ You may assume that the operation $*$ is commutative and associative.

Explain briefly the meanings of the terms 'commutative' and 'associative'.
Show that $x * y = 2 \left( x + \frac { 1 } { 2 } \right) \left( y + \frac { 1 } { 2 } \right) - \frac { 1 } { 2 }$. The set $S$ consists of all real numbers greater than $- \frac { 1 } { 2 }$.
(A) Use the result in part (ii) to show that $S$ is closed under the operation *.
(B) Show that $S$, with the operation $*$, is a group.
Show that $S$ contains no element of order 2 . The group $G = \{ 0,1,2,4,5,6 \}$ has binary operation ∘ defined by $$x \circ y \text { is the remainder when } x * y \text { is divided by } 7 \text {. }$$
Show that $4 \circ 6 = 2$. The composition table for $G$ is as follows.
$\circ$ 0 1 2 4 5 6
0 0 1 2 4 5 6
1 1 4 0 6 2 5
2 2 0 5 1 6 4
4 4 6 1 5 0 2
5 5 2 6 0 4 1
6 6 5 4 2 1 0
Find the order of each element of $G$.
List all the subgroups of $G$.

OCR MEI FP3 2008 June Q5

5 Every day, a security firm transports a large sum of money from one bank to another. There are three possible routes $A , B$ and $C$. The route to be used is decided just before the journey begins, by a computer programmed as follows. On the first day, each of the three routes is equally likely to be used.
If route $A$ was used on the previous day, route $A$, $B$ or $C$ will be used, with probabilities $0.1,0.4,0.5$ respectively.
If route $B$ was used on the previous day, route $A , B$ or $C$ will be used, with probabilities $0.7,0.2,0.1$ respectively.
If route $C$ was used on the previous day, route $A , B$ or $C$ will be used, with probabilities $0.1,0.6,0.3$ respectively. The situation is modelled as a Markov chain with three states.

Write down the transition matrix $\mathbf { P }$.
Find the probability that route $B$ is used on the 7th day.
Find the probability that the same route is used on the 7th and 8th days.
Find the probability that the route used on the 10th day is the same as that used on the 7th day.
Given that $\mathbf { P } ^ { n } \rightarrow \mathbf { Q }$ as $n \rightarrow \infty$, find the matrix $\mathbf { Q }$ (give the elements to 4 decimal places). Interpret the probabilities which occur in the matrix $\mathbf { Q }$. The computer program is now to be changed, so that the long-run probabilities for routes $A , B$ and $C$ will become $0.4,0.2$ and 0.4 respectively. The transition probabilities after routes $A$ and $B$ remain the same as before.
Find the new transition probabilities after route $C$.
A long time after the change of program, a day is chosen at random. Find the probability that the route used on that day is the same as on the previous day. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

OCR MEI FP3 2010 June Q1

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.

Find the perpendicular distance from C to the line AB .
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 $$\frac { 21 | p - 5 | } { \sqrt { 17 p ^ { 2 } - 2 p + 26 } }$$
Find, in terms of $p$, the volume of the tetrahedron ABCD .
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

OCR MEI FP3 2010 June Q2

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 $\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }$.

Find $\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }$ and $\frac { \partial \mathrm { g } } { \partial z }$.
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 $\mathrm { g } ( x , y , z ) = 0$.
Show that one of these points is $\mathrm { P } ( 0,3,0 )$, and find the coordinates of the other point Q .
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.
There is a point on $L$, close to P , at which $\mathrm { g } ( x , y , z ) = h$. Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
Find the approximate coordinates of the point on $L$, close to Q , at which $\mathrm { g } ( x , y , z ) = h$.

OCR MEI FP3 2010 June Q3

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

Show that the arc of $C$ for which $0 \leqslant x \leqslant a$ has length $a ^ { \frac { 1 } { 2 } } + \frac { 1 } { 3 } a ^ { \frac { 3 } { 2 } }$.
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.
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 $p > 0$ ).
Find the equation of the envelope of this family of curves.

OCR MEI FP3 2010 June Q4

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.

Show that st $= \mathrm { r }$ and find ts.
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
Give the inverse of each element of $F$.
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.
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.
Show that $G$ is a cyclic group which can be generated by the element 2 .
Explain why $G$ has no subgroup which is isomorphic to $F$.
Find a subgroup of $G$ which is isomorphic to $M$.

OCR MEI FP3 2012 June Q1

1 A mine contains several underground tunnels beneath a hillside. The hillside is a plane, all the tunnels are straight and the width of the tunnels may be neglected. A coordinate system is chosen with the $z$-axis pointing vertically upwards and the units are metres. Three points on the hillside have coordinates $\mathrm { A } ( 15 , - 60,20 )$, $B ( - 75,100,40 )$ and $C ( 18,138,35.6 )$.

Find the vector product $\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }$ and hence show that the equation of the hillside is $2 x - 2 y + 25 z = 650$. The tunnel $T _ { \mathrm { A } }$ begins at A and goes in the direction of the vector $15 \mathbf { i } + 14 \mathbf { j } - 2 \mathbf { k }$; the tunnel $T _ { \mathrm { C } }$ begins at C and goes in the direction of the vector $8 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }$. Both these tunnels extend a long way into the ground.
Find the least possible length of a tunnel which connects B to a point in $T _ { \mathrm { A } }$.
Find the least possible length of a tunnel which connects a point in $T _ { \mathrm { A } }$ to a point in $T _ { \mathrm { C } }$.
A tunnel starts at B , passes through the point ( $18,138 , p$ ) vertically below C , and intersects $T _ { \mathrm { A } }$ at the point Q . Find the value of $p$ and the coordinates of Q .

OCR MEI FP3 2012 June Q2

2 You are given that $\mathrm { g } ( x , y , z ) = x ^ { 2 } + 2 y ^ { 2 } - z ^ { 2 } + 2 x z + 2 y z + 4 z - 3$.

Find $\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }$ and $\frac { \partial \mathrm { g } } { \partial z }$. The surface $S$ has equation $\mathrm { g } ( x , y , z ) = 0$, and $\mathrm { P } ( - 2 , - 1,1 )$ is a point on $S$.
Find an equation for the normal line to the surface $S$ at the point P .
A point Q lies on this normal line and is close to P . At $\mathrm { Q } , \mathrm { g } ( x , y , z ) = h$, where $h$ is small. Find the constant $c$ such that $\mathrm { PQ } \approx c | h |$.
Show that there is no point on $S$ at which the normal line is parallel to the $z$-axis.
Given that $x + y + z = k$ is a tangent plane to the surface $S$, find the two possible values of $k$.

OCR MEI FP3 2012 June Q3

3 A curve has parametric equations $$x = a \left( 1 - \cos ^ { 3 } \theta \right) , \quad y = a \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ where $a$ is a positive constant.
The arc length from the origin to a general point on the curve is denoted by $s$, and $\psi$ is the acute angle defined by $\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }$.

Express $s$ and $\psi$ in terms of $\theta$, and hence show that the intrinsic equation of the curve is $$s = \frac { 3 } { 2 } a \sin ^ { 2 } \psi$$
For the point on the curve given by $\theta = \frac { \pi } { 6 }$, find the radius of curvature and the coordinates of the centre of curvature.
Find the area of the curved surface generated when the curve is rotated through $2 \pi$ radians about the $y$-axis.

OCR MEI FP3 2012 June Q4

Show that the set $P = \{ 1,5,7,11 \}$, under the binary operation of multiplication modulo 12, is a group. You may assume associativity. A group $Q$ has identity element $e$. The result of applying the binary operation of $Q$ to elements $x$ and $y$ is written $x y$, and the inverse of $x$ is written $x ^ { - 1 }$.
Verify that the inverse of $x y$ is $y ^ { - 1 } x ^ { - 1 }$. Three elements $a , b$ and $c$ of $Q$ all have order 2, and $a b = c$.
By considering the inverse of $c$, or otherwise, show that $b a = c$.
Show that $b c = a$ and $a c = b$. Find $c b$ and $c a$.

Complete the composition table for $R = \{ e , a , b , c \}$. Hence show that $R$ is a subgroup of $Q$ and that $R$ is isomorphic to $P$. The group $T$ of symmetries of a square contains four reflections $A , B , C , D$, the identity transformation $E$ and three rotations $F , G , H$. The binary operation is composition of transformations. The composition table for $T$ is given below.

	A	$B$	$C$	D	$E$	$F$	$G$	$H$
A	E	$G$	$H$	$F$	$A$	D	$B$	$C$
B	G	E	$F$	$H$	$B$	C	A	D
C	$F$	H	E	G	C	A	D	$B$
D	$H$	$F$	$G$	E	$D$	$B$	C	$A$
E	A	$B$	C	D	$E$	$F$	$G$	$H$
F	C	D	$B$	A	$F$	G	$H$	$E$
$G$	B	$A$	$D$	C	$G$	H	E	$F$
$H$	D	C	A	B	$H$	E	$F$	G

Find the order of each element of $T$.
List all the proper subgroups of $T$.

OCR MEI FP3 2013 June Q1

1 Three points have coordinates $\mathrm { A } ( 3,2,10 ) , \mathrm { B } ( 11,0 , - 3 ) , \mathrm { C } ( 5,18,0 )$, and $L$ is the straight line through A with equation $$\frac { x - 3 } { - 1 } = \frac { y - 2 } { 4 } = \frac { z - 10 } { 1 }$$

Find the shortest distance between the lines $L$ and BC .
Find the shortest distance from A to the line BC . A straight line passes through B and the point $\mathrm { P } ( 5,18 , k )$, and intersects the line $L$.
Find $k$, and the point of intersection of the lines BP and $L$. The point D is on the line $L$, and AD has length 12 .
Find the volume of the tetrahedron ABCD .

OCR MEI FP3 2013 June Q2

2 A surface has equation $z = 2 \left( x ^ { 3 } + y ^ { 3 } \right) + 3 \left( x ^ { 2 } + y ^ { 2 } \right) + 12 x y$.

For a point on the surface at which $\frac { \partial z } { \partial x } = \frac { \partial z } { \partial y }$, show that either $y = x$ or $y = 1 - x$.
Show that there are exactly two stationary points on the surface, and find their coordinates.
The point $\mathrm { P } \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , 5 \right)$ is on the surface, and $\mathrm { Q } \left( \frac { 1 } { 2 } + h , \frac { 1 } { 2 } + h , 5 + w \right)$ is a point on the surface close to P . Find an approximate expression for $h$ in terms of $w$.
Find the four points on the surface at which the normal line is parallel to the vector $24 \mathbf { i } + 24 \mathbf { j } - \mathbf { k }$.

OCR MEI FP3 2013 June Q3

Find the length of the arc of the polar curve $r = a ( 1 + \cos \theta )$ for which $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
A curve $C$ has cartesian equation $y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x }$.
1. The arc of $C$ for which $1 \leqslant x \leqslant 2$ is rotated through $2 \pi$ radians about the $x$-axis to form a surface of revolution. Find the area of this surface. For the point on $C$ at which $x = 2$,
2. show that the radius of curvature is $\frac { 289 } { 64 }$,
3. find the coordinates of the centre of curvature.

OCR MEI FP3 2013 June Q4

The composition table for a group $G$ of order 8 is given below.

	$a$	$b$	$c$	$d$	$e$	$f$	$g$	$h$
$a$	$c$	$e$	$b$	$f$	$a$	$h$	$d$	$g$
$b$	$e$	$c$	$a$	$g$	$b$	$d$	h	$f$
$c$	$b$	$a$	$e$	$h$	$c$	$g$	$f$	$d$
$d$	$f$	$g$	$h$	$a$	$d$	$c$	$e$	$b$
$e$	$a$	$b$	$c$	$d$	$e$	$f$	$g$	$h$
$f$	$h$	$d$	$g$	$c$	$f$	$b$	$a$	$e$
$g$	$d$	$h$	$f$	$e$	$g$	$a$	$b$	$c$
$h$	$g$	$f$	$d$	$b$	$h$	$e$	$c$	$a$

State which is the identity element, and give the inverse of each element of $G$.
Show that $G$ is cyclic.
Specify an isomorphism between $G$ and the group $H$ consisting of $\{ 0,2,4,6,8,10,12,14 \}$ under addition modulo 16 .
Show that $G$ is not isomorphic to the group of symmetries of a square.

The set $S$ consists of the functions $\mathrm { f } _ { n } ( x ) = \frac { x } { 1 + n x }$, for all integers $n$, and the binary operation is composition of functions.
1. Show that $\mathrm { f } _ { m } \mathrm { f } _ { n } = \mathrm { f } _ { m + n }$.
2. Hence show that the binary operation is associative.
3. Prove that $S$ is a group.
4. Describe one subgroup of $S$ which contains more than one element, but which is not the whole of $S$.

OCR MEI FP3 2013 June Q5

5 In this question, give probabilities correct to 4 decimal places.
A contestant in a game-show starts with one, two or three 'lives', and then performs a series of tasks. After each task, the number of lives either decreases by one, or remains the same, or increases by one. The game ends when the number of lives becomes either four or zero. If the number of lives is four, the contestant wins a prize; if the number of lives is zero, the contestant loses and leaves with nothing. At the start, the number of lives is decided at random, so that the contestant is equally likely to start with one, two or three lives. The tasks do not involve any skill, and after every task:

the probability that the number of lives decreases by one is 0.5 ,
the probability that the number of lives remains the same is 0.05 ,
the probability that the number of lives increases by one is 0.45 .

This is modelled as a Markov chain with five states corresponding to the possible numbers of lives. The states corresponding to zero lives and four lives are absorbing states.

Write down the transition matrix $\mathbf { P }$.
Show that, after 8 tasks, the probability that the contestant has three lives is 0.0207 , correct to 4 decimal places.
Find the probability that, after 10 tasks, the game has not yet ended.
Find the probability that the game ends after exactly 10 tasks.
Find the smallest value of $N$ for which the probability that the game has not yet ended after $N$ tasks is less than 0.01 .
Find the limit of $\mathbf { P } ^ { n }$ as $n$ tends to infinity.
Find the probability that the contestant wins a prize. The beginning of the game is now changed, so that the probabilities of starting with one, two or three lives can be adjusted.
State the maximum possible probability that the contestant wins a prize, and how this can be achieved.
Given that the probability of starting with one life is 0.1 , and the probability of winning a prize is 0.6 , find the probabilities of starting with two lives and starting with three lives. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR MEI FP3 2014 June Q1

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 .

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 $L$.
Find the distance AD .
Find an equation for the line $L$.
Find the shortest distance from A to the line $L$.
Find the volume of the tetrahedron ABCD .

OCR MEI FP3 2014 June Q2

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 $S$.

Find $\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }$ and $\frac { \partial \mathrm { g } } { \partial z }$.
Find the equation of the normal line to the surface $S$ at the point P .
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 $S$.
Find the coordinates of the two points on the surface at which the normal line is parallel to the $y$-axis.
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.

	A	\(B\)	\(C\)	D	\(E\)	\(F\)	\(G\)	\(H\)
A	E	\(G\)	\(H\)	\(F\)	\(A\)	D	\(B\)	\(C\)
B	G	E	\(F\)	\(H\)	\(B\)	C	A	D
C	\(F\)	H	E	G	C	A	D	\(B\)
D	\(H\)	\(F\)	\(G\)	E	\(D\)	\(B\)	C	\(A\)
E	A	\(B\)	C	D	\(E\)	\(F\)	\(G\)	\(H\)
F	C	D	\(B\)	A	\(F\)	G	\(H\)	\(E\)
\(G\)	B	\(A\)	\(D\)	C	\(G\)	H	E	\(F\)
\(H\)	D	C	A	B	\(H\)	E	\(F\)	G

	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)	\(f\)	\(g\)	\(h\)
\(a\)	\(c\)	\(e\)	\(b\)	\(f\)	\(a\)	\(h\)	\(d\)	\(g\)
\(b\)	\(e\)	\(c\)	\(a\)	\(g\)	\(b\)	\(d\)	h	\(f\)
\(c\)	\(b\)	\(a\)	\(e\)	\(h\)	\(c\)	\(g\)	\(f\)	\(d\)
\(d\)	\(f\)	\(g\)	\(h\)	\(a\)	\(d\)	\(c\)	\(e\)	\(b\)
\(e\)	\(a\)	\(b\)	\(c\)	\(d\)	\(e\)	\(f\)	\(g\)	\(h\)
\(f\)	\(h\)	\(d\)	\(g\)	\(c\)	\(f\)	\(b\)	\(a\)	\(e\)
\(g\)	\(d\)	\(h\)	\(f\)	\(e\)	\(g\)	\(a\)	\(b\)	\(c\)
\(h\)	\(g\)	\(f\)	\(d\)	\(b\)	\(h\)	\(e\)	\(c\)	\(a\)

	I	J	K	L	-I	-J	-K	\(- \mathbf { L }\)
I	I	J	K	L	-I	-J	-K	-L
J	J	-I	L	-K	-J	I	-L	K
K	K	-L	-I
L	L	K
-I	-I	-J
-J	-J	I
-K	-K	L
-L	-L	-K

\(\circ\)	0	1	2	4	5	6
0	0	1	2	4	5	6
1	1	4	0	6	2	5
2	2	0	5	1	6	4
4	4	6	1	5	0	2
5	5	2	6	0	4	1
6	6	5	4	2	1	0

Questions — OCR MEI FP3 (53 questions)

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