Questions FP3 - A-Level Maths

OCR MEI FP3 2014 June Q5

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 .

Write down the transition matrix $\mathbf { P }$.
Find the probabilities that the 10th vehicle is in each of the three categories.
Find the probability that the 12th and 13th vehicles are in the same category.
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.
Find the expected number of vehicles (including the first vehicle) in category A before a vehicle in a different category.
Find the limit of $\mathbf { P } ^ { n }$ as $n$ tends to infinity, and hence write down the equilibrium probabilities for the three categories.
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.
Find the transition matrix for the new stretch of road.

OCR MEI FP3 2009 June Q1

1 The point $\mathrm { A } ( - 1,12,5 )$ lies on the plane $P$ with equation $8 x - 3 y + 10 z = 6$. The point $\mathrm { B } ( 6 , - 2,9 )$ lies on the plane $Q$ with equation $3 x - 4 y - 2 z = 8$. The planes $P$ and $Q$ intersect in the line $L$.

Find an equation for the line $L$.
Find the shortest distance between $L$ and the line AB . The lines $M$ and $N$ are both parallel to $L$, with $M$ passing through A and $N$ passing through B .
Find the distance between the parallel lines $M$ and $N$. The point C has coordinates $( k , 0,2 )$, and the line AC intersects the line $N$ at the point D .
Find the value of $k$, and the coordinates of D .

OCR MEI FP3 2009 June Q2

2 A surface has equation $z = 3 x ( x + y ) ^ { 3 } - 2 x ^ { 3 } + 24 x$.

Find $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$.
Find the coordinates of the three stationary points on the surface.
Find the equation of the normal line at the point $\mathrm { P } ( 1 , - 2,19 )$ on the surface.
The point $\mathrm { Q } ( 1 + k , - 2 + h , 19 + 3 h )$ is on the surface and is close to P . Find an approximate expression for $k$ in terms of $h$.
Show that there is only one point on the surface at which the tangent plane has an equation of the form $27 x - z = d$. Find the coordinates of this point and the corresponding value of $d$.

OCR MEI FP3 2009 June Q3

3 A curve has parametric equations $x = a ( \theta + \sin \theta ) , y = a ( 1 - \cos \theta )$, for $0 \leqslant \theta \leqslant \pi$, where $a$ is a positive constant.

Show that the arc length $s$ from the origin to a general point on the curve is given by $s = 4 a \sin \frac { 1 } { 2 } \theta$.
Find the intrinsic equation of the curve giving $s$ in terms of $a$ and $\psi$, where $\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence, or otherwise, show that the radius of curvature at a point on the curve is $4 a \cos \frac { 1 } { 2 } \theta$.
Find the coordinates of the centre of curvature corresponding to the point on the curve where $\theta = \frac { 2 } { 3 } \pi$.
Find the area of the surface generated when the curve is rotated through $2 \pi$ radians about the $x$-axis.

OCR MEI FP3 2009 June Q4

4 The group $G = \{ 1,2,3,4,5,6 \}$ has multiplication modulo 7 as its operation. The group $H = \{ 1,5,7,11,13,17 \}$ has multiplication modulo 18 as its operation.

Show that the groups $G$ and $H$ are both cyclic.
List all the proper subgroups of $G$.
Specify an isomorphism between $G$ and $H$. The group $S = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } , \mathrm { f } \}$ consists of functions with domain $\{ 1,2,3 \}$ given by $$\begin{array} { l l l } \mathrm { a } ( 1 ) = 2 & \mathrm { a } ( 2 ) = 3 & \mathrm { a } ( 3 ) = 1
\mathrm {~b} ( 1 ) = 3 & \mathrm {~b} ( 2 ) = 1 & \mathrm {~b} ( 3 ) = 2
\mathrm { c } ( 1 ) = 1 & \mathrm { c } ( 2 ) = 3 & \mathrm { c } ( 3 ) = 2
\mathrm {~d} ( 1 ) = 3 & \mathrm {~d} ( 2 ) = 2 & \mathrm {~d} ( 3 ) = 1
\mathrm { e } ( 1 ) = 1 & \mathrm { e } ( 2 ) = 2 & \mathrm { e } ( 3 ) = 3
\mathrm { f } ( 1 ) = 2 & \mathrm { f } ( 2 ) = 1 & \mathrm { f } ( 3 ) = 3 \end{array}$$ and the group operation is composition of functions.
Show that ad $= \mathrm { c }$ and find da.
Give a reason why $S$ is not isomorphic to $G$.
Find the order of each element of $S$.
List all the proper subgroups of $S$.

OCR MEI FP3 2009 June Q5

5 Each level of a fantasy computer game is set in a single location, Alphaworld, Betaworld, Chiworld or Deltaworld. After completing a level, a player goes on to the next level, which could be set in the same location as the previous level, or in a different location. In the first version of the game, the initial and transition probabilities are as follows.
Level 1 is set in Alphaworld or Betaworld, with probabilities 0.6, 0.4 respectively.
After a level set in Alphaworld, the next level will be set in Betaworld, Chiworld or Deltaworld, with probabilities $0.7,0.1,0.2$ respectively.
After a level set in Betaworld, the next level will be set in Alphaworld, Betaworld or Deltaworld, with probabilities $0.1,0.8,0.1$ respectively.
After a level set in Chiworld, the next level will also be set in Chiworld.
After a level set in Deltaworld, the next level will be set in Alphaworld, Betaworld or Chiworld, with probabilities $0.3,0.6,0.1$ respectively. The situation is modelled as a Markov chain with four states.

Write down the transition matrix.
Find the probabilities that level 14 is set in each location.
Find the probability that level 15 is set in the same location as level 14 .
Find the level at which the probability of being set in Chiworld first exceeds 0.5.
Following a level set in Betaworld, find the expected number of further levels which will be set in Betaworld before changing to a different location. In the second version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are all the same as in the first version; but after a level set in Chiworld, the next level will be set in Chiworld or Deltaworld, with probabilities $0.9,0.1$ respectively.
By considering powers of the new transition matrix, or otherwise, find the equilibrium probabilities for the four locations. In the third version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are again all the same as in the first version; but the transition probabilities after Chiworld have changed again. The equilibrium probabilities for Alphaworld, Betaworld, Chiworld and Deltaworld are now 0.11, 0.75, 0.04, 0.1 respectively.
Find the new transition probabilities after a level set in Chiworld. 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, is given to all schools that receive assessment material 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 FP3 2009 January Q1

1 In this question $G$ is a group of order $n$, where $3 \leqslant n < 8$.

In each case, write down the smallest possible value of $n$ :
(a) if $G$ is cyclic,
(b) if $G$ has a proper subgroup of order 3,
(c) if $G$ has at least two elements of order 2 .
Another group has the same order as $G$, but is not isomorphic to $G$. Write down the possible value(s) of $n$.

OCR FP3 2009 January Q2

2

Express $\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $0 \leqslant \theta < 2 \pi$.
Hence find the smallest positive value of $n$ for which $\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }$ is real and positive.

OCR FP3 2009 January Q3

3 Two skew lines have equations $$\frac { x } { 2 } = \frac { y + 3 } { 1 } = \frac { z - 6 } { 3 } \quad \text { and } \quad \frac { x - 5 } { 3 } = \frac { y + 1 } { 1 } = \frac { z - 7 } { 5 } .$$

Find the direction of the common perpendicular to the lines.
Find the shortest distance between the lines.

OCR FP3 2009 January Q4

4 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$

OCR FP3 2009 January Q5

5 The variables $x$ and $y$ are related by the differential equation $$x ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x y + x + 1 .$$

Use the substitution $y = u - \frac { 1 } { x }$, where $u$ is a function of $x$, to show that the differential equation may be written as $$x ^ { 2 } \frac { \mathrm {~d} u } { \mathrm {~d} x } = u .$$
Hence find the general solution of the differential equation (A), giving your answer in the form $y = \mathrm { f } ( x )$.

OCR FP3 2009 January Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{bc975428-c594-427b-a32e-268412b3cd26-3_554_825_264_660} The cuboid $O A B C D E F G$ shown in the diagram has $\overrightarrow { O A } = 4 \mathbf { i } , \overrightarrow { O C } = 2 \mathbf { j } , \overrightarrow { O D } = 3 \mathbf { k }$, and $M$ is the mid-point of $G F$.

Find the equation of the plane $A C G E$, giving your answer in the form r.n $= p$.
The plane $O E F C$ has equation $\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { k } ) = 0$. Find the acute angle between the planes $O E F C$ and $A C G E$.
The line $A M$ meets the plane $O E F C$ at the point $W$. Find the ratio $A W : W M$.

OCR FP3 2009 January Q7

7

The operation $*$ is defined by $x * y = x + y - a$, where $x$ and $y$ are real numbers and $a$ is a real constant.
(a) Prove that the set of real numbers, together with the operation $*$, forms a group.
(b) State, with a reason, whether the group is commutative.
(c) Prove that there are no elements of order 2.
The operation $\circ$ is defined by $x \circ y = x + y - 5$, where $x$ and $y$ are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied.

OCR FP3 2009 January Q8

8

By expressing $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$, show that $$\sin ^ { 6 } \theta \equiv - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )$$
Replace $\theta$ by ( $\frac { 1 } { 2 } \pi - \theta$ ) in the identity in part (i) to obtain a similar identity for $\cos ^ { 6 } \theta$.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sin ^ { 6 } \theta - \cos ^ { 6 } \theta \right) \mathrm { d } \theta$.

OCR FP3 2010 January Q1

1 Determine whether the lines $$\frac { x - 1 } { 1 } = \frac { y + 2 } { - 1 } = \frac { z + 4 } { 2 } \quad \text { and } \quad \frac { x + 3 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 5 } { 4 }$$ intersect or are skew.
$2 \quad H$ denotes the set of numbers of the form $a + b \sqrt { 5 }$, where $a$ and $b$ are rational. The numbers are combined under multiplication.

Show that the product of any two members of $H$ is a member of $H$. It is now given that, for $a$ and $b$ not both zero, $H$ forms a group under multiplication.
State the identity element of the group.
Find the inverse of $a + b \sqrt { 5 }$.
With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse.

OCR FP3 2010 January Q3

3 Use the integrating factor method to find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \mathrm { e } ^ { - 3 x }$$ for which $y = 1$ when $x = 0$. Express your answer in the form $y = \mathrm { f } ( x )$.

OCR FP3 2010 January Q4

4

Write down, in cartesian form, the roots of the equation $z ^ { 4 } = 16$.
Hence solve the equation $w ^ { 4 } = 16 ( 1 - w ) ^ { 4 }$, giving your answers in cartesian form.

OCR FP3 2010 January Q5

5 A regular tetrahedron has vertices at the points $$A \left( 0,0 , \frac { 2 } { 3 } \sqrt { 6 } \right) , \quad B \left( \frac { 2 } { 3 } \sqrt { 3 } , 0,0 \right) , \quad C \left( - \frac { 1 } { 3 } \sqrt { 3 } , 1,0 \right) , \quad D \left( - \frac { 1 } { 3 } \sqrt { 3 } , - 1,0 \right) .$$

Obtain the equation of the face $A B C$ in the form $$x + \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$ (Answers which only verify the given equation will not receive full credit.)
Give a geometrical reason why the equation of the face $A B D$ can be expressed as $$x - \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$
Hence find the cosine of the angle between two faces of the tetrahedron.

OCR FP3 2010 January Q6

6 The variables $x$ and $y$ satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 y = 8 \cos 4 x$$

Find the complementary function of the differential equation.
Given that there is a particular integral of the form $y = p x \sin 4 x$, where $p$ is a constant, find the general solution of the equation.
Find the solution of the equation for which $y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$.

OCR FP3 2010 January Q7

7

Solve the equation $\cos 6 \theta = 0$, for $0 < \theta < \pi$.
By using de Moivre's theorem, show that $$\cos 6 \theta \equiv \left( 2 \cos ^ { 2 } \theta - 1 \right) \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 1 \right)$$
Hence find the exact value of $$\cos \left( \frac { 1 } { 12 } \pi \right) \cos \left( \frac { 5 } { 12 } \pi \right) \cos \left( \frac { 7 } { 12 } \pi \right) \cos \left( \frac { 11 } { 12 } \pi \right)$$ justifying your answer.

OCR FP3 2010 January Q8

8 The function f is defined by $\mathrm { f } : x \mapsto \frac { 1 } { 2 - 2 x }$ for $x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1$. The function g is defined by $\mathrm { g } ( x ) = \mathrm { ff } ( x )$.

Show that $\mathrm { g } ( x ) = \frac { 1 - x } { 1 - 2 x }$ and that $\operatorname { gg } ( x ) = x$. It is given that f and g are elements of a group $K$ under the operation of composition of functions. The element e is the identity, where e : $x \mapsto x$ for $x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1$.
State the orders of the elements f and g .
The inverse of the element f is denoted by h . Find $\mathrm { h } ( x )$.
Construct the operation table for the elements e, f, g, h of the group $K$.

OCR FP3 2011 January Q1

1

Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = x \mathrm { e } ^ { \frac { 1 } { 2 } x ^ { 2 } }$$ giving your answer in the form $y = \mathrm { f } ( x )$.
Find the particular solution for which $y = 1$ when $x = 0$.

OCR FP3 2011 January Q2

2 Two intersecting lines, lying in a plane $p$, have equations $$\frac { x - 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z - 4 } { - 3 } \quad \text { and } \quad \frac { x - 1 } { - 1 } = \frac { y - 3 } { 2 } = \frac { z - 4 } { 4 } .$$

Obtain the equation of $p$ in the form $2 x - y + z = 3$.
Plane $q$ has equation $2 x - y + z = 21$. Find the distance between $p$ and $q$.

OCR FP3 2011 January Q3

3

Express $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$ and show that $$\sin ^ { 4 } \theta \equiv \frac { 1 } { 8 } ( \cos 4 \theta - 4 \cos 2 \theta + 3 )$$
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta$.

OCR FP3 2011 January Q4

4 The cube roots of 1 are denoted by $1 , \omega$ and $\omega ^ { 2 }$, where the imaginary part of $\omega$ is positive.

Show that $1 + \omega + \omega ^ { 2 } = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{d12573dd-c0c2-4f0d-8e49-8fdf8d5864a5-2_616_748_1676_699} In the diagram, $A B C$ is an equilateral triangle, labelled anticlockwise. The points $A , B$ and $C$ represent the complex numbers $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ respectively.
State the geometrical effect of multiplication by $\omega$ and hence explain why $z _ { 1 } - z _ { 3 } = \omega \left( z _ { 3 } - z _ { 2 } \right)$.
Hence show that $z _ { 1 } + \omega z _ { 2 } + \omega ^ { 2 } z _ { 3 } = 0$.

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