Questions - OCR MEI - A-Level Maths

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.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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 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.

OCR MEI FP3 2014 June Q3

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 .
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.

OCR MEI FP3 2014 June Q4

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 $b ^ { 2 } = e$.

Show that the element $a ^ { 2 } b$ has order 6 .
Show that $\left\{ e , a ^ { 3 } , b , a ^ { 3 } b \right\}$ is a subgroup of $G$.
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.
Determine the order of each of the elements 11, 17 and 19 .
Give a cyclic subgroup of $H$ with order 4.
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.

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 S1 Q1

1 A drug for treating a particular minor illness cures, on average, $78 \%$ of patients. Twenty people with this minor illness are selected at random and treated with the drug.

(A) Find the probability that exactly 19 patients are cured.
(B) Find the probability that at most 18 patients are cured.
(C) Find the expected number of patients who are cured.
A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the $1 \%$ significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
If the researchers had chosen to carry out the hypothesis test at the $5 \%$ significance level, what would the result have been? Justify your answer.

OCR MEI S1 Q2

2 It is known that on average 85\% of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.

(A) Find the probability that exactly 12 germinate.
(B) Find the probability that fewer than 12 germinate The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting $n$ of these seeds and sowing them. He then carries out a hypothesis test at the $1 \%$ significance level to investigate whether he is correct.
Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
In a trial with $n = 20$, Ramesh finds that 13 seeds germinate. Carry out the test.
Suppose instead that Ramesh conducts the trial with $n = 50$, and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35 , complete the test.
If $n$ is small, there is no point in carrying out the test at the $1 \%$ significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of $n$ for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer.

OCR MEI S1 Q3

3 At a dog show, three out of eleven dogs are to be selected for a national competition.

Find the number of possible selections.
Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.

OCR MEI S1 Q1

1 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.

10 customers are selected at random.
(A) Find the probability that exactly 5 of them are accessing the internet.
(B) Find the probability that at least 5 of them are accessing the internet.
(C) Find the expected number of these customers who are accessing the internet. Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
Carry out a hypothesis test at the $5 \%$ significance level to investigate whether the probability for this coffee shop is different from 0.35 . Give a reason for your choice of alternative hypothesis.
To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if $X$ has the binomial distribution with parameters $n = 200$ and $p = 0.35$, then $\mathrm { P } ( X \geqslant 90 ) = 0.0022$. Using the same hypotheses and significance level which you used in part (ii), complete this test.

OCR MEI S1 Q2

2 A manufacturer produces titanium bicycle frames. The bicycle frames are tested before use and on average $5 \%$ of them are found to be faulty. A cheaper manufacturing process is introduced and the manufacturer wishes to check whether the proportion of faulty bicycle frames has increased. A random sample of 18 bicycle frames is selected and it is found that 4 of them are faulty. Carry out a hypothesis test at the $5 \%$ significance level to investigate whether the proportion of faulty bicycle frames has increased.

OCR MEI S1 Q3

3 It is known that $25 \%$ of students in a particular city are smokers. A random sample of 20 of the students is selected.

(A) Find the probability that there are exactly 4 smokers in the sample.
(B) Find the probability that there are at least 3 but no more than 6 smokers in the sample
(C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
(A) Write down suitable null and alternative hypotheses for the test.
(B) Explain why the alternative hypothesis has the form that it does
Find the critical region for the test at the $5 \%$ level, showing all of your calculations.
In fact there are 3 smokers in the sample. Complete the test, stating your conclusion clearly.

OCR MEI S1 Q1

1 Any patient who fails to turn up for an outpatient appointment at a hospital is described as a 'no-show'. At a particular hospital, on average $15 \%$ of patients are no-shows. A random sample of 20 patients who have outpatient appointments is selected.

Find the probability that
(A) there is exactly 1 no-show in the sample,
(B) there are at least 2 no-shows in the sample. The hospital management introduces a policy of telephoning patients before appointments. It is hoped that this will reduce the proportion of no-shows. In order to check this, a random sample of $n$ patients is selected. The number of no-shows in the sample is recorded and a hypothesis test is carried out at the 5\% level.
Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
In the case that $n = 20$ and the number of no-shows in the sample is 1 , carry out the test.
In another case, where $n$ is large, the number of no-shows in the sample is 6 and the critical value for the test is 8 . Complete the test.
In the case that $n \leqslant 18$, explain why there is no point in carrying out the test at the $5 \%$ level.

OCR MEI S1 Q2

2 Mark is playing solitaire on his computer. The probability that he wins a game is 0.2 , independently of all other games that he plays.

Find the expected number of wins in 12 games.
Find the probability that
(A) he wins exactly 2 out of the next 12 games that he plays,
(B) he wins at least 2 out of the next 12 games that he plays.
Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the $5 \%$ level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test.

OCR MEI S1 Q3

3 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.

(A) Find the probability that there are exactly 2 faulty tiles in the sample.
(B) Find the probability that there are more than 2 faulty tiles in the sample.
(C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
(A) Write down suitable null and alternative hypotheses for the test.
(B) Explain why the alternative hypothesis has the form that it does.
Find the critical region for the test at the $5 \%$ level, showing all of your calculations.
In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.

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

Questions — OCR MEI (4301 questions)

Browse by module

Browse by board

OCR MEI FP3 2010 June Q3

OCR MEI FP3 2010 June Q4

OCR MEI FP3 2012 June Q1

OCR MEI FP3 2012 June Q2

OCR MEI FP3 2012 June Q3

OCR MEI FP3 2012 June Q4

OCR MEI FP3 2013 June Q1

OCR MEI FP3 2013 June Q2

OCR MEI FP3 2013 June Q3

OCR MEI FP3 2013 June Q4

OCR MEI FP3 2013 June Q5

OCR MEI FP3 2014 June Q1

OCR MEI FP3 2014 June Q2

OCR MEI FP3 2014 June Q3

OCR MEI FP3 2014 June Q4

OCR MEI FP3 2014 June Q5

OCR MEI S1 Q1

OCR MEI S1 Q2

OCR MEI S1 Q3

OCR MEI S1 Q1

OCR MEI S1 Q2

OCR MEI S1 Q3

OCR MEI S1 Q1

OCR MEI S1 Q2

OCR MEI S1 Q3