Questions - OCR MEI - A-Level Maths

Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = 0.5 , \quad \lambda = 3 \quad \text { and } \quad \lambda = 5 .$$
Given that the curve is a conic, name the type of conic.
Show that $y$ has a maximum value of $\sqrt { 2 }$ when $\theta = \frac { 1 } { 4 } \pi$.
Show that $x ^ { 2 } + y ^ { 2 } = \left( 1 + \lambda ^ { 2 } \right) + \left( \frac { 1 } { \lambda ^ { 2 } } - \lambda ^ { 2 } \right) \sin ^ { 2 } \theta$, and deduce that the distance from the origin of any point on the curve is between $\sqrt { 1 + \frac { 1 } { \lambda ^ { 2 } } }$ and $\sqrt { 1 + \lambda ^ { 2 } }$.
For the case $\lambda = 1$, show that the curve is a circle, and find its radius.
For the case $\lambda = 2$, draw a sketch of the curve, and label the points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H }$ on the curve corresponding to $\theta = 0 , \frac { 1 } { 4 } \pi , \frac { 1 } { 2 } \pi , \frac { 3 } { 4 } \pi , \pi , \frac { 5 } { 4 } \pi , \frac { 3 } { 2 } \pi , \frac { 7 } { 4 } \pi$ respectively. You should make clear what is special about each of these points. \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 FP2 2010 June Q1

1. Given that $\mathrm { f } ( t ) = \arcsin t$, write down an expression for $\mathrm { f } ^ { \prime } ( t )$ and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
2. Show that the Maclaurin expansion of the function $\arcsin \left( x + \frac { 1 } { 2 } \right)$ begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in $x ^ { 2 }$.
Sketch the curve with polar equation $r = \frac { \pi a } { \pi + \theta }$, where $a > 0$, for $0 \leqslant \theta < 2 \pi$. Find, in terms of $a$, the area of the region bounded by the part of the curve for which $0 \leqslant \theta \leqslant \pi$ and the lines $\theta = 0$ and $\theta = \pi$.
Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

OCR MEI FP2 2010 June Q2

Given that $z = \cos \theta + \mathrm { j } \sin \theta$, express $z ^ { n } + \frac { 1 } { z ^ { n } }$ and $z ^ { n } - \frac { 1 } { z ^ { n } }$ in simplified trigonometric form.
Hence find the constants $A , B , C$ in the identity $$\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta$$
1. Find the 4th roots of - 9 j in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$ and $0 < \theta < 2 \pi$. Illustrate the roots on an Argand diagram.
2. Let the points representing these roots, taken in order of increasing $\theta$, be $\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }$. The mid-points of the sides of PQRS represent the 4th roots of a complex number $w$. Find the modulus and argument of $w$. Mark the point representing $w$ on your Argand diagram.

OCR MEI FP2 2010 June Q3

1. A $3 \times 3$ matrix $\mathbf { M }$ has characteristic equation $$2 \lambda ^ { 3 } + \lambda ^ { 2 } - 13 \lambda + 6 = 0$$ Show that $\lambda = 2$ is an eigenvalue of $\mathbf { M }$. Find the other eigenvalues.
2. An eigenvector corresponding to $\lambda = 2$ is $\left( \begin{array} { r } 3
  - 3
  1 \end{array} \right)$. Evaluate $\mathbf { M } \left( \begin{array} { r } 3
  - 3
  1 \end{array} \right)$ and $\mathbf { M } ^ { 2 } \left( \begin{array} { r } 1
  - 1
  \frac { 1 } { 3 } \end{array} \right)$.
  Solve the equation $\mathbf { M } \left( \begin{array} { l } x
  y
  z \end{array} \right) = \left( \begin{array} { r } 3
  - 3
  1 \end{array} \right)$.
3. Find constants $A , B , C$ such that $$\mathbf { M } ^ { 4 } = A \mathbf { M } ^ { 2 } + B \mathbf { M } + C \mathbf { I }$$
A $2 \times 2$ matrix $\mathbf { N }$ has eigenvalues -1 and 2, with eigenvectors $\binom { 1 } { 2 }$ and $\binom { - 1 } { 1 }$ respectively. Find $\mathbf { N }$. Section B (18 marks)

OCR MEI FP2 2010 June Q4

Prove, using exponential functions, that $$\sinh 2 x = 2 \sinh x \cosh x$$ Differentiate this result to obtain a formula for $\cosh 2 x$.
Sketch the curve with equation $y = \cosh x - 1$. The region bounded by this curve, the $x$-axis, and the line $x = 2$ is rotated through $2 \pi$ radians about the $x$-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.)
Show that the curve with equation $$y = \cosh 2 x + \sinh x$$ has exactly one stationary point.
Determine, in exact logarithmic form, the $x$-coordinate of the stationary point.

OCR MEI FP2 2010 June Q5

5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of $k$.

Firstly consider cases in which $k$ is a positive even integer.
(A) State the shape of the curve when $k = 2$.
(B) Sketch, on the same axes, the curves for $k = 2$ and $k = 4$.
(C) Describe the shape that the curve tends to as $k$ becomes very large.
(D) State the range of possible values of $x$ and $y$.
Now consider cases in which $k$ is a positive odd integer.
(A) Explain why $x$ and $y$ may take any value.
(B) State the shape of the curve when $k = 1$.
(C) Sketch the curve for $k = 3$. State the equation of the asymptote of this curve.
(D) Sketch the shape that the curve tends to as $k$ becomes very large.
Now let $k = \frac { 1 } { 2 }$. Sketch the curve, indicating the range of possible values of $x$ and $y$.
Now consider the modified equation $| x | ^ { k } + | y | ^ { k } = 1$.
(A) Sketch the curve for $k = \frac { 1 } { 2 }$.
(B) Investigate the shape of the curve for $k = \frac { 1 } { n }$ as the positive integer $n$ becomes very large.

OCR MEI S1 2005 January Q1

1 The number of minutes of recorded music on a sample of 100 CDs is summarised below.

Time ( $t$ minutes)	$40 \leqslant t < 45$	$45 \leqslant t < 50$	$50 \leqslant t < 60$	$60 \leqslant t < 70$	$70 \leqslant t < 90$
Number of CDs	26	18	31	16	9

Illustrate the data by means of a histogram.
Identify two features of the distribution.

OCR MEI S1 2005 January Q2

2 A sprinter runs many 100 -metre trials, and the time, $x$ seconds, for each is recorded. A sample of eight of these times is taken, as follows. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$

Calculate the sample mean, $\bar { x }$, and sample standard deviation, $s$, of these times.
Show that the time of 10.04 seconds may be regarded as an outlier.
Discuss briefly whether or not the time of 10.04 seconds should be discarded.

OCR MEI S1 2005 January Q3

3 The Venn diagram illustrates the occurrence of two events $A$ and $B$.
\includegraphics[max width=\textwidth, alt={}, center]{b35b2b3b-0d26-4a35-b4d2-110bf270d5dc-2_513_826_1713_658} You are given that $\mathrm { P } ( A \cap B ) = 0.3$ and that the probability that neither $A$ nor $B$ occurs is 0.1 . You are also given that $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$. Find $\mathrm { P } ( B )$.

OCR MEI S1 2005 January Q4

4 The number, $X$, of children per family in a certain city is modelled by the probability distribution $\mathrm { P } ( X = r ) = k ( 6 - r ) ( 1 + r )$ for $r = 0,1,2,3,4$.

Copy and complete the following table and hence show that the value of $k$ is $\frac { 1 } { 50 }$.
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ $6 k$ $10 k$
Calculate $\mathrm { E } ( X )$.
Hence write down the probability that a randomly selected family in this city has more than the mean number of children.

OCR MEI S1 2005 January Q5

5 A rugby union team consists of 15 players made up of 8 forwards and 7 backs. A manager has to select his team from a squad of 12 forwards and 11 backs.

In how many ways can the manager select the forwards?
In how many ways can the manager select the team?

OCR MEI S1 2005 January Q6

6 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table.

Weather Forecast

\multirow{2}{*}{Total}

\cline { 3 - 6 } \multicolumn{2}{|c|}{}

A day is selected at random from that year.

Show that the probability that the forecast is correct is $\frac { 264 } { 365 }$. Find the probability that
the forecast is correct, given that the forecast is sunny,
the forecast is correct, given that the weather is wet,
the weather is cloudy, given that the forecast is correct.

OCR MEI S1 2005 January Q7

7 The cumulative frequency graph below illustrates the distances that 176 children live from their primary school. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Distance from school} \includegraphics[alt={},max width=\textwidth]{b35b2b3b-0d26-4a35-b4d2-110bf270d5dc-4_1073_1571_580_340}

\end{figure}

Use the graph to estimate, to the nearest 10 metres,
(A) the median distance from school,
(B) the lower quartile, upper quartile and interquartile range.
Draw a box and whisker plot to illustrate the data. The graph on page 4 used the following grouped data.
Distance (metres) 200 400 600 800 1000 1200
Cumulative frequency 20 64 118 150 169 176
Copy and complete the grouped frequency table below describing the same data.
Distance ( $d$ metres) Frequency
$0 < d \leqslant 200$ 20
$200 < d \leqslant 400$
Hence estimate the mean distance these children live from school. It is subsequently found that none of the 176 children lives within 100 metres of the school.
Calculate the revised estimate of the mean distance.
Describe what change needs to be made to the cumulative frequency graph.

OCR MEI S1 2005 January Q8

8 At a doctor's surgery, records show that $20 \%$ of patients who make an appointment fail to turn up. During afternoon surgery the doctor has time to see 16 patients. There are 16 appointments to see the doctor one afternoon.

Find the probability that all 16 patients turn up.
Find the probability that more than 3 patients do not turn up. To improve efficiency, the doctor decides to make more than 16 appointments for afternoon surgery, although there will still only be enough time to see 16 patients. There must be a probability of at least 0.9 that the doctor will have enough time to see all the patients who turn up.
The doctor makes 17 appointments for afternoon surgery. Find the probability that at least one patient does not turn up. Hence show that making 17 appointments is satisfactory.
Now find the greatest number of appointments the doctor can make for afternoon surgery and still have a probability of at least 0.9 of having time to see all patients who turn up. A computerised appointment system is introduced at the surgery. It is decided to test, at the 5\% level, whether the proportion of patients failing to turn up for their appointments has changed. There are always 20 appointments to see the doctor at morning surgery. On a randomly chosen morning, 1 patient does not turn up.
Write down suitable hypotheses and carry out the test.

OCR MEI S1 2006 January Q1

1 The times taken, in minutes, by 80 people to complete a crossword puzzle are summarised by the box and whisker plot below.
\includegraphics[max width=\textwidth, alt={}, center]{acb05873-e441-4b95-9732-6ebd5ae79fa6-2_147_848_507_612}

Write down the range and the interquartile range of the times.
Determine whether any of the times can be regarded as outliers.
Describe the shape of the distribution of the times.

OCR MEI S1 2006 January Q2

2 Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, $X$, which are now in the correct envelope is given in the following table.

$r$	0	1	2	3	4
$\mathrm { P } ( X = r )$	$\frac { 3 } { 8 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 4 }$	0	$\frac { 1 } { 24 }$

Explain why the case $X = 3$ is impossible.
Explain why $\mathrm { P } ( X = 4 ) = \frac { 1 } { 24 }$.
Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2006 January Q3

3 Over a long period of time, 20\% of all bowls made by a particular manufacturer are imperfect and cannot be sold.

Find the probability that fewer than 4 bowls from a random sample of 10 made by the manufacturer are imperfect. The manufacturer introduces a new process for producing bowls. To test whether there has been an improvement, each of a random sample of 20 bowls made by the new process is examined. From this sample, 2 bowls are found to be imperfect.
Show that this does not provide evidence, at the $5 \%$ level of significance, of a reduction in the proportion of imperfect bowls. You should show your hypotheses and calculations clearly.

OCR MEI S1 2006 January Q4

4 A company sells sugar in bags which are labelled as containing 450 grams.
Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.

State two adjustments the company could make. The weights, $x$ grams, of a random sample of 25 bags are now recorded.
Given that $\sum x = 11409$ and $\sum x ^ { 2 } = 5206937$, calculate the sample mean and sample standard deviation of these weights.

OCR MEI S1 2006 January Q5

5 A school athletics team has 10 members. The table shows which competitions each of the members can take part in.

		Competiton
		100 m	200 m	110 m hurdles	400 m	Long jump
\multirow{10}{*}{Athlete}	Abel	✓	✓			✓
	Bernoulli		✓		✓
	Cauchy	✓		✓		✓
	Descartes	✓	✓
	Einstein		✓		✓
	Fermat	✓		✓
	Galois				✓	✓
	Hardy	✓	✓			✓
	Iwasawa		✓		✓
	Jacobi			✓

An athlete is selected at random. Events $A , B , C , D$ are defined as follows.
A: the athlete can take part in exactly 2 competitions.
$B$ : the athlete can take part in the 200 m .
$C$ : the athlete can take part in the 110 m hurdles.
$D$ : the athlete can take part in the long jump.

Write down the value of $\mathrm { P } ( A \cap B )$.
Write down the value of $\mathrm { P } ( C \cup D )$.
Which two of the four events $A , B , C , D$ are mutually exclusive?
Show that events $B$ and $D$ are not independent.

OCR MEI S1 2006 January Q6

6 A band has a repertoire of 12 songs suitable for a live performance. From these songs, a selection of 7 has to be made.

Calculate the number of different selections that can be made.
Once the 7 songs have been selected, they have to be arranged in playing order. In how many ways can this be done?

OCR MEI S1 2006 January Q7

7 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.

Grade	A*	A	B	C	D	E	F	G	U
Points	8	7	6	5	4	3	2	1	0

Calculate the mean GCSE score, $X$, of a student who has the following GCSE grades: $$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$ 60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.
Mean GCSE score Number of students
$4.5 \leqslant X < 5.5$ 8
$5.5 \leqslant X < 6.0$ 14
$6.0 \leqslant X < 6.5$ 19
$6.5 \leqslant X < 7.0$ 13
$7.0 \leqslant X \leqslant 8.0$ 6
Draw a histogram to illustrate this information.
Calculate estimates of the sample mean and the sample standard deviation. The scoring system for AS grades is shown in the table below.
AS Grade A B C D E U
Score 60 50 40 30 20 0
The Mathematics department at the college predicts each student's AS score, $Y$, using the formula $Y = 13 X - 46$, where $X$ is the student's average GCSE score.
What AS grade would the department predict for a student with an average GCSE score of 7.4 ?
What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.
Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.

OCR MEI S1 2006 January Q8

8 Jane buys 5 jam doughnuts, 4 cream doughnuts and 3 plain doughnuts.
On arrival home, each of her three children eats one of the twelve doughnuts. The different kinds of doughnut are indistinguishable by sight and so selection of doughnuts is random. Calculate the probabilities of the following events.

All 3 doughnuts eaten contain jam.
All 3 doughnuts are of the same kind.
The 3 doughnuts are all of a different kind.
The 3 doughnuts contain jam, given that they are all of the same kind. On 5 successive Saturdays, Jane buys the same combination of 12 doughnuts and her three children eat one each. Find the probability that all 3 doughnuts eaten contain jam on
exactly 2 Saturdays out of the 5 ,
at least 1 Saturday out of the 5 .

OCR MEI S1 2007 January Q1

1 The total annual emissions of carbon dioxide, $x$ tonnes per person, for 13 European countries are given below. $$\begin{array} { c c c c c c c c c c c c c } 6.2 & 6.7 & 6.8 & 8.1 & 8.1 & 8.5 & 8.6 & 9.0 & 9.9 & 10.1 & 11.0 & 11.8 & 22.8 \end{array}$$

Find the mean, median and midrange of these data.
Comment on how useful each of these is as a measure of central tendency for these data, giving a brief reason for each of your answers.

OCR MEI S1 2007 January Q2

2 The numbers of absentees per day from Mrs Smith’s reception class over a period of 50 days are summarised below.

Number of absentees	0	1	2	3	4	5	6	$> 6$
Frequency	8	15	11	8	3	4	1	0

Illustrate these data by means of a vertical line chart.
Calculate the mean and root mean square deviation of these data.
There are 30 children in Mrs Smith's class altogether. Find the mean and root mean square deviation of the number of children who are present during the 50 days.

OCR MEI S1 2007 January Q3

3 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

Time $( t$ minutes $)$	$0 \leqslant t < 5$	$5 \leqslant t < 10$	$10 \leqslant t < 20$	$20 \leqslant t < 30$	$30 \leqslant t < 40$	$40 \leqslant t < 60$
Frequency	34	153	188	73	27	5

Illustrate these data by means of a histogram.
Identify the type of skewness of the distribution.

Time ( \(t\) minutes)	\(40 \leqslant t < 45\)	\(45 \leqslant t < 50\)	\(50 \leqslant t < 60\)	\(60 \leqslant t < 70\)	\(70 \leqslant t < 90\)
Number of CDs	26	18	31	16	9

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	\(6 k\)	\(10 k\)

Distance ( \(d\) metres)	Frequency
\(0 < d \leqslant 200\)	20
\(200 < d \leqslant 400\)

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	\(\frac { 3 } { 8 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 4 }\)	0	\(\frac { 1 } { 24 }\)

Time \(( t\) minutes \()\)	\(0 \leqslant t < 5\)	\(5 \leqslant t < 10\)	\(10 \leqslant t < 20\)	\(20 \leqslant t < 30\)	\(30 \leqslant t < 40\)	\(40 \leqslant t < 60\)
Frequency	34	153	188	73	27	5

Distance (metres)	200	400	600	800	1000	1200
Cumulative frequency	20	64	118	150	169	176

Mean GCSE score	Number of students
\(4.5 \leqslant X < 5.5\)	8
\(5.5 \leqslant X < 6.0\)	14
\(6.0 \leqslant X < 6.5\)	19
\(6.5 \leqslant X < 7.0\)	13
\(7.0 \leqslant X \leqslant 8.0\)	6

AS Grade	A	B	C	D	E	U
Score	60	50	40	30	20	0

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