Questions - OCR - A-Level Maths

$x$	$10 \leqslant x \leqslant 19$	$20 \leqslant x \leqslant 24$	$25 \leqslant x \leqslant 29$	$30 \leqslant x \leqslant 49$
Number of stones	25	22	29	24

These data are to be presented on a statistical diagram.

For a histogram, find the frequency density of the $10 \leqslant x \leqslant 19$ class.
For a cumulative frequency graph, state the coordinates of the first two points that should be plotted.
Why is it not possible to draw an exact box-and-whisker plot to illustrate the data?

OCR S1 2009 June Q6

6 Last year Eleanor played 11 rounds of golf. Her scores were as follows: $79 , \quad 71 , \quad 80 , \quad 67 , \quad 67 , \quad 74 , \quad 66 , \quad 65 , \quad 71 , \quad 66 , \quad 64$.

Calculate the mean of these scores and show that the standard deviation is 5.31 , correct to 3 significant figures.
Find the median and interquartile range of the scores. This year, Eleanor also played 11 rounds of golf. The standard deviation of her scores was 4.23, correct to 3 significant figures, and the interquartile range was the same as last year.
Give a possible reason why the standard deviation of her scores was lower than last year although her interquartile range was unchanged. In golf, smaller scores mean a better standard of play than larger scores. Ken suggests that since the standard deviation was smaller this year, Eleanor's overall standard has improved.
Explain why Ken is wrong.
State what the smaller standard deviation does show about Eleanor's play.

OCR S1 2009 June Q7

7 Three letters are selected at random from the 8 letters of the word COMPUTER, without regard to order.

Find the number of possible selections of 3 letters.
Find the probability that the letter P is included in the selection. Three letters are now selected at random, one at a time, from the 8 letters of the word COMPUTER, and are placed in order in a line.
Find the probability that the 3 letters form the word TOP.

OCR S1 2009 June Q8

8 A game at a charity event uses a bag containing 19 white counters and 1 red counter. To play the game once a player takes counters at random from the bag, one at a time, without replacement. If the red counter is taken, the player wins a prize and the game ends. If not, the game ends when 3 white counters have been taken. Niko plays the game once.

(a) Copy and complete the tree diagram showing the probabilities for Niko. \section*{First counter} \includegraphics[max width=\textwidth, alt={}, center]{c985b9cc-a202-4d5d-a6b3-591b0560f570-4_293_426_1231_532}
(b) Find the probability that Niko will win a prize.
The number of counters that Niko takes is denoted by $X$.
(a) Find $\mathrm { P } ( X = 3 )$.
(b) Find $\mathrm { E } ( X )$.

OCR S1 2009 June Q9

9 Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is 0.12 .

Find the smallest value of $n$ such that the probability of at least one success in $n$ trials is more than 0.95.
Find the probability that the 3rd success occurs on the 7th trial.

OCR S1 2010 June Q1

1 The marks of some students in a French examination were summarised in a grouped frequency distribution and a cumulative frequency diagram was drawn, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{18d9bdc5-2758-47ec-8698-d90c1c6ac224-02_846_1404_356_367}

Estimate how many students took the examination.
How can you tell that no student scored more than 55 marks?
Find the greatest possible range of the marks.
The minimum mark for Grade C was 27 . The number of students who gained exactly Grade C was the same as the number of students who gained a grade lower than C. Estimate the maximum mark for Grade C.
In a German examination the marks of the same students had an interquartile range of 16 marks. What does this result indicate about the performance of the students in the German examination as compared with the French examination?

OCR S1 2010 June Q2

2 Three skaters, $A , B$ and $C$, are placed in rank order by four judges. Judge $P$ ranks skater $A$ in 1st place, skater $B$ in 2nd place and skater $C$ in 3rd place.

Without carrying out any calculation, state the value of Spearman's rank correlation coefficient for the following ranks. Give a reason for your answer. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR S1 2010 June Q5

3 (ii) (b)





3 (ii) (c)





3 (ii) (d)

\href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR S1 2010 June Q7

7 The menu below shows all the dishes available at a certain restaurant.

Rice dishes	Main dishes	Vegetable dishes
Boiled rice	Chicken	Mushrooms
Fried rice	Beef	Cauliflower
Pilau rice	Lamb	Spinach
Keema rice	Mixed grill	Lentils
	Prawn	Potatoes
	Vegetarian

A group of friends decide that they will share a total of 2 different rice dishes, 3 different main dishes and 4 different vegetable dishes from this menu. Given these restrictions,

find the number of possible combinations of dishes that they can choose to share,
assuming that all choices are equally likely, find the probability that they choose boiled rice. The friends decide to add a further restriction as follows. If they choose boiled rice, they will not choose potatoes.
Find the number of possible combinations of dishes that they can now choose.

OCR S1 2010 June Q8

8 The proportion of people who watch West Street on television is $30 \%$. A market researcher interviews people at random in order to contact viewers of West Street. Each day she has to contact a certain number of viewers of West Street.

Near the end of one day she finds that she needs to contact just one more viewer of West Street. Find the probability that the number of further interviews required is
(a) 4 ,

OCR S1 2010 June Q10

(a) 8

(b) \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT

OCR FP3 2007 January Q1

Show that the set of numbers $\{ 3,5,7 \}$, under multiplication modulo 8, does not form a group.
The set of numbers $\{ 3,5,7 , a \}$, under multiplication modulo 8 , forms a group. Write down the value of $a$.
State, justifying your answer, whether or not the group in part (ii) is isomorphic to the multiplicative group $\left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}$, where $e$ is the identity and $r ^ { 4 } = e$.

OCR FP3 2007 January Q2

2 Find the equation of the line of intersection of the planes with equations $$\mathbf { r } . ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 4 \quad \text { and } \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } ) = 6 ,$$ giving your answer in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.

OCR FP3 2007 January Q3

Solve the equation $z ^ { 2 } - 6 z + 36 = 0$, and give your answers in the form $r ( \cos \theta \pm \mathrm { i } \sin \theta )$, where $r > 0$ and $0 \leqslant \theta \leqslant \pi$.
Given that $Z$ is either of the roots found in part (i), deduce the exact value of $Z ^ { - 3 }$.

OCR FP3 2007 January Q4

4 The variables $x$ and $y$ are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - y ^ { 2 } } { x y }$$

Use the substitution $y = x z$, where $z$ is a function of $x$, to obtain the differential equation $$x \frac { \mathrm {~d} z } { \mathrm {~d} x } = \frac { 1 - 2 z ^ { 2 } } { z }$$
Hence show by integration that the general solution of the differential equation (A) may be expressed in the form $x ^ { 2 } \left( x ^ { 2 } - 2 y ^ { 2 } \right) = k$, where $k$ is a constant.

OCR FP3 2007 January Q5

5 A multiplicative group $G$ of order 9 has distinct elements $p$ and $q$, both of which have order 3 . The group is commutative, the identity element is $e$, and it is given that $q \neq p ^ { 2 }$.

Write down the elements of a proper subgroup of $G$
(a) which does not contain $q$,
(b) which does not contain $p$.
Find the order of each of the elements $p q$ and $p q ^ { 2 }$, justifying your answers.
State the possible order(s) of proper subgroups of $G$.
Find two proper subgroups of $G$ which are distinct from those in part (i), simplifying the elements.

OCR FP3 2007 January Q6

6 The variables $x$ and $y$ satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y = 2 x + 1$$ Find

the complementary function,
the general solution. In a particular case, it is given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$.
Find the solution of the differential equation in this case.
Write down the function to which $y$ approximates when $x$ is large and positive.

OCR FP3 2007 January Q7

7 The position vectors of the points $A , B , C , D , G$ are given by $$\mathbf { a } = 6 \mathbf { i } + 4 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { b } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { c } = \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { d } = 3 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { g } = 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }$$ respectively.

The line through $A$ and $G$ meets the plane $B C D$ at $M$. Write down the vector equation of the line through $A$ and $G$ and hence show that the position vector of $M$ is $2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }$.
Find the value of the ratio $A G : A M$.
Find the position vector of the point $P$ on the line through $C$ and $G$, such that $\overrightarrow { C P } = \frac { 4 } { 3 } \overrightarrow { C G }$.
Verify that $P$ lies in the plane $A B D$.

OCR FP3 2007 January Q8

Use de Moivre's theorem to find an expression for $\tan 4 \theta$ in terms of $\tan \theta$.
Deduce that $\cot 4 \theta = \frac { \cot ^ { 4 } \theta - 6 \cot ^ { 2 } \theta + 1 } { 4 \cot ^ { 3 } \theta - 4 \cot \theta }$.
Hence show that one of the roots of the equation $x ^ { 2 } - 6 x + 1 = 0$ is $\cot ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)$.
Hence find the value of $\operatorname { cosec } ^ { 2 } \left( \frac { 1 } { 8 } \pi \right) + \operatorname { cosec } ^ { 2 } \left( \frac { 3 } { 8 } \pi \right)$, justifying your answer.

OCR FP3 2008 January Q1

A group $G$ of order 6 has the combination table shown below.

	$e$	$a$	$b$	$p$	$q$	$r$
$e$	$e$	$a$	$b$	$p$	$q$	$r$
$a$	$a$	$b$	$e$	$r$	$p$	$q$
$b$	$b$	$e$	$a$	$q$	$r$	$p$
$p$	$p$	$q$	$r$	$e$	$a$	$b$
$q$	$q$	$r$	$p$	$b$	$e$	$a$
$r$	$r$	$p$	$q$	$a$	$b$	$e$

State, with a reason, whether or not $G$ is commutative.
State the number of subgroups of $G$ which are of order 2 .
List the elements of the subgroup of $G$ which is of order 3 .

A multiplicative group $H$ of order 6 has elements $e , c , c ^ { 2 } , c ^ { 3 } , c ^ { 4 } , c ^ { 5 }$, where $e$ is the identity. Write down the order of each of the elements $c ^ { 3 } , c ^ { 4 }$ and $c ^ { 5 }$.

OCR FP3 2008 January Q2

2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 4 x .$$

OCR FP3 2008 January Q3

3 Two fixed points, $A$ and $B$, have position vectors $\mathbf { a }$ and $\mathbf { b }$ relative to the origin $O$, and a variable point $P$ has position vector $\mathbf { r }$.

Give a geometrical description of the locus of $P$ when $\mathbf { r }$ satisfies the equation $\mathbf { r } = \lambda \mathbf { a }$, where $0 \leqslant \lambda \leqslant 1$.
Given that $P$ is a point on the line $A B$, use a property of the vector product to explain why $( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }$.
Give a geometrical description of the locus of $P$ when $\mathbf { r }$ satisfies the equation $\mathbf { r } \times ( \mathbf { a } - \mathbf { b } ) = \mathbf { 0 }$.

OCR FP3 2008 January Q4

4 The integrals $C$ and $S$ are defined by $$C = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x \quad \text { and } \quad S = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin 3 x \mathrm {~d} x$$ By considering $C + \mathrm { i } S$ as a single integral, show that $$C = - \frac { 1 } { 13 } \left( 2 + 3 \mathrm { e } ^ { \pi } \right) ,$$ and obtain a similar expression for $S$.
(You may assume that the standard result for $\int \mathrm { e } ^ { k x } \mathrm {~d} x$ remains true when $k$ is a complex constant, so that $\left. \int \mathrm { e } ^ { ( a + \mathrm { i } b ) x } \mathrm {~d} x = \frac { 1 } { a + \mathrm { i } b } \mathrm { e } ^ { ( a + \mathrm { i } b ) x } .\right)$

OCR FP3 2008 January Q5

Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \sin 2 x$$ expressing $y$ in terms of $x$ in your answer. In a particular case, it is given that $y = \frac { 2 } { \pi }$ when $x = \frac { 1 } { 4 } \pi$.
Find the solution of the differential equation in this case.
Write down a function to which $y$ approximates when $x$ is large and positive.

	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
\(e\)	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
\(a\)	\(a\)	\(b\)	\(e\)	\(r\)	\(p\)	\(q\)
\(b\)	\(b\)	\(e\)	\(a\)	\(q\)	\(r\)	\(p\)
\(p\)	\(p\)	\(q\)	\(r\)	\(e\)	\(a\)	\(b\)
\(q\)	\(q\)	\(r\)	\(p\)	\(b\)	\(e\)	\(a\)
\(r\)	\(r\)	\(p\)	\(q\)	\(a\)	\(b\)	\(e\)

\(x\)	\(10 \leqslant x \leqslant 19\)	\(20 \leqslant x \leqslant 24\)	\(25 \leqslant x \leqslant 29\)	\(30 \leqslant x \leqslant 49\)
Number of stones	25	22	29	24

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