Questions - A-Level Maths

Edexcel C4 Q4

11 marks Challenging +1.2

Show that the volume of the solid formed is $\frac { 1 } { 4 } \pi ( \pi + 2 )$.
Find a cartesian equation for the curve.

OCR S1 2011 January Q5

10 marks Moderate -0.8

The number of free gifts that Jan receives in a week is denoted by $X$. Name a suitable probability distribution with which to model $X$, giving the value(s) of any parameter(s). State any assumption(s) necessary for the distribution to be a valid model. Assume now that your model is valid.
Find
(a) $\mathrm { P } ( X \leqslant 2 )$,
(b) $\mathrm { P } ( X = 2 )$.
Find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts. 6
The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
3
2
2
2
2

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}
\multirow[t]{4}{*}{3
}
3
\href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR S1 2012 January Q7

8 marks Moderate -0.8

State a suitable distribution that can be used as a model for $X$, giving the value(s) of any parameter(s). State also any necessary condition(s) for this distribution to be a good model. Use the distribution stated in part (i) to find
$\mathrm { P } ( X = 4 )$,
$\mathrm { P } ( X \geqslant 4 )$.

OCR MEI S1 2005 June Q5

6 marks Moderate -0.8

On the insert, complete the table giving the lowest common multiples of all pairs of integers between 1 and 6 .
[0pt] [1]
\multirow{2}{*}{} Second integer
1 2 3 4 5 6
\multirow{6}{*}{First integer} 1 1 2 3 4 5 6
2 2 2 6 4 10 6
3 3 6 3 12 15 6
4 4 4 12 12
5 5 10 15
6 6 6 6 12
Two fair dice are thrown and the lowest common multiple of the two scores is found.
Use the table to find the probabilities of the following events.
(A) The lowest common multiple is greater than 6 .
(B) The lowest common multiple is a multiple of 5 .
(C) The lowest common multiple is both greater than 6 and a multiple of 5 .
Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.

Edexcel S1 Q2

9 marks Moderate -0.8

Plot a scatter diagram showing these data. The student wanted to investigate further whether or not her data provided evidence of an increase in temperature in June each year. Using $Y$ for the number of years since 1993 and $T$ for the mean temperature, she calculated the following summary statistics. $$\Sigma Y = 28 , \quad \Sigma T = 182.5 , \quad \Sigma Y ^ { 2 } = 140 , \quad \Sigma T ^ { 2 } = 4173.93 , \quad \Sigma Y T = 644.7 .$$
Calculate the product moment correlation coefficient for these data.
Comment on your result in relation to the student's enquiry.

OCR MEI S1 Q3

6 marks Easy -1.8

On die insert, complete the lable giving due lowest common multiples of all pairs of integers between 1 and 6 .

Second integer

\cline { 2 - 8 } \multicolumn{2}{|c|}{}

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6

\multirow{5}{*}{

First

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6

\cline { 2 - 8 }

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10

6

\cline { 2 - 8 }

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6

3

12

15

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\cline { 2 - 8 }

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12

\cline { 2 - 8 }

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6

12

Two fair dice are thrown and the lowest common multiple of the two scores is found.

Use the table to find the probabilities of the following events.
(A) The lowest common multiple is greater than 6 .
(B) The lowest common multiple is a multiple of 5 .
(C) The lowest common multiple is both greater than 6 and a multiple of 5.
Use your answers to part (ii) to show that the events "the lowest common multiple is greater than 6 " and "the lowest common multiple is a multiple of 5 " are not independent.

AQA S2 2011 June Q3

10 marks Standard +0.3

State the null hypothesis that Emily used.
Find the value of the test statistic, $X ^ { 2 }$, giving your answer to one decimal place.
State, in context, the conclusion that Emily should reach based on the results of her $\chi ^ { 2 }$ test.
Make one comment on the GCSE performances of 16-year-old students attending Bailey Language School.
Emily's friend, Joanna, used the same data to correctly conduct a $\chi ^ { 2 }$ test using the $10 \%$ level of significance. State, with justification, the conclusion that Joanna should reach.

Edexcel S2 Q2

4 marks Standard +0.3

Specify a suitable model for the distribution of $X$.
Find the mean and the standard deviation of $X$. \item A secretarial agency carefully assesses the work of a new recruit, with the following results after 150 pages: \end{enumerate}
No of errors 0 1 2 3 4 5 6
No of pages 16 38 41 29 17 7 2
Find the mean and variance of the number of errors per page.
Explain how these results support the idea that the number of errors per page follows a Poisson distribution.
After two weeks at the agency, the secretary types a fresh piece of work, six pages long, which is found to contain 15 errors.
The director suspects that the secretary was trying especially hard during the early period and that she is now less conscientious. Using a Poisson distribution with the mean found in part (a), test this hypothesis at the $5 \%$ significance level.

Edexcel S2 Q2

5 marks Moderate -0.3

Name the distribution of $L$, the length of the longer part of string, and sketch the probability density function for $L$.
Find the probability that one part of the string is more than twice as long as the other. \item A supplier of widgets claims that only $10 \%$ of his widgets have faults.
In a consignment of 50 widgets, 9 are faulty. Test, at the $5 \%$ significance level, whether this suggests that the supplier's claim is false.
Find how many faulty widgets would be needed to provide evidence against the claim at the $1 \%$ significance level. \item In a survey of 22 families, the number of children, $X$, in each family was given by the following table, where $f$ denotes the frequency: \end{enumerate}
$X$ 0 1 2 3 4 5
$f$ 3 8 5 3 2 1
Find the mean and variance of $X$.
Explain why these results suggest that $X$ may follow a Poisson distribution.
State another feature of the data that suggests a Poisson distribution. It is sometimes suggested that the number of children in a family follows a Poisson distribution with mean 2.4. Assuming that this is correct,
find the probability that a family has less than two children.
Use this result to find the probability that, in a random sample of 22 families, exactly 11 of the families have less than two children. \section*{STATISTICS 2 (A) TEST PAPER 7 Page 2}

OCR S2 2007 June Q4

6 marks Moderate -0.3

State two conditions needed for $X$ to be well modelled by a normal distribution.
It is given that $X \sim \mathrm {~N} \left( 50.0,8 ^ { 2 } \right)$. The mean of 20 random observations of $X$ is denoted by $\bar { X }$. Find $\mathrm { P } ( \bar { X } > 47.0 )$. 5 The number of system failures per month in a large network is a random variable with the distribution $\operatorname { Po } ( \lambda )$. A significance test of the null hypothesis $\mathrm { H } _ { 0 } : \lambda = 2.5$ is carried out by counting $R$, the number of system failures in a period of 6 months. The result of the test is that $\mathrm { H } _ { 0 }$ is rejected if $R > 23$ but is not rejected if $R \leqslant 23$.
State the alternative hypothesis.
Find the significance level of the test.
Given that $\mathrm { P } ( R > 23 ) < 0.1$, use tables to find the largest possible actual value of $\lambda$. You should show the values of any relevant probabilities. 6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter $e$ appears $19 \%$ of the time. A certain encoded message of 20 letters contains one letter $e$.
Using an exact binomial distribution, test at the $10 \%$ significance level whether there is evidence that the proportion of the letter $e$ in the language from which this message is a sample is less than in German, i.e., less than $19 \%$.
Give a reason why a binomial distribution might not be an appropriate model in this context. 7 Two continuous random variables $S$ and $T$ have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \\ T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
Sketch on the same axes the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$. [You should not use graph paper or attempt to plot points exactly.]
Explain in everyday terms the difference between the two random variables.
Find the value of $t$ such that $\mathrm { P } ( T > t ) = 0.2$. 8 A random variable $Y$ is normally distributed with mean $\mu$ and variance 12.25. Two statisticians carry out significance tests of the hypotheses $\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0$.
Statistician $A$ uses the mean $\bar { Y }$ of a sample of size 23, and the critical region for his test is $\bar { Y } > 64.20$. Find the significance level for $A$ 's test.
Statistician $B$ uses the mean of a sample of size 50 and a significance level of $5 \%$.
(a) Find the critical region for $B$ 's test.
(b) Given that $\mu = 65.0$, find the probability that $B$ 's test results in a Type II error.
Given that, when $\mu = 65.0$, the probability that $A$ 's test results in a Type II error is 0.1365 , state with a reason which test is better. 9 (a) The random variable $G$ has the distribution $\mathrm { B } ( n , 0.75 )$. Find the set of values of $n$ for which the distribution of $G$ can be well approximated by a normal distribution.
(b) The random variable $H$ has the distribution $\mathrm { B } ( n , p )$. It is given that, using a normal approximation, $\mathrm { P } ( H \geqslant 71 ) = 0.0401$ and $\mathrm { P } ( H \leqslant 46 ) = 0.0122$.
Find the mean and standard deviation of the approximating normal distribution.
Hence find the values of $n$ and $p$.

OCR S4 2010 June Q3

7 marks Challenging +1.8

Assuming that all rankings are equally likely, show that $\mathrm { P } ( R \leqslant 17 ) = \frac { 2 } { 231 }$. The marks of 5 randomly chosen students from School $A$ and 6 randomly chosen students from School $B$, who took the same examination, achieving different marks, were ranked. The rankings are shown in the table.
Rank 1 2 3 4 5 6 7 8 9 10 11
School $A$ $A$ $A$ $B$ $A$ $A$ $B$ $B$ $B$ $B$ $B$
For a Wilcoxon rank-sum test, obtain the exact smallest significance level for which there is evidence of a difference in performance at the two schools.

OCR M1 2010 January Q5

11 marks Moderate -0.3

Find the value of $t$ when $A$ and $B$ have the same speed.
Calculate the value of $t$ when $B$ overtakes $A$.
On a single diagram, sketch the $( t , x )$ graphs for the two cyclists for the time from $t = 0$ until after $B$ has overtaken $A$.

OCR M1 2015 June Q3

8 marks Standard +0.3

Calculate the distance $A$ cycles, and hence find the period of time for which $B$ walks before finding the bicycle.
Find $T$.
Calculate the distance $A$ and $B$ each travel.

OCR MEI M1 2008 June Q7

17 marks Moderate -0.3

What information in the question indicates that the tension in the string section CB is also 60 N ?
Show that the string sections AC and CB are equally inclined to the horizontal (so that $\alpha = \beta$ in Fig. 7.1).
Calculate the angle of the string sections AC and CB to the horizontal. In a different situation the same box is supported by two separate light strings, PC and QC, that are tied to the box at C . There is also a horizontal force of 10 N acting at C . This force and the angles between these strings and the horizontal are shown in Fig. 7.2. The box is in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-4_323_503_1649_822} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
\end{figure}
Calculate the tensions in the two strings.

OCR MEI M1 2013 June Q7

18 marks Moderate -0.3

Represent the forces acting on the object as a fully labelled triangle of forces.
Find $F$ and $\theta$. Show that the distance between the object and the vertical section of rope A is 3 m . Abi then pulls harder and the object moves upwards. Bob adjusts the tension in rope B so that the object moves along a vertical line. Fig. 7.2 shows the situation when the object is part of the way up. The tension in rope A is $S \mathrm {~N}$ and the tension in rope B is $T \mathrm {~N}$. The ropes make angles $\alpha$ and $\beta$ with the vertical as shown in the diagram. Abi and Bob are taking a rest and holding the object stationary and in equilibrium. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-5_383_360_534_854} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
\end{figure}
Give the equations, involving $S , T , \alpha$ and $\beta$, for equilibrium in the vertical and horizontal directions.
Find the values of $S$ and $T$ when $\alpha = 8.5 ^ { \circ }$ and $\beta = 35 ^ { \circ }$.
Abi's mass is 40 kg . Explain why it is not possible for her to raise the object to a position in which $\alpha = 60 ^ { \circ }$.

Edexcel M1 Q2

6 marks Standard +0.3

Find the speed of $B$.
Find the velocity of $B$ relative to $A$.
Find the acute angle between the relative velocity found in part (b) and the vector $\mathbf { i }$, giving your answer in degrees correct to 1 decimal place.
(2 marks) \item \end{enumerate} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2108a1be-0214-42c4-9cb4-8622cc0fa496-2_360_1018_1242_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a uniform plank $A B$ of length 8 m and mass 30 kg . It is supported in a horizontal position by two pivots, one situated at $A$ and the other 2 m from $B$. A man whose mass is 80 kg is standing on the plank 2 m from $A$ when his dog steps onto the plank at $B$. Given that the plank remains in equilibrium and that the magnitude of the forces exerted by each of the pivots on the plank are equal,
calculate the magnitude of the force exerted on the plank by the pivot at $A$,
find the dog's mass. If the dog was heavier and the plank was on the point of tilting,
explain how the force exerted on the plank by each of the pivots would be changed.
(2 marks)

OCR MEI M1 Q2

19 marks Standard +0.3

Obtain expressions, in terms of $U$ and $t$, for
(A) $x$,
(B) $y$.
The ball takes $T$ s to travel from O to P . Show that $T = \frac { U \sin 68.5 ^ { \circ } } { 4.9 }$ and write down a second equation connecting $U$ and $T$.
Hence show that $U = 12.0$ (correct to three significant figures).
Calculate the horizontal distance of the ball from the platform when the ball lands on the ground.
Use the expressions you found in part (i) to show that the cartesian equation of the trajectory of the ball in terms of $U$ is $$y = x \tan 68.5 ^ { \circ } - \frac { 4.9 x ^ { 2 } } { U ^ { 2 } \left( \cos 68.5 ^ { \circ } \right) ^ { 2 } }$$ Use this equation to show again that $U = 12.0$ (correct to three significant figures). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7bcde451-5c86-4ed6-b6f5-62c1ad77618c-3_391_1480_248_364} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows the graph of $y = \frac { 1 } { 100 } \left( 100 + 15 x - x ^ { 2 } \right)$.
For $0 \leqslant x \leqslant 20$, this graph shows the trajectory of a small stone projected from the point Q where $y \mathrm {~m}$ is the height of the stone above horizontal ground and $x \mathrm {~m}$ is the horizontal displacement of the stone from O . The stone hits the ground at the point R .
Write down the height of Q above the ground.
Find the horizontal distance from O of the highest point of the trajectory and show that this point is 1.5625 m above the ground.
Show that the time taken for the stone to fall from its highest point to the ground is 0.565 seconds, correct to 3 significant figures.
Show that the horizontal component of the velocity of the stone is $22.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 3 significant figures. Deduce the time of flight from Q to R .
Calculate the speed at which the stone hits the ground.

AQA M2 2012 January Q7

11 marks Standard +0.3

Show that $v ^ { 2 } = u ^ { 2 } - 4 a g$.
The ratio of the tensions in the string when the bead is at the two points $A$ and $B$ is $2 : 5$.
1. Find $u$ in terms of $g$ and $a$.
2. Find the ratio $u : v$.

AQA M2 2010 June Q7

12 marks Standard +0.8

Draw a diagram to show the forces acting on the rod.
Find the magnitude of the normal reaction force between the rod and the ground.
1. Find the normal reaction acting on the rod at $C$.
2. Find the friction force acting on the rod at $C$.
In this position, the rod is on the point of slipping. Calculate the coefficient of friction between the rod and the peg.
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-15_2484_1709_223_153}

Edexcel M2 Q2

5 marks Standard +0.3

Find the magnitude of the impulse exerted on $B$ by $A$, stating the units of your answer.
Find the speed of $B$ immediately after the collision. \item A small car, of mass 850 kg , moves on a straight horizontal road. Its engine is working at its maximum rate of 25 kW , and a constant resisting force of magnitude 900 N opposes the car's motion.
Find the acceleration of the car when it is moving with speed $15 \mathrm {~ms} ^ { - 1 }$.
Find the maximum speed of the car on the horizontal road. \end{enumerate} With the engine still working at 25 kW and the non-gravitational resistance remaining at 900 N , the car now climbs a hill inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 10 }$.
Find the maximum speed of the car on this hill. \section*{MECHANICS 2 (A)TEST PAPER 2 Page 2}

Edexcel M2 Q2

6 marks Standard +0.3

Find the value of $k$ if $P$ takes 4 seconds to reach $Y$.
Show that $Q$ has constant acceleration and find the magnitude and direction of this acceleration. \item Three particles $A , B$ and $C$, of equal size and each of mass $m$, are at rest on the same straight line on a smooth horizontal surface. The coefficient of restitution between $A$ and $B$, and between $B$ and $C$, is $e$. $A$ is projected with speed $7 \mathrm {~ms} ^ { - 1 }$ and strikes $B$ directly. $B$ then collides with $C$, which starts to move with speed $4 \mathrm {~ms} ^ { - 1 }$.
Calculate the value of $e$. \end{enumerate} \section*{MECHANICS 2 (A) TEST PAPER 4 Page 2}

Edexcel M2 Q2

7 marks Standard +0.3

Find the velocity vector of $P$ at time $t$ seconds.
Show that the direction of the acceleration of $P$ is constant.
Find the value of $t$ when the acceleration of $P$ has magnitude $12 \mathrm {~ms} ^ { - 2 }$. \item A uniform plank of wood $X Y$, of mass 1.4 kg , rests with its upper end $X$ against a rough vertical wall and its lower end $Y$ on rough horizontal ground. The coefficient of friction between the plank and both the wall and the ground is $\mu$. The plank is in limiting equilibrium at both ends and the vertical component of the force exerted on the plank by the ground has magnitude 12 N .
Find the value of $\mu$, to 2 decimal places. \item A motor-cycle and its rider have a total mass of 460 kg . The maximum rate at which the cycle's engine can work is 25920 W and the maximum speed of the cycle on a horizontal road is $36 \mathrm {~ms} ^ { - 1 }$. A variable resisting force acts on the cycle and has magnitude $k v ^ { 2 }$, where $v$ is the speed of the cycle in $\mathrm { ms } ^ { - 1 }$.
Show that $k = \frac { 5 } { 9 }$.
Find the acceleration of the cycle when it is moving at $25 \mathrm {~ms} ^ { - 1 }$ on the horizontal road, with its engine working at full power. \end{enumerate} \section*{MECHANICS 2 (A)TEST PAPER 8 Page 2}

OCR M2 2008 January Q6

11 marks Standard +0.3

Show that the tension in the string is 4.16 N , correct to 3 significant figures.
Calculate $\omega$.
(ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_510_417_1238_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The lower part of the string is now attached to a point $R$, vertically below $P$. $P B$ makes an angle $30 ^ { \circ }$ with the vertical and $R B$ makes an angle $60 ^ { \circ }$ with the vertical. The bead $B$ now moves in a horizontal circle of radius 1.5 m with constant speed $v _ { \mathrm { m } } \mathrm { m } ^ { - 1 }$ (see Fig. 2).
Calculate the tension in the string.
Calculate $v$.

OCR M2 2006 June Q6

11 marks Standard +0.3

Calculate the tension in the string and hence find the angular speed of $Q$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The particle $Q$ on the plane is now fixed to a point 0.2 m from the hole at $A$ and the particle $P$ rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
Calculate the tension in the string.
Calculate the speed of $P$.

OCR M2 2008 June Q5

8 marks Standard +0.3

Show that the distance from the ball to the centre of mass of the toy is 10.7 cm , correct to 1 decimal place.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_312_1051_1509_587} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The toy lies on horizontal ground in a position such that the ball is touching the ground (see Fig. 2). Determine whether the toy is lying in equilibrium or whether it will move to a position where the rod is vertical.

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\multirow{2}{*}{}		Second integer
		1	2	3	4	5	6
\multirow{6}{*}{First integer}	1	1	2	3	4	5	6
	2	2	2	6	4	10	6
	3	3	6	3	12	15	6
	4	4	4	12			12
	5	5	10	15
	6	6	6	6	12

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No of errors	0	1	2	3	4	5	6
No of pages	16	38	41	29	17	7	2

\(X\)	0	1	2	3	4	5
\(f\)	3	8	5	3	2	1