Questions - OCR - A-Level Maths

Find the number of days on which the daily income was between $\pounds 4000$ and $\pounds 6000$.
Calculate an estimate of the number of days on which the daily income was between $\pounds 2700$ and $\pounds 3600$.
Use the midpoints of the classes to show that an estimate of the mean daily income is $\pounds 3275$. An estimate of the standard deviation of the daily income is $\pounds 1060$. The finance department uses the distribution $\mathrm { N } \left( 3275,1060 ^ { 2 } \right)$ to model the daily income, in pounds.
Calculate the number of days on which, according to this model, the daily income would be between $\pounds 4000$ and $\pounds 6000$.
It is given that approximately $95 \%$ of values of the distribution $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$ lie within the range $\mu \pm 2 \sigma$. Without further calculation, use this fact to comment briefly on whether the proposed model is a good fit to the data illustrated in the histogram.

OCR Stats 1 2018 September Q10

10 The table shows information, derived from the 2011 UK census, about the percentage of employees who used various methods of travel to work in four Local Authorities.

Local Authority	Underground, metro, light rail or tram	Train	Bus	Drive	Walk or cycle
A	0.3\%	4.5\%	17\%	52.8\%	11\%
B	0.2\%	1.7\%	1.7\%	63.4\%	11\%
C	35.2\%	3.0\%	12\%	11.7\%	16\%
D	8.9\%	1.4\%	9\%	54.7\%	10\%

One of the Local Authorities is a London borough and two are metropolitan boroughs, not in London.

Which one of the Local Authorities is a London borough? Give a reason for your answer.
Which two of the Local Authorities are metropolitan boroughs outside London? In each case give a reason for your answer.
Describe one difference between the public transport available in the two metropolitan boroughs, as suggested by the table.
Comment on the availability of public transport in Local Authority B as suggested by the table.

OCR Stats 1 2018 September Q11

11 In an experiment involving a bivariate distribution ( $X , Y$ ) a random sample of 7 pairs of values was obtained and Pearson's product-moment correlation coefficient $r$ was calculated for these values.

The value of $r$ was found to be 0.894 . Use the table below to test, at the $5 \%$ significance level, whether there is positive linear correlation in the population, stating your hypotheses and conclusion clearly.

1-tail test 2-tail test	5\%	2.5\%	1\%	0.5\%
	10\%	5\%	2\%	1\%
$n$
1	-	-	-	-
2	-	-	-	-
3	0.9877	0.9969	0.9995	0.9999
4	0.9000	0.9500	0.9800	0.9900
5	0.8054	0.8783	0.9343	0.9587
6	0.7293	0.8114	0.8822	0.9587
7	0.6694	0.7545	0.8329	0.9745
8	0.6215	0.7067	0.7887	0.8343
9	0.5882	0.6664	0.7498	0.7977
10	0.5494	0.6319	0.7155	0.7646

Scatter diagrams for four sets of bivariate data, are shown. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_380_371_301_191} \captionsetup{labelformat=empty} \caption{Diagram A}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_373_373_301_628} \captionsetup{labelformat=empty} \caption{Diagram B}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1064} \captionsetup{labelformat=empty} \caption{Diagram C}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1503} \captionsetup{labelformat=empty} \caption{Diagram D}

\end{figure} It is given that $r = 0.894$ for one of these diagrams.

For each of the other diagrams, state how you can tell that $r \neq 0.894$.

OCR Stats 1 2018 September Q12

12 In the past, the time spent by customers in a certain shop had mean 10.5 minutes and standard deviation 4.2 minutes. Following a change of layout in the shop, the mean time spent in the shop by a random sample of 50 customers is found to be 12.0 minutes.

Assuming that the standard deviation is unchanged, test at the $1 \%$ significance level whether the mean time spent by customers in the shop has changed.
Another random sample of 50 customers is chosen and a similar test at the $1 \%$ significance level is carried out. Given that the population mean time has not changed, state the probability that the conclusion of the test will be that the population mean time has changed.

OCR Stats 1 2018 September Q13

13 Bag A contains 3 black discs and 2 white discs only. Initially Bag B is empty. Discs are removed at random from bag A, and are placed in bag B, one at a time, until all 5 discs are in bag B.

Write down the probability that the last disc that is placed in bag B is black.
Find the probability that the first disc and the last disc that are placed in bag B are both black.
Find the probability that, starting from when the first disc is placed in bag B , the number of black discs in bag B is always greater than the number of white discs in bag B.

OCR Stats 1 2018 September Q14

14 A counter is initially at point $O$ on the $x$-axis. A fair coin is thrown 6 times. Each time the coin shows heads, the counter is moved one unit in the positive $x$-direction. Each time the coin shows tails, the counter is moved one unit in the negative $x$-direction. The final distance of the counter from $O$, in either direction, is denoted by $D$. Determine the most probable value of $D$. \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}

OCR Mechanics 1 2018 September Q1

Show that $4 x ^ { 2 } - 12 x + 3 = 4 \left( x - \frac { 3 } { 2 } \right) ^ { 2 } - 6$.
State the coordinates of the minimum point of the curve $y = 4 x ^ { 2 } - 12 x + 3$.

OCR Mechanics 1 2018 September Q2

2 A curve has equation $y = a x ^ { 4 } + b x ^ { 3 } - 2 x + 3$.

Given that the curve has a stationary point where $x = 2$, show that $16 a + 6 b = 1$.
Given also that this stationary point is a point of inflection, determine the values of $a$ and $b$.

OCR Mechanics 1 2018 September Q3

Given that $\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta$, show that $6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$.
In this question you must show detailed reasoning. Solve the equation $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ giving all values of $\theta$ between $0 ^ { \circ }$ and $360 ^ { \circ }$ correct to 1 decimal place.
Explain why not all the solutions from part (ii) are solutions of the equation $$\sqrt { 2 \sin ^ { 2 } \theta + \cos \theta } = 2 \cos \theta$$

OCR Mechanics 1 2018 September Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-05_787_892_267_568} The diagram shows the graph of $y = \mathrm { f } ( x )$, where $$f ( x ) = \begin{cases} 4 - 4 x , & x \leqslant a ,
\ln ( b x - 8 ) - 2 , & x \geqslant a . \end{cases}$$ The range of f is $\mathrm { f } ( x ) \geqslant - 2$.

Show that $a = \frac { 3 } { 2 }$.
Find the value of $b$.
Find the exact value of $\mathrm { ff } ( - 1 )$.
Explain why the function f does not have an inverse.

OCR Mechanics 1 2018 September Q5

5 The curve $C$ has equation $$3 x ^ { 2 } - 5 x y + \mathrm { e } ^ { 2 y - 4 } + 6 = 0$$ The point $P$ with coordinates $( 1,2 )$ lies on $C$. The tangent to $C$ at $P$ meets the $y$-axis at the point $A$ and the normal to $C$ at $P$ meets the $y$-axis at the point $B$. Find the exact area of triangle $A B P$.

OCR Mechanics 1 2018 September Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-06_463_702_264_685} The diagram shows the curve $C$ with parametric equations $$x = \frac { 1 } { 4 } \sin t , \quad y = t \cos t$$ where $0 \leqslant t \leqslant k$.

Find the value of $k$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} t }$ in terms of $t$. The maximum point on $C$ is denoted by $P$.
Using your answer to part (ii) and the standard small angle approximations, find an approximation for the $x$-coordinate of $P$.
(a) Show that the area of the finite region bounded by $C$ and the $x$-axis is given by $$b \int _ { 0 } ^ { a } t ( 1 + \cos 2 t ) \mathrm { d } t$$ where $a$ and $b$ are constants to be determined.
(b) In this question you must show detailed reasoning. Hence find the exact area of the finite region bounded by $C$ and the $x$-axis.

OCR Mechanics 1 2018 September Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-07_512_1072_484_502} The diagram shows the velocity-time graph for a train travelling on a straight level track between stations $A$ and $B$ that are 2 km apart. The train leaves $A$, accelerating uniformly from rest for 400 m until reaching a speed of $32 \mathrm {~ms} ^ { - 1 }$. The train then travels at this steady speed for $T$ seconds before decelerating uniformly at $1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, coming to rest at $B$. Find the total time for the journey.

OCR Mechanics 1 2018 September Q8

8 At time $t$ seconds a particle $P$ has position vector $\mathbf { r }$ metres, with respect to a fixed origin $O$, where $$\mathbf { r } = \left( 4 t ^ { 2 } - k t + 5 \right) \mathbf { i } + \left( 4 t ^ { 3 } + 2 k t ^ { 2 } - 8 t \right) \mathbf { j } , \quad t \geqslant 0 .$$ When $t = 2 , P$ is moving parallel to the vector $\mathbf { i }$.

Show that $k = - 5$.
Find the values of $t$ when the magnitude of the acceleration of $P$ is $10 \mathrm {~ms} ^ { - 2 }$.

OCR Mechanics 1 2018 September Q9

9
\includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-08_302_992_260_539} The diagram shows a plank of wood $A B$, of mass 10 kg and length 6 m , resting with its end $A$ on rough horizontal ground and its end $B$ in contact with a fixed cylindrical oil drum. The plank is in a vertical plane perpendicular to the axis of the drum, and the line $A B$ is a tangent to the circular cross-section of the drum, with the point of contact at $B$. The plank is inclined at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 5 } { 12 }$. The plank is modelled as a uniform rod and the oil drum is modelled as being smooth.

Find, in terms of $g$, the normal contact force between the drum and the plank.
Given that the plank is in limiting equilibrium, find the coefficient of friction between the plank and the ground.

OCR Mechanics 1 2018 September Q10

10 A small ball $P$ is projected with speed $5 \mathrm {~ms} ^ { - 1 }$ at an angle $\theta$ above the horizontal from a point $O$ and moves freely under gravity. The horizontal and vertically upwards displacements of the ball from $O$ at any subsequent time $t$ seconds are $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively. The ball is modelled as a particle and the acceleration due to gravity is taken to be $10 \mathrm {~ms} ^ { - 2 }$.

Show that the equation of the trajectory of $P$ is $$y = x \tan \theta - \frac { x ^ { 2 } } { 5 } \left( 1 + \tan ^ { 2 } \theta \right)$$ It is given that $\tan \theta = 3$.
Using part (i), find the maximum height above the level of $O$ of $P$ in the subsequent motion.
Find the values of $t$ when $P$ is moving at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 1 } { 3 }$.
Give two possible reasons why the values of $t$ found in part (iii) may not be accurate.
\includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-09_435_714_267_678} Two particles $P$ and $Q$, with masses 2 kg and 8 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. Plane $\Pi _ { 1 }$ is inclined at an angle of $30 ^ { \circ }$ to the horizontal and plane $\Pi _ { 2 }$ is inclined at an angle of $\theta$ to the horizontal. Particle $P$ is on $\Pi _ { 1 }$ and $Q$ is on $\Pi _ { 2 }$ with the string taut (see diagram).
$\Pi _ { 1 }$ is rough and the coefficient of friction between $P$ and $\Pi _ { 1 }$ is $\frac { \sqrt { 3 } } { 3 }$.
$\Pi _ { 2 }$ is smooth.
The particles are released from rest and $P$ begins to move towards the pulley with an acceleration of $g \cos \theta \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Show that $\theta$ satisfies the equation $$4 \sin \theta - 5 \cos \theta = 1 .$$
By expressing $4 \sin \theta - 5 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$, find, correct to 3 significant figures, the tension in the string.

OCR Pure 1 2018 December Q1

1 In this question you must show detailed reasoning. Andrea is comparing the prices charged by two different taxi firms.
Firm A charges $\pounds 20$ for a 5 mile journey and $\pounds 30$ for a 10 mile journey, and there is a linear relationship between the price and the length of the journey.
Firm B charges a pick-up fee of $\pounds 3$ and then $\pounds 2.40$ for each mile travelled.
Find the length of journey for which both firms would charge the same amount.

OCR Pure 1 2018 December Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-4_661_579_831_246} The diagram shows a patio. The perimeter of the patio has to be less than 44 m .
The area of the patio has to be at least $45 \mathrm {~m} ^ { 2 }$.

Write down, in terms of $x$, an inequality satisfied by
1. the perimeter of the patio,
2. the area of the patio.
Hence determine the set of possible values of $x$.

OCR Pure 1 2018 December Q3

3 In this question you must show detailed reasoning.

Given that $\sin \alpha = \frac { 2 } { 3 }$, find the exact values of $\cos \alpha$.
Given that $2 \tan ^ { 2 } \beta - 7 \sec \beta + 5 = 0$, find the exact value of $\sec \beta$.

OCR Pure 1 2018 December Q4

4 In this question you must show detailed reasoning. Solve the simultaneous equations
$\mathrm { e } ^ { x } - 2 \mathrm { e } ^ { y } = 3$
$\mathrm { e } ^ { 2 x } - 4 \mathrm { e } ^ { 2 y } = 33$. Give your answer in an exact form.

OCR Pure 1 2018 December Q5

Given that $\mathrm { f } ( x ) = x ^ { 2 } - 4 x$, use differentiation from first principles to show that $\mathrm { f } ^ { \prime } ( x ) = 2 x - 4$.
Find the equation of the curve through $( 2,7 )$ for which $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x - 4$.

OCR Pure 1 2018 December Q6

6 In this question you must show detailed reasoning.
A sequence $S$ has terms $u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots$ defined by $u _ { 1 } = 500$ and $u _ { n + 1 } = 0.8 u _ { n }$.

State whether $S$ is an arithmetic sequence or a geometric sequence, giving a reason for your answer.
Find $u _ { 20 }$.
Find $\sum _ { n = 1 } ^ { 20 } u _ { n }$.
Given that $\sum _ { n = k } ^ { \infty } u _ { n } = 1024$, find the value of $k$.

OCR Pure 1 2018 December Q7

7 As a spherical snowball melts its volume decreases. The rate of decrease of the volume of the snowball at any given time is modelled as being proportional to its volume at that time. Initially the volume of the snowball is $500 \mathrm {~cm} ^ { 3 }$ and the rate of decrease of its volume is $20 \mathrm {~cm} ^ { 3 }$ per hour.

Find the time that this model would predict for the snowball's volume to decrease to $250 \mathrm {~cm} ^ { 3 }$.
Write down one assumption made when using this model.
Comment on how realistic this model would be in the long term.

OCR Pure 1 2018 December Q8

Expand $\sqrt { 1 + 2 x }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Hence expand $\frac { \sqrt { 1 + 2 x } } { 1 + 9 x ^ { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Determine the range of values of $x$ for which the expansion in part (b) is valid.

OCR Pure 1 2018 December Q9

9 A function f is defined for $x > 0$ by $\mathrm { f } ( x ) = \frac { 6 } { x ^ { 2 } + a }$, where $a$ is a positive constant.

Show that f is a decreasing function.
Find, in terms of $a$, the coordinates of the point of inflection on the curve $y = \mathrm { f } ( x )$.

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