Questions Stats 1 - A-Level Maths

Find $\mathrm { f } ( - 1 )$.
Write down three linear factors of $\mathrm { f } ( x )$. It is given that $\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d$.
Show that $a = - 2$.
Find $b , c$ and $d$.

OCR Stats 1 2018 March Q2

2 Angela makes the following claim. \begin{displayquote} " $n$ is an odd positive integer greater than $1 \Rightarrow 2 ^ { n } - 1$ is prime" \end{displayquote} Prove that Angela's claim is false.

OCR Stats 1 2018 March Q3

3 On a particular voyage, a ship sails 500 km at a constant speed of $v \mathrm {~km} / \mathrm { h }$. The cost for the voyage is $\pounds R$ per hour. The total cost of the voyage is $\pounds T$.

Show that $T = \frac { 500 R } { v }$. The running cost is modelled by the following formula. $$R = 270 + \frac { v ^ { 3 } } { 200 }$$ The ship's owner wishes to sail at a speed that will minimise the total cost for the voyage. It is given that the graph of $T$ against $v$ has exactly one stationary point, which is a minimum.
Find the speed that gives the minimum value of $T$.
Find the minimum value of the total cost.

OCR Stats 1 2018 March Q4

4 The diagram shows part of the graph of $y = \cos x$, where $x$ is measured in radians.
\includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-05_609_846_294_607}

Use the copy of this diagram in the Printed Answer Booklet to find an approximate solution to the equation $x = \cos x$.
Use an iterative method to find the solution to the equation $x = \cos x$ correct to 3 significant figures. You should show your first, second and last two iterations, writing down all the figures on your calculator.

OCR Stats 1 2018 March Q5

5 Points $A , B$ and $C$ have position vectors $\left( \begin{array} { l } 1
2
3 \end{array} \right) , \left( \begin{array} { c } 2
- 1
5 \end{array} \right)$ and $\left( \begin{array} { c } - 4
0
3 \end{array} \right)$ respectively.

Find the exact distance between the midpoint of $A B$ and the midpoint of $B C$. Point $D$ has position vector $\left( \begin{array} { c } x
- 6
z \end{array} \right)$ and the line $C D$ is parallel to the line $A B$.
Find all the possible pairs of $x$ and $z$.

OCR Stats 1 2018 March Q6

6 In this question you must show detailed reasoning.

Use the formula for $\tan ( A - B )$ to show that $\tan \frac { \pi } { 12 } = 2 - \sqrt { 3 }$.
Solve the equation $2 \sqrt { 3 } \sin 3 A - 2 \cos 3 A = 1$ for $0 ^ { \circ } \leqslant A < 180 ^ { \circ }$.

OCR Stats 1 2018 March Q7

7 A tank is shaped as a cuboid. The base has dimensions 10 cm by 10 cm . Initially the tank is empty. Water flows into the tank at $25 \mathrm {~cm} ^ { 3 }$ per second. Water also leaks out of the tank at $4 h ^ { 2 } \mathrm {~cm} ^ { 3 }$ per second, where $h \mathrm {~cm}$ is the depth of the water after $t$ seconds. Find the time taken for the water to reach a depth of 2 cm .

OCR Stats 1 2018 March Q8

8 The masses, $X$ grams, of tomatoes are normally distributed. Half of the tomatoes have masses greater than 56.0 g and $70 \%$ of the tomatoes have masses greater than 53.0 g .

Find the percentage of tomatoes with masses greater than 59.0 g .
Find the percentage of tomatoes with masses greater than 65.0 g .
Given that $\mathrm { P } ( a < X < 50 ) = 0.1$, find $a$.

OCR Stats 1 2018 March Q9

9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number $X$ of these discs that are black.

Find $\mathrm { P } ( X = 3 )$. Paula decides to use the binomial distribution as a model for the distribution of $X$.
Explain why this model will give probabilities that are approximately, but not exactly, correct.
Paula uses the binomial model to find an approximate value for $\mathrm { P } ( X = 3 )$. Calculate the percentage by which her answer will differ from the answer in part (ii). Paula now assumes that the binomial distribution is a good model for $X$. She uses a computer simulation to generate 1000 values of $X$. The number of times that $X = 3$ occurs is denoted by $Y$.
Calculate estimates of the limits between which two thirds of the values of $Y$ will lie.

OCR Stats 1 2018 March Q10

7 marks

10 A researcher is investigating the actual lengths of time that patients spend at their appointments with the doctors at a certain clinic. There are 12 doctors at the clinic, and each doctor has 24 appointments per day. The researcher plans to choose a sample of 24 appointments on a particular day.

The researcher considers the following two methods for choosing the sample. Method A: Choose a random sample of 24 appointments from the 288 on that day.
Method B: Choose one doctor's 1st and 2nd appointments. Choose another doctor's 3rd and 4th appointments and so on until the last doctor's 23rd and 24th appointments. For each of A and B state a disadvantage of using this method. Appointments are scheduled to last 10 minutes. The researcher suspects that the actual times that patients spend are more than 10 minutes on average. To test this suspicion, he uses method A , and takes a random sample of 24 appointments. He notes the actual time spent for each appointment and carries out a hypothesis test at the $1 \%$ significance level.
Explain why a 1-tail test is appropriate. The population mean of the actual times that patients spend at their appointments is denoted by $\mu$ minutes.
Assuming that $\mu = 10$, state the probability that the conclusion of the test will be that $\mu$ is not greater than 10 . The actual lengths of time, in minutes, that patients spend for their appointments may be assumed to have a normal distribution with standard deviation 3.4.
[0pt]
Given that the total length of time spent for the 24 appointments is 285 minutes, carry out the test. [7]
In part (iv) it was necessary to use the fact that the sample mean is normally distributed. Give a reason why you know that this is true in this case.

OCR Stats 1 2018 March Q11

11 The scatter diagram shows data, taken from the pre-release data set, for several Local Authorities in one region of the UK in 2011. The diagram shows, for each Local Authority, the number of workers who drove to work, and the number of workers who walked to work. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{2011} \includegraphics[alt={},max width=\textwidth]{6a6316e4-7b2d-4533-988a-4863d79ce668-08_483_956_479_557}

\end{figure}

Four students calculated the value of Pearson's product-moment correlation coefficient for the data in the diagram. Their answers were $0.913,0.124 , - 0.913$ and - 0.124 . One of these values is correct. Without calculation state, with a reason, which is the correct value.
Sanjay makes the following statement.
"The diagram shows that, in any Local Authority, if there are a large number of people who drive to work there will be a large number who walk to work." Give a reason why this statement is incorrect.
Rosie makes the following statement.
"The diagram must be wrong because it shows good positive correlation. If there are more people driving to work, there will be fewer people walking to work, so there would be negative correlation." Explain briefly why Rosie's statement is incorrect.
The diagram shows a fairly close relationship between the two variables. One point on the diagram represents a Local Authority where this relationship is less strong than for the others. On the diagram in the Printed Answer Booklet, label this point A.
Given that the point A represents a metropolitan borough, suggest a reason why the relationship is less strong for this Local Authority than for the others in the region. The scatter diagram below shows the corresponding data for the same region in 2001. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{2001} \includegraphics[alt={},max width=\textwidth]{6a6316e4-7b2d-4533-988a-4863d79ce668-09_481_885_388_591}
\end{figure}
(a) State a change that has taken place in the metropolitan borough represented by the point A between 2001 and 2011.
(b) Suggest a possible reason for this change.

OCR Stats 1 2018 March Q12

12 Rob has two six-sided dice, each with sides numbered 1, 2, 3, 4, 5, 6.
One dice is fair. The other dice is biased, with probabilities as shown in the table.

Biased die
$y$	1	2	3	4	5	6
$\mathrm { P } ( Y = y )$	0.3	0.25	0.2	0.14	0.1	0.01

Rob throws each dice once and notes the two scores, $X$ on the fair dice and $Y$ on the biased dice. He then calculates the value of the variable $S$ which is defined as follows.

If $X \leqslant 3$, then $S = X + 2 Y$.
If $X > 3$, then $S = X + Y$.
1. (a) Draw up a sample space diagram showing all the possible outcomes and the corresponding values of $S$.
  (b) On your diagram, circle the four cells where the value $S = 10$ occurs.
2. Explain the mistake in the following calculation.

$$\mathrm { P } ( S = 10 ) = \frac { \text { Number of outcomes giving } S = 10 } { \text { Total number of outcomes } } = \frac { 4 } { 36 } = \frac { 1 } { 9 } .$$

Find the correct value of $\mathrm { P } ( S = 10 )$.

Given that $S = 10$, find the probability that the score on one of the dice is 4 .

The events " $X = 1$ or 2 " and " $S = n$ " are mutually exclusive. Given that $\mathrm { P } ( S = n ) \neq 0$, find the value of $n$.

OCR Stats 1 2018 September Q1

Differentiate the following with respect to $x$.
(a) $\frac { 1 } { ( 3 x - 4 ) ^ { 2 } }$
(b) $\frac { \ln ( x + 2 ) } { x }$
Find $\int \mathrm { e } ^ { ( 2 x + 3 ) } \mathrm { d } x$.

OCR Stats 1 2018 September Q2

Ben saves his pocket money as follows.
Each week he puts money into his piggy bank (which pays no interest). In the first week he puts in 10p. In the second week he puts in 12p. In the third week he puts in 14p, and so on. How much money does Ben have in his piggy bank after 25 weeks?
On January 1st Shirley invests $\pounds 500$ in a savings account that pays compound interest at $3 \%$ per annum. She makes no further payments into this account. The interest is added on 31st December each year.
(a) Find the number of years after which her investment will first be worth more than $\pounds 600$.
(b) State an assumption that you have made in answering part (ii)(a).

OCR Stats 1 2018 September Q3

3 Use small angle approximations to estimate the solution of the equation $\frac { \cos \frac { 1 } { 2 } \theta } { 1 + \sin \theta } = 0.825$, if $\theta$ is small enough to neglect terms in $\theta ^ { 3 }$ or above.

OCR Stats 1 2018 September Q4

4 Prove that the sum of the squares of any two consecutive integers is of the form $4 k + 1$, where $k$ is an integer.

OCR Stats 1 2018 September Q5

5 The diagram shows the graph of $y = \sin x ^ { \circ }$ for $0 \leqslant x \leqslant 360$.
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-4_597_965_1909_539} Find an equation for the transformed curve when the curve $y = \sin x ^ { \circ }$ is reflected in

the $x$-axis,
the line $y = 0.5$.

OCR Stats 1 2018 September Q6

Find the coefficient of $x ^ { 4 }$ in the expansion of $( 3 x - 2 ) ^ { 10 }$.
In the expansion of $( 1 + 2 x ) ^ { n }$, where $n$ is a positive integer, the coefficients of $x ^ { 7 }$ and $x ^ { 8 }$ are equal. Find the value of $n$.
Find the coefficient of $x ^ { 3 }$ in the expansion of $\frac { 1 } { \sqrt { 4 + x } }$.

OCR Stats 1 2018 September Q7

7 The diagram shows part of the curve $y = x ^ { 2 }$ for $0 \leqslant x \leqslant p$, where $p$ is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_736_543_669_762} The area $A$ of the region enclosed by the curve, the $x$-axis and the line $x = p$ is given approximately by the sum $S$ of the areas of $n$ rectangles, each of width $h$, where $h$ is small and $n h = p$. The first three such rectangles are shown in the diagram.

Find an expression for $S$ in terms of $n$ and $h$.
Use the identity $\sum _ { r = 1 } ^ { n } r ^ { 2 } \equiv \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$ to show that $S = \frac { 1 } { 6 } p ( p + h ) ( 2 p + h )$.
Show how to use this result to find $A$ in terms of $p$.

OCR Stats 1 2018 September Q8

8 The points $A , B$ and $C$ have position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$, relative to an origin $O$, in three dimensions. The figure $O A P B S C T U$ is a cuboid, with vertices labelled as in the following diagram. $M$ is the midpoint of $A U$.
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_557_1221_2087_420}

OCR Stats 1 2018 September Q9

9 The finance department of a retail firm recorded the daily income each day for 300 days. The results are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-6_689_1575_488_246}

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.

Biased die
\(y\)	1	2	3	4	5	6
\(\mathrm { P } ( Y = y )\)	0.3	0.25	0.2	0.14	0.1	0.01

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