Questions - AQA - A-Level Maths

Show that the exact value of $k$ is $q - \ln 2$, where $q$ is a rational number.
Find the gradient of the curve at $P$.

AQA C4 2015 June Q8

12 marks Standard +0.3

A pond is initially empty and is then filled gradually with water. After $t$ minutes, the depth of the water, $x$ metres, satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \sqrt { 4 + 5 x } } { 5 ( 1 + t ) ^ { 2 } }$$ Solve this differential equation to find $x$ in terms of $t$.
Another pond is gradually filling with water. After $t$ minutes, the surface of the water forms a circle of radius $r$ metres. The rate of change of the radius is inversely proportional to the area of the surface of the water.
1. Write down a differential equation, in the variables $r$ and $t$ and a constant of proportionality, which represents how the radius of the surface of the water is changing with time.
  (You are not required to solve your differential equation.)
2. When the radius of the pond is 1 metre, the radius is increasing at a rate of 4.5 metres per second. Find the radius of the pond when the radius is increasing at a rate of 0.5 metres per second.
  [0pt] [2 marks]
  \includegraphics[max width=\textwidth, alt={}]{fdd3905e-11f7-4b20-adfe-4c686018a221-18_1277_1709_1430_153}
  
  \includegraphics[max width=\textwidth, alt={}]{fdd3905e-11f7-4b20-adfe-4c686018a221-20_2288_1707_221_153}

AQA C4 2016 June Q1

11 marks Moderate -0.3

Express $\frac { 19 x - 3 } { ( 1 + 2 x ) ( 3 - 4 x ) }$ in the form $\frac { A } { 1 + 2 x } + \frac { B } { 3 - 4 x }$.
1. Find the binomial expansion of $\frac { 19 x - 3 } { ( 1 + 2 x ) ( 3 - 4 x ) }$ up to and including the term in $x ^ { 2 }$.
2. State the range of values of $x$ for which this expansion is valid.
  [0pt] [1 mark]

AQA C4 2016 June Q3

8 marks Moderate -0.3

Express $\frac { 3 + 13 x - 6 x ^ { 2 } } { 2 x - 3 }$ in the form $A x + B + \frac { C } { 2 x - 3 }$.
Show that $\int _ { 3 } ^ { 6 } \frac { 3 + 13 x - 6 x ^ { 2 } } { 2 x - 3 } \mathrm {~d} x = p + q \ln 3$, where $p$ and $q$ are rational numbers.
[0pt] [4 marks]

AQA C4 2016 June Q4

7 marks Moderate -0.3

4 The mass of radioactive atoms in a substance can be modelled by the equation $$m = m _ { 0 } k ^ { t }$$ where $m _ { 0 }$ grams is the initial mass, $m$ grams is the mass after $t$ days and $k$ is a constant. The value of $k$ differs from one substance to another.

1. A sample of radioactive iodine reduced in mass from 24 grams to 12 grams in 8 days. Show that the value of the constant $k$ for this substance is 0.917004 , correct to six decimal places.
2. A similar sample of radioactive iodine reduced in mass to 1 gram after 60 days. Calculate the initial mass of this sample, giving your answer to the nearest gram.
The half-life of a radioactive substance is the time it takes for a mass of $m _ { 0 }$ to reduce to a mass of $\frac { 1 } { 2 } m _ { 0 }$. A sample of radioactive vanadium reduced in mass from exactly 10 grams to 8.106 grams in 100 days. Find the half-life of radioactive vanadium, giving your answer to the nearest day. [4 marks]
\includegraphics[max width=\textwidth, alt={}]{c42685e9-bfa4-48d4-8abb-13e88a4b765e-08_1182_1707_1525_153}

AQA C4 2016 June Q5

10 marks Moderate -0.3

5 It is given that $\sin A = \frac { \sqrt { 5 } } { 3 }$ and $\sin B = \frac { 1 } { \sqrt { 5 } }$, where the angles $A$ and $B$ are both acute.

1. Show that the exact value of $\cos B = \frac { 2 } { \sqrt { 5 } }$.
2. Hence show that the exact value of $\sin 2 B$ is $\frac { 4 } { 5 }$.
1. Show that the exact value of $\sin ( A - B )$ can be written as $p ( 5 - \sqrt { 5 } )$, where $p$ is a rational number.
2. Find the exact value of $\cos ( A - B )$ in the form $r + s \sqrt { 5 }$, where $r$ and $s$ are rational numbers.

AQA C4 2016 June Q6

15 marks Challenging +1.2

6 The line $l _ { 1 }$ passes through the point $A ( 0,6,9 )$ and the point $B ( 4 , - 6 , - 11 )$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } - 1 \\ 5 \\ - 2 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ - 5 \\ 1 \end{array} \right]$.

The acute angle between the lines $l _ { 1 }$ and $l _ { 2 }$ is $\theta$. Find the value of $\cos \theta$ as a fraction in its lowest terms.
Show that the lines $l _ { 1 }$ and $l _ { 2 }$ intersect and find the coordinates of the point of intersection.
The points $C$ and $D$ lie on line $l _ { 2 }$ such that $A C B D$ is a parallelogram.
\includegraphics[max width=\textwidth, alt={}, center]{c42685e9-bfa4-48d4-8abb-13e88a4b765e-12_392_949_1018_548} The length of $A B$ is three times the length of $C D$.
Find the coordinates of the points $C$ and $D$.
[0pt] [5 marks] $7 \quad$ A curve $C$ is defined by the parametric equations $$x = \frac { 4 - \mathrm { e } ^ { 2 - 6 t } } { 4 } , \quad y = \frac { \mathrm { e } ^ { 3 t } } { 3 t } , \quad t \neq 0$$

AQA C4 2016 June Q8

10 marks Standard +0.3

8 It is given that $\theta = \tan ^ { - 1 } \left( \frac { 3 x } { 2 } \right)$.

By writing $\theta = \tan ^ { - 1 } \left( \frac { 3 x } { 2 } \right)$ as $2 \tan \theta = 3 x$, use implicit differentiation to show that $\frac { \mathrm { d } \theta } { \mathrm { d } x } = \frac { k } { 4 + 9 x ^ { 2 } }$, where $k$ is an integer.
[0pt] [3 marks]
Hence solve the differential equation $$9 y \left( 4 + 9 x ^ { 2 } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } 3 y$$ given that $x = 0$ when $y = \frac { \pi } { 3 }$. Give your answer in the form $\mathrm { g } ( y ) = \mathrm { h } ( x )$.
[0pt] [7 marks]

AQA S1 2006 January Q1

11 marks Moderate -0.8

1 At a certain small restaurant, the waiting time is defined as the time between sitting down at a table and a waiter first arriving at the table. This waiting time is dependent upon the number of other customers already seated in the restaurant. Alex is a customer who visited the restaurant on 10 separate days. The table shows, for each of these days, the number, $x$, of customers already seated and his waiting time, $y$ minutes.

$\boldsymbol { x }$	9	3	4	10	8	12	7	11	2	6
$\boldsymbol { y }$	11	6	5	11	9	13	9	12	4	7

Calculate the equation of the least squares regression line of $y$ on $x$ in the form $y = a + b x$.
Give an interpretation, in context, for each of your values of $a$ and $b$.
Use your regression equation to estimate Alex's waiting time when the number of customers already seated in the restaurant is:
1. 5 ;
2. 25 .
Comment on the likely reliability of each of your estimates in part (c), given that, for the regression line calculated in part (a), the values of the 10 residuals lie between + 1.1 minutes and - 1.1 minutes.

AQA S1 2006 January Q2

10 marks Moderate -0.8

2 Xavier, Yuri and Zara attend a sports centre for their judo club's practice sessions. The probabilities of them arriving late are, independently, $0.3,0.4$ and 0.2 respectively.

Calculate the probability that for a particular practice session:
1. all three arrive late;
2. none of the three arrives late;
3. only Zara arrives late.
Zara's friend, Wei, also attends the club's practice sessions. The probability that Wei arrives late is 0.9 when Zara arrives late, and is 0.25 when Zara does not arrive late. Calculate the probability that for a particular practice session:
1. both Zara and Wei arrive late;
2. either Zara or Wei, but not both, arrives late.

AQA S1 2006 January Q3

8 marks Easy -1.2

3 When an alarm is raised at a market town's fire station, the fire engine cannot leave until at least five fire-fighters arrive at the station. The call-out time, $X$ minutes, is the time between an alarm being raised and the fire engine leaving the station. The value of $X$ was recorded on a random sample of 50 occasions. The results are summarised below, where $\bar { x }$ denotes the sample mean. $$\sum x = 286.5 \quad \sum ( x - \bar { x } ) ^ { 2 } = 45.16$$

Find values for the mean and standard deviation of this sample of 50 call-out times.
Hence construct a $99 \%$ confidence interval for the mean call-out time.
The fire and rescue service claims that the station's mean call-out time is less than 5 minutes, whereas a parish councillor suggests that it is more than $6 \frac { 1 } { 2 }$ minutes. Comment on each of these claims.

AQA S1 2006 January Q4

10 marks Moderate -0.3

4 The time, $x$ seconds, spent by each of a random sample of 100 customers at an automatic teller machine (ATM) is recorded. The times are summarised in the table.

Time (seconds)	Number of customers
$20 < x \leqslant 30$	2
$30 < x \leqslant 40$	7
$40 < x \leqslant 60$	18
$60 < x \leqslant 80$	27
$80 < x \leqslant 100$	23
$100 < x \leqslant 120$	13
$120 < x \leqslant 150$	7
$150 < x \leqslant 180$	3
Total	100

Calculate estimates for the mean and standard deviation of the time spent at the ATM by a customer.
The mean time spent at the ATM by a random sample of $\mathbf { 3 6 }$ customers is denoted by $\bar { Y }$.
1. State why the distribution of $\bar { Y }$ is approximately normal.
2. Write down estimated values for the mean and standard error of $\bar { Y }$.
3. Hence estimate the probability that $\bar { Y }$ is less than $1 \frac { 1 } { 2 }$ minutes.

AQA S1 2006 January Q5

11 marks Easy -1.2

5 [Figure 1, printed on the insert, is provided for use in this question.]
The table shows the times, in seconds, taken by a random sample of 10 boys from a junior swimming club to swim 50 metres freestyle and 50 metres backstroke.

Boy	A	B	C	D	E	F	G	H	I	J
Freestyle ( $\boldsymbol { x }$ seconds)	30.2	32.8	25.1	31.8	31.2	35.6	32.4	38.0	36.1	34.1
Backstroke ( $y$ seconds)	33.5	35.4	37.4	27.2	34.7	38.2	37.7	41.4	42.3	38.4

On Figure 1, complete the scatter diagram for these data.
Hence:
1. give two distinct comments on what your scatter diagram reveals;
2. state, without calculation, which of the following 3 values is most likely to be the value of the product moment correlation coefficient for the data in your scatter diagram. $$0.912 \quad 0.088 \quad 0.462$$
In the sample of 10 boys, one boy is a junior-champion freestyle swimmer and one boy is a junior-champion backstroke swimmer. Identify the two most likely boys.
Removing the data for the two boys whom you identified in part (c):
1. calculate the value of the product moment correlation coefficient for the remaining 8 pairs of values of $x$ and $y$;
2. comment, in context, on the value that you obtain.

AQA S1 2006 January Q6

11 marks Standard +0.3

6 Plastic clothes pegs are made in various colours.
The number of red pegs may be modelled by a binomial distribution with parameter $p$ equal to 0.2 . The contents of packets of 50 pegs of mixed colours may be considered to be random samples.

Determine the probability that a packet contains:
1. less than or equal to 15 red pegs;
2. exactly 10 red pegs;
3. more than 5 but fewer than 15 red pegs.
Sly, a student, claims to have counted the number of red pegs in each of 100 packets of 50 pegs. From his results the following values are calculated. Mean number of red pegs per packet $= 10.5$
Variance of number of red pegs per packet $= 20.41$
Comment on the validity of Sly's claim.

AQA S1 2006 January Q7

14 marks Standard +0.3

The weight, $X$ grams, of soup in a carton may be modelled by a normal random variable with mean 406 and standard deviation 4.2. Find the probability that the weight of soup in a carton:
1. is less than 400 grams;
2. is between 402.5 grams and 407.5 grams.
The weight, $Y$ grams, of chopped tomatoes in a tin is a normal random variable with mean $\mu$ and standard deviation $\sigma$.
1. Given that $\mathrm { P } ( Y < 310 ) = 0.975$, explain why: $$310 - \mu = 1.96 \sigma$$
2. Given that $\mathrm { P } ( Y < 307.5 ) = 0.86$, find, to two decimal places, values for $\mu$ and $\sigma$.
  (4 marks)

AQA S1 2008 January Q1

12 marks Moderate -0.8

1 In large-scale tree-felling operations, a machine cuts down trees, strips off the branches and then cuts the trunks into logs of length $X$ metres for transporting to a sawmill. It may be assumed that values of $X$ are normally distributed with mean $\mu$ and standard deviation 0.16 , where $\mu$ can be set to a specific value.

Given that $\mu$ is set to 3.3 , determine:
1. $\mathrm { P } ( X < 3.5 )$;
2. $\mathrm { P } ( X > 3.0 )$;
3. $\mathrm { P } ( 3.0 < X < 3.5 )$.
The sawmill now requires a batch of logs such that there is a probability of 0.025 that any given log will have a length less than 3.1 metres. Determine, to two decimal places, the new value of $\mu$.

AQA S1 2008 January Q2

7 marks Moderate -0.8

2 The head and body length, $x$ millimetres, and tail length, $y$ millimetres, of each of a sample of 20 adult dormice were measured. The following statistics are derived from the results. $$S _ { x x } = 1280.55 \quad S _ { y y } = 281.8 \quad S _ { x y } = 416.3$$

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
Interpret your value in the context of this question.
Write down the value of the product moment correlation coefficient if the measurements had been recorded in centimetres.
Give a reason why it is not generally advisable to calculate the value of the product moment correlation coefficient without first viewing a scatter diagram of the data. Illustrate your answer with a sketch.

AQA S1 2008 January Q3

8 marks Moderate -0.3

3 The height, in metres, of adult male African elephants may be assumed to be normally distributed with mean $\mu$ and standard deviation 0.20 . The heights of a sample of 12 such elephants were measured with the following results, in metres. $$\begin{array} { l l l l l l l l l l l l } 3.37 & 3.45 & 2.93 & 3.42 & 3.49 & 3.67 & 2.96 & 3.57 & 3.36 & 2.89 & 3.22 & 2.91 \end{array}$$

Stating a necessary assumption, construct a $98 \%$ confidence interval for $\mu$. (6 marks)
The mean height of adult male Asian elephants is known to be 2.90 metres. Using your confidence interval, state, with a reason, what can be concluded about the mean heights of adult males in these two types of elephant.

AQA S1 2008 January Q4

12 marks Moderate -0.3

4 [Figure 1, printed on the insert, is provided for use in this question.]
Roseen is a self-employed decorator who wishes to estimate the times that it will take her to decorate bedrooms based upon their floor areas. She records the floor area, $x \mathrm {~m} ^ { 2 }$, and the decorating time, $y$ hours, for each of 10 bedrooms she has recently decorated.

$\boldsymbol { x }$	11.0	22.0	7.5	21.0	13.0	16.5	14.0	16.0	18.5	20.5
$\boldsymbol { y }$	15.0	35.0	16.0	23.5	24.0	17.5	14.5	27.5	22.5	34.5

On Figure 1, plot a scatter diagram of these data.
Calculate the equation of the least squares regression line of $y$ on $x$.
Draw your regression line on Figure 1.
1. Use your regression equation to estimate the time that Roseen will take to decorate a bedroom with a floor area of $15 \mathrm {~m} ^ { 2 }$.
2. Making reference to Figure 1, comment on the likely reliability of your estimate in part (d)(i).

AQA S1 2008 January Q5

12 marks Easy -1.2

5 A health club has a number of facilities which include a gym and a sauna. Andrew and his wife, Heidi, visit the health club together on Tuesday evenings. On any visit, Andrew uses either the gym or the sauna or both, but no other facilities. The probability that he uses the gym, $\mathrm { P } ( G )$, is 0.70 . The probability that he uses the sauna, $\mathrm { P } ( S )$, is 0.55 . The probability that he uses both the gym and the sauna is 0.25 .

Calculate the probability that, on a particular visit:
1. he does not use the gym;
2. he uses the gym but not the sauna;
3. he uses either the gym or the sauna but not both.
Assuming that Andrew's decision on what facility to use is independent from visit to visit, calculate the probability that, during a month in which there are exactly four Tuesdays, he does not use the gym.
The probability that Heidi uses the gym when Andrew uses the gym is 0.6 , but is only 0.1 when he does not use the gym. Calculate the probability that, on a particular visit, Heidi uses the gym.
On any visit, Heidi uses exactly one of the club's facilities. The probability that she uses the sauna is 0.35 .
Calculate the probability that, on a particular visit, she uses a facility other than the gym or the sauna.

AQA S1 2008 January Q6

12 marks Moderate -0.8

6 For each of the Premiership football seasons 2004/05 and 2005/06, a count is made of the number of goals scored in each of the 380 matches. The results are shown in the table.

\multirow{2}{*}{Number of goals scored in a match}	Number of matches
	2004/05	2005/06
0	30	32
1	79	82
2	99	95
3	68	78
4	60	48
5	24	30
6	11	9
7	6	6
8	2	0
9	1	0
Total	380	380

For the number of goals scored in a match during the 2004/05 season:
1. determine the median and the interquartile range;
2. calculate the mean and the standard deviation.
Two statistics students, Jole and Katie, independently analyse the data on the number of goals scored in a match during the 2005/06 season.
- Jole determines correctly that the median is 2 and that the interquartile range is also 2.
- Katie calculates correctly, to two decimal places, that the mean is 2.48 and that the standard deviation is 1.59 .
  1. Use your answers from part (a), together with Jole's and Katie's results, to compare briefly the two seasons with regard to the average and the spread of the number of goals scored in a match.
  2. Jole claims that Katie's results must be wrong as $95 \%$ of values always lie within 2 standard deviations of the mean and $( 2.48 - 2 \times 1.59 ) < 0$ which is nonsense.
Explain why Jole's claim is incorrect. (You are not expected to confirm Katie's results.)

AQA S1 2008 January Q7

12 marks Moderate -0.3

7 A travel agency in Tunisia offers customers a 3-day tour into the Sahara desert by either coach or minibus.

The agency accepts bookings from 50 customers for seats on the coach. The probability that a customer, who has booked a seat on the coach, will not turn up to claim the seat is 0.08 , and may be assumed to be independent of the behaviour of other customers. Determine the probability that, of the customers who have booked a seat on the coach:
1. two or more will not turn up;
2. three or more will not turn up.
The agency accepts bookings from 15 customers for seats on the minibus. The probability that a customer, who has booked a seat on the minibus, will not turn up to claim the seat is 0.025 , and may be assumed to be independent of the behaviour of other customers. Calculate the probability that, of the customers who have booked a seat on the minibus:
1. all will turn up;
2. one or more will not turn up.
The coach has 48 seats and the minibus has 14 seats. If 14 or fewer customers who have booked seats on the minibus turn up, they will be allocated a seat on the minibus. If all 15 customers who have booked seats on the minibus turn up, one will be allocated a seat on the coach. This will leave only 47 seats available for the 50 customers who have booked seats on the coach. Use your results from parts (a) and (b) to calculate the probability that there will be seats available on the coach for all those who turn up having booked such seats.
(4 marks)

AQA S1 2009 January Q1

7 marks Easy -1.8

1 Ms N Parker always reads the columns of announcements in her local weekly newspaper. During each week of 2008, she notes the number of births announced. Her results are summarised in the table.

Number of births	1	2	3	4	5	6	7	8
Number of weeks	1	2	9	13	7	13	6	1

Calculate the mean, median and modes of these data.
State, with a reason, which of the three measures of average in part (a) you consider to be the least appropriate for summarising the number of births.

AQA S1 2009 January Q2

7 marks Moderate -0.3

2 A greengrocer sells bunches of 9 carrots at his Saturday market stall. Tom and Geri are two Statistics students who work on the stall. Each selects a bunch of carrots at random.

At home, Tom measures the length, $x$ centimetres, and the maximum diameter, $y$ centimetres, of each carrot in his selected bunch with the following results.
$\boldsymbol { x }$ 16.2 13.1 10.4 12.1 14.6 9.7 11.8 13.6 17.3
$\boldsymbol { y }$ 4.2 3.9 4.7 3.3 3.7 2.4 3.1 3.5 2.7
1. Calculate the value of the product moment correlation coefficient.
2. Interpret your value in context.
At her home, Geri measures the length, in centimetres, and the weight, in grams, of each carrot in her selected bunch and then obtains a value of - 0.986 for the product moment correlation coefficient. Comment, with a reason, on the likely validity of Geri's value.

AQA S1 2009 January Q3

14 marks Standard +0.3

3 UPVC facia board is supplied in lengths labelled as 5 metres. The actual length, $X$ metres, of a board may be modelled by a normal distribution with a mean of 5.08 and a standard deviation of 0.05 .

Determine:
1. $\mathrm { P } ( X < 5 )$;
2. $\mathrm { P } ( 5 < X < 5.10 )$.
Determine the probability that the mean length of a random sample of 4 boards:
1. exceeds 5.05 metres;
2. is exactly 5 metres.
Assuming that the value of the standard deviation remains unchanged, determine the mean length necessary to ensure that only 1 per cent of boards have lengths less than 5 metres.

\(\boldsymbol { x }\)	9	3	4	10	8	12	7	11	2	6
\(\boldsymbol { y }\)	11	6	5	11	9	13	9	12	4	7

Time (seconds)	Number of customers
\(20 < x \leqslant 30\)	2
\(30 < x \leqslant 40\)	7
\(40 < x \leqslant 60\)	18
\(60 < x \leqslant 80\)	27
\(80 < x \leqslant 100\)	23
\(100 < x \leqslant 120\)	13
\(120 < x \leqslant 150\)	7
\(150 < x \leqslant 180\)	3
Total	100

Boy	A	B	C	D	E	F	G	H	I	J
Freestyle ( \(\boldsymbol { x }\) seconds)	30.2	32.8	25.1	31.8	31.2	35.6	32.4	38.0	36.1	34.1
Backstroke ( \(y\) seconds)	33.5	35.4	37.4	27.2	34.7	38.2	37.7	41.4	42.3	38.4

\(\boldsymbol { x }\)	16.2	13.1	10.4	12.1	14.6	9.7	11.8	13.6	17.3
\(\boldsymbol { y }\)	4.2	3.9	4.7	3.3	3.7	2.4	3.1	3.5	2.7

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