Questions - A-Level Maths

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AQA AS Paper 2 Specimen Q15

2 marks Moderate -0.3

A school took 225 children on a trip to a theme park. After the trip the children had to write about their favourite ride at the park from a choice of three. The table shows the number of children who wrote about each ride.

	Ride written about
	The Drop	The Beanstalk	The Giant	Total
Year 7	24	45	23	92
Year 8	36	17	22	75
Year 9	20	13	25	58
Total	80	75	70	225

Three children were randomly selected from those who went on the trip. Calculate the probability that one wrote about 'The Drop', one wrote about 'The Beanstalk' and one wrote about The Giant'. [2 marks]

AQA AS Paper 2 Specimen Q16

2 marks Easy -1.8

The boxplot below represents the time spent in hours by students revising for a history exam. \includegraphics{figure_16}

Use the information in the boxplot to state the value of a measure of central tendency of the revision times, stating clearly which measure you are using. [1 mark]
Use the information in the boxplot to explain why the distribution of revision times is negatively skewed. [1 mark]

AQA AS Paper 2 Specimen Q17

6 marks Moderate -0.8

The table below is an extract from the Large Data Set.

Make	Region	Engine size	Mass	CO2	CO
VAUXHALL	South West	1398	1163	118	0.463
VOLKSWAGEN	London	999	1055	106	0.407
VAUXHALL	South West	1248	1225	85	0.141
BMW	South West	2979	1635	194	0.139
TOYOTA	South West	1995	1650	123	0.274
BMW	South West	2979	0	244	0.447
FORD	South West	1596	0	165	0.518
TOYOTA	South West	1299	1050	144
VAUXHALL	London	1398	1361	140	0.695
FORD	North West	4951	1799	299	0.621

1. Calculate the standard deviation of the engine sizes in the table. [1 mark]
2. The mean of the engine sizes is 2084 Any value more than 2 standard deviations from the mean can be identified as an outlier. Using this definition of an outlier, show that the sample of engine sizes has exactly one outlier. Fully justify your answer. [3 marks]
Rajan calculates the mean of the masses of the cars in this extract and states that it is 1094 kg. Use your knowledge of the Large Data Set to suggest what error Rajan is likely to have made in his calculation. [1 mark]
Rajan claims there is an error in the data recorded in the table for one of the Toyotas from the South West, because there is no value for its carbon monoxide emissions. Use your knowledge of the Large Data Set to comment on Rajan's claim. [1 mark]

AQA AS Paper 2 Specimen Q18

4 marks Easy -1.8

Neesha wants to open an Indian restaurant in her town. Her cousin, Ranji, has an Indian restaurant in a neighbouring town. To help Neesha plan her menu, she wants to investigate the dishes chosen by a sample of Ranji's customers. Ranji has a list of the 750 customers who dined at his restaurant during the past month and the dish that each customer chose. Describe how Neesha could obtain a simple random sample of size 50 from Ranji's customers. [4 marks]

AQA AS Paper 2 Specimen Q19

11 marks Standard +0.3

Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day. Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size $n$, can be modelled by the binomial distribution B($n$, 0.20).

There are 10 students in Ellie's statistics class. Using the distributional model suggested by Ellie, find the probability that, of the students in her class:
1. two or fewer eat at least five portions of fruit and vegetables every day; [1 mark]
2. at least one but fewer than four eat at least five portions of fruit and vegetables every day; [2 marks]
Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day.
1. Name the sampling method used by Declan. [1 mark]
2. Describe one weakness of this sampling method. [1 mark]
3. Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the 5\% significance level to investigate whether Declan's belief is supported by this evidence. [6 marks]

Show that $4a + 30d = 65$ [2 marks]
Given that the sum of the first 60 terms is 315, find the sum of the first 41 terms. [3 marks]
$S_n$ is the sum of the first $n$ terms of the sequence. Explain why the value you found in part (b) is the maximum value of $S_n$ [2 marks]

AQA Paper 1 2019 June Q6

8 marks Moderate -0.3

The function f is defined by $$f(x) = \frac{1}{2}(x^2 + 1), \quad x \geq 0$$

Find the range of f. [1 mark]
1. Find $f^{-1}(x)$ [3 marks]
2. State the range of $f^{-1}(x)$ [1 mark]
State the transformation which maps the graph of $y = f(x)$ onto the graph of $y = f^{-1}(x)$ [1 mark]
Find the coordinates of the point of intersection of the graphs of $y = f(x)$ and $y = f^{-1}(x)$ [2 marks]

AQA Paper 1 2019 June Q7

11 marks Standard +0.3

By sketching the graphs of $y = \frac{1}{x}$ and $y = \sec 2x$ on the axes below, show that the equation $$\frac{1}{x} = \sec 2x$$ has exactly one solution for $x > 0$ [3 marks] \includegraphics{figure_7a}
By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6 [2 marks]
Show that the equation can be rearranged to give $$x = \frac{1}{2}\cos^{-1}x$$ [2 marks]
1. Use the iterative formula $$x_{n+1} = \frac{1}{2}\cos^{-1}x_n$$ with $x_1 = 0.4$, to find $x_2$, $x_3$ and $x_4$, giving your answers to four decimal places. [2 marks]
2. On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x_2$, $x_3$ and $x_4$. [2 marks] \includegraphics{figure_7d}

AQA Paper 1 2019 June Q8

4 marks Standard +0.3

$$P(n) = \sum_{k=0}^{n} k^3 - \sum_{k=0}^{n-1} k^3 \text{ where } n \text{ is a positive integer.}$$

Find P(3) and P(10) [2 marks]
Solve the equation $P(n) = 1.25 \times 10^8$ [2 marks]

AQA Paper 1 2019 June Q9

5 marks Standard +0.8

Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]

AQA Paper 1 2019 June Q10

4 marks Standard +0.3

The volume of a spherical bubble is increasing at a constant rate. Show that the rate of increase of the radius, $r$, of the bubble is inversely proportional to $r^2$ Volume of a sphere = $\frac{4}{3}\pi r^3$ [4 marks]

AQA Paper 1 2019 June Q11

4 marks Moderate -0.8

Jodie is attempting to use differentiation from first principles to prove that the gradient of $y = \sin x$ is zero when $x = \frac{\pi}{2}$ Jodie's teacher tells her that she has made mistakes starting in Step 4 of her working. Her working is shown below. \includegraphics{figure_11} Complete Steps 4 and 5 of Jodie's working below, to correct her proof. [4 marks] Step 4 \quad For gradient of curve at A, Step 5 \quad Hence the gradient of the curve at A is given by

AQA Paper 1 2019 June Q12

7 marks Standard +0.3

Show that the equation $$2\cot^2 x + 2\cosec^2 x = 1 + 4\cosec x$$ can be written in the form $$a\cosec^2 x + b\cosec x + c = 0$$ [2 marks]
Hence, given $x$ is obtuse and $$2\cot^2 x + 2\cosec^2 x = 1 + 4\cosec x$$ find the exact value of $\tan x$ Fully justify your answer. [5 marks]

AQA Paper 1 2019 June Q13

7 marks Challenging +1.2

A curve, C, has equation $$y = \frac{e^{3x-5}}{x^2}$$ Show that C has exactly one stationary point. Fully justify your answer. [7 marks]

AQA Paper 1 2019 June Q14

10 marks Standard +0.3

The graph of $y = \frac{2x^3}{x^2 + 1}$ is shown for $0 \leq x \leq 4$

[diagram]

Caroline is attempting to approximate the shaded area, A, under the curve using the trapezium rule by splitting the area into $n$ trapezia.

When $n = 4$
1. State the number of ordinates that Caroline uses. [1 mark]
2. Calculate the area that Caroline should obtain using this method. Give your answer correct to two decimal places. [3 marks]
Show that the exact area of $A$ is $$16 - \ln 17$$ Fully justify your answer. [5 marks]
Explain what would happen to Caroline's answer to part (a)(ii) as $n \to \infty$ [1 mark]

AQA Paper 1 2019 June Q15

13 marks Standard +0.3

At time $t$ hours after a high tide, the height, $h$ metres, of the tide and the velocity, $v$ knots, of the tidal flow can be modelled using the parametric equations $$v = 4 - \left(\frac{2t}{3} - 2\right)^2$$ $$h = 3 - 2\sqrt[3]{t - 3}$$ High tides and low tides occur alternately when the velocity of the tidal flow is zero. A high tide occurs at 2am.

1. Use the model to find the height of this high tide. [1 mark]
2. Find the time of the first low tide after 2am. [3 marks]
3. Find the height of this low tide. [1 mark]
Use the model to find the height of the tide when it is flowing with maximum velocity. [3 marks]
Comment on the validity of the model. [2 marks]

AQA Paper 1 2019 June Q16

16 marks Standard +0.8

$y = e^{-x}(\sin x + \cos x)$ Find $\frac{dy}{dx}$ Simplify your answer. [3 marks]
Hence, show that $$\int e^{-x}\sin x \, dx = ae^{-x}(\sin x + \cos x) + c$$ where $a$ is a rational number. [2 marks]
A sketch of the graph of $y = e^{-x}\sin x$ for $x \geq 0$ is shown below. The areas of the finite regions bounded by the curve and the $x$-axis are denoted by $A_1, A_2, \ldots, A_n, \ldots$ \includegraphics{figure_16c}
1. Find the exact value of the area $A_1$ [3 marks]
2. Show that $$\frac{A_2}{A_1} = e^{-\pi}$$ [4 marks]
3. Given that $$\frac{A_{n+1}}{A_n} = e^{-\pi}$$ show that the exact value of the total area enclosed between the curve and the $x$-axis is $$\frac{1 + e^\pi}{2(e^\pi - 1)}$$ [4 marks]