Questions - AQA - A-Level Maths

Find the range of f . 6
1. Find $\mathrm { f } ^ { - 1 } ( x )$
  6
(ii) State the range of $\mathrm { f } ^ { - 1 } ( x )$
6
State the transformation which maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { f } ^ { - 1 } ( x )$ 6
Find the coordinates of the point of intersection of the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$

AQA Paper 1 2019 June Q7

11 marks Standard +0.3

By sketching the graphs of $y = \frac { 1 } { x }$ and $y = \sec 2 x$ on the axes below, show that the equation $$\frac { 1 } { x } = \sec 2 x$$ has exactly one solution for $x > 0$
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-08_675_771_689_639} 7
By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6
7
Show that the equation can be rearranged to give $$x = \frac { 1 } { 2 } \cos ^ { - 1 } x$$ 7
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.
  7
(ii) 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 $x _ { 4 }$.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-09_954_1600_1717_223}

AQA Paper 1 2019 June Q8

4 marks Standard +0.3

8 $\quad \mathrm { P } ( n ) = \sum _ { k = 0 } ^ { n } k ^ { 3 } - \sum _ { k = 0 } ^ { n - 1 } k ^ { 3 }$ where $n$ is a positive integer.
8

Find $\mathrm { P } ( 3 )$ and $\mathrm { P } ( 10 )$
8
Solve the equation $\mathrm { P } ( n ) = 1.25 \times 10 ^ { 8 }$

AQA Paper 1 2019 June Q9

5 marks Standard +0.3

9 Prove that the sum of a rational number and an irrational number is always irrational.

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 $r ^ { 2 }$ $\text { Volume of a sphere } = \frac { 4 } { 3 } \pi r ^ { 3 }$

\includegraphics[max width=\textwidth, alt={}]{6b1312f4-9a5c-4465-8129-7d37e99efefe-13_2488_1716_219_153}

AQA Paper 1 2019 June Q12

7 marks Standard +0.3

Show that the equation $$2 \cot ^ { 2 } x + 2 \operatorname { cosec } ^ { 2 } x = 1 + 4 \operatorname { cosec } x$$ can be written in the form $$a \operatorname { cosec } ^ { 2 } x + b \operatorname { cosec } x + c = 0$$ 12
Hence, given $x$ is obtuse and $$2 \cot ^ { 2 } x + 2 \operatorname { cosec } ^ { 2 } x = 1 + 4 \operatorname { cosec } x$$ find the exact value of $\tan x$ Fully justify your answer.

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

\includegraphics[max width=\textwidth, alt={}]{6b1312f4-9a5c-4465-8129-7d37e99efefe-19_2488_1716_219_153}

AQA Paper 1 2019 June Q14

10 marks Standard +0.3

14 The graph of $y = \frac { 2 x ^ { 3 } } { x ^ { 2 } + 1 }$ is shown for $0 \leq x \leq 4$
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-20_1022_640_411_701} Caroline is attempting to approximate the shaded area, $A$, under the curve using the trapezium rule by splitting the area into $n$ trapezia. 14

When $n = 4$
14
1. State the number of ordinates that Caroline uses. 14
(ii) Calculate the area that Caroline should obtain using this method.
Give your answer correct to two decimal places.
14
Show that the exact area of $A$ is $$16 - \ln 17$$ Fully justify your answer.

14
Explain what would happen to Caroline's answer to part (a)(ii) as $n \rightarrow \infty$ [1 mark] Do not write outside the box

\includegraphics[max width=\textwidth, alt={}]{6b1312f4-9a5c-4465-8129-7d37e99efefe-23_2488_1716_219_153}

AQA Paper 1 2019 June Q15

10 marks Moderate -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 $$\begin{aligned} & v = 4 - \left( \frac { 2 t } { 3 } - 2 \right) ^ { 2 } \\ & h = 3 - 2 \sqrt [ 3 ] { t - 3 } \end{aligned}$$ High tides and low tides occur alternately when the velocity of the tidal flow is zero.
A high tide occurs at 2 am.
15
1. Use the model to find the height of this high tide.
  15
(ii) Find the time of the first low tide after 2 am.
15
(iii) Find the height of this low tide.
15
Use the model to find the height of the tide when it is flowing with maximum velocity.
15
Comment on the validity of the model.

AQA Paper 1 2019 June Q16

16 marks Challenging +1.2

$\quad y = \mathrm { e } ^ { - x } ( \sin x + \cos x )$ Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
Simplify your answer.
16
Hence, show that $$\int \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = a \mathrm { e } ^ { - x } ( \sin x + \cos x ) + c$$ where $a$ is a rational number.
16
A sketch of the graph of $y = \mathrm { e } ^ { - x } \sin x$ for $x \geq 0$ is shown below. $A _ { 1 } , A _ { 2 } , \ldots , A _ { n } , \ldots$
The areas of the finite regions bounded by the curve and the $x$-axis are denoted by
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-27_974_1507_502_262} 16
1. Find the exact value of the area $A _ { 1 }$
  16
(ii) Show that $$\frac { A _ { 2 } } { A _ { 1 } } = \mathrm { e } ^ { - \pi }$$ 16
(iii) Given that $$\frac { A _ { n + 1 } } { A _ { n } } = \mathrm { e } ^ { - \pi }$$ show that the exact value of the total area enclosed between the curve and the $x$-axis is $$\frac { 1 + \mathrm { e } ^ { \pi } } { 2 \left( \mathrm { e } ^ { \pi } - 1 \right) }$$ \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-30_2488_1719_219_150}
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-31_2488_1719_219_150}
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-32_2496_1721_214_148}

AQA Paper 1 2020 June Q1

2 marks Easy -1.2

1 The first three terms, in ascending powers of $x$, of the binomial expansion of $( 9 + 2 x ) ^ { \frac { 1 } { 2 } }$ are given by $$( 9 + 2 x ) ^ { \frac { 1 } { 2 } } \approx a + \frac { x } { 3 } - \frac { x ^ { 2 } } { 54 }$$ where $a$ is a constant. 1

State the range of values of $x$ for which this expansion is valid.
Circle your answer.
$| x | < \frac { 2 } { 9 }$
$| x | < \frac { 2 } { 3 }$
$| x | < 1$
$| x | < \frac { 9 } { 2 }$ 1
Find the value of $a$.
Circle your answer.
[0pt] [1 mark]
1239

AQA Paper 1 2020 June Q2

1 marks Easy -1.8

2 A student is searching for a solution to the equation $\mathrm { f } ( x ) = 0$ He correctly evaluates $$f ( - 1 ) = - 1 \text { and } f ( 1 ) = 1$$ and concludes that there must be a root between - 1 and 1 due to the change of sign.
Select the function $\mathrm { f } ( x )$ for which the conclusion is incorrect.
Circle your answer. $$\mathrm { f } ( x ) = \frac { 1 } { x } \quad \mathrm { f } ( x ) = x \quad \mathrm { f } ( x ) = x ^ { 3 } \quad \mathrm { f } ( x ) = \frac { 2 x + 1 } { x + 2 }$$

AQA Paper 1 2020 June Q3

1 marks Easy -1.2

3 The diagram shows a sector $O A B$ of a circle with centre $O$ and radius 2
\includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-03_374_455_1187_790} The angle $A O B$ is $\theta$ radians and the perimeter of the sector is 6
Find the value of $\theta$ Circle your answer.
[0pt] [1 mark]
1
$\sqrt { 3 }$
2
3

AQA Paper 1 2020 June Q4

5 marks Easy -1.2

Sketch the graph of
\includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-04_933_1093_349_475} 4
Solve the inequality $$4 - | 2 x - 6 | > 2$$

AQA Paper 1 2020 June Q5

2 marks Easy -1.8

5 Prove that, for integer values of $n$ such that $0 \leq n < 4$ $$2 ^ { n + 2 } > 3 ^ { n }$$

AQA Paper 1 2020 June Q6

4 marks Easy -1.2

6 Four students, Tom, Josh, Floella and Georgia are attempting to complete the indefinite integral $$\int \frac { 1 } { x } \mathrm {~d} x \quad \text { for } x > 0$$ Each of the students' solutions is shown below: $$\begin{array} { l l } \text { Tom } & \int \frac { 1 } { x } \mathrm {~d} x = \ln x \\ \text { Josh } & \int \frac { 1 } { x } \mathrm {~d} x = k \ln x \\ \text { Floella } & \int \frac { 1 } { x } \mathrm {~d} x = \ln A x \\ \text { Georgia } & \int \frac { 1 } { x } \mathrm {~d} x = \ln x + c \end{array}$$ 6

1. Explain what is wrong with Tom's answer. 6
(ii) Explain what is wrong with Josh's answer.
6
Explain why Floella and Georgia's answers are equivalent.

AQA Paper 1 2020 June Q7

4 marks Moderate -0.8

7 Consecutive terms of a sequence are related by $$u _ { n + 1 } = 3 - \left( u _ { n } \right) ^ { 2 }$$ 7

In the case that $u _ { 1 } = 2$
7
1. Find $u _ { 3 }$ 7
(ii) Find $u _ { 50 }$ 7
State a different value for $u _ { 1 }$ which gives the same value for $u _ { 50 }$ as found in part (a)(ii).

AQA Paper 1 2020 June Q8

7 marks Standard +0.3

8 Mike, an amateur astronomer who lives in the South of England, wants to know how the number of hours of darkness changes through the year. On various days between February and September he records the length of time, $H$ hours, of darkness along with $t$, the number of days after 1 January. His results are shown in Figure 1 below. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-08_940_1541_696_246}

\end{figure} Mike models this data using the equation $$H = 3.87 \sin \left( \frac { 2 \pi ( t + 101.75 ) } { 365 } \right) + 11.7$$ 8

Find the minimum number of hours of darkness predicted by Mike's model. Give your answer to the nearest minute.
[0pt] [2 marks] 8
Find the maximum number of consecutive days where the number of hours of darkness predicted by Mike's model exceeds 14
8
Mike's friend Sofia, who lives in Spain, also records the number of hours of darkness on various days throughout the year. Her results are shown in Figure 2 below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-10_933_1537_561_248}
\end{figure} Sofia attempts to model her data by refining Mike's model.
She decides to increase the 3.87 value, leaving everything else unchanged.
Explain whether Sofia's refinement is appropriate.
\includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-11_2488_1730_219_141}
$9 \quad$ Chloe is attempting to write $\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } }$ as partial fractions, with constant numerators. Her incorrect attempt is shown below. Step 1 $$\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } } \equiv \frac { A } { x + 1 } + \frac { B } { ( x + 2 ) ^ { 2 } }$$ Step 2 $$2 x ^ { 2 } + x \equiv A ( x + 2 ) ^ { 2 } + B ( x + 1 )$$ Step 3 $$\begin{aligned} & \text { Let } x = - 1 \Rightarrow A = 1 \\ & \text { Let } x = - 2 \Rightarrow B = - 6 \end{aligned}$$ Answer $$\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } } \equiv \frac { 1 } { x + 1 } - \frac { 6 } { ( x + 2 ) ^ { 2 } }$$

AQA Paper 1 2020 June Q9

7 marks Moderate -0.3

1. By using a counter example, show that the answer obtained by Chloe cannot be correct.
  9
(ii) Explain her mistake in Step 1.
9
Write $\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } }$ as partial fractions, with constant numerators.

AQA Paper 1 2020 June Q10

12 marks Moderate -0.8

An arithmetic series is given by $$\sum _ { r = 5 } ^ { 20 } ( 4 r + 1 )$$ 10
1. Write down the first term of the series.
  10
(ii) Write down the common difference of the series.
10
(iii) Find the number of terms of the series.

10
A different arithmetic series is given by $\sum _ { r = 10 } ^ { 100 } ( b r + c )$
where $b$ and $c$ are constants.
The sum of this series is 7735
10
(ii) The 40th term of the series is 4 times the 2nd term. Find the values of $b$ and $c$.
[0pt] [4 marks]

AQA Paper 1 2020 June Q11

9 marks Moderate -0.3

11 The region $R$ enclosed by the lines $x = 1 , x = 6 , y = 0$ and the curve $$y = \ln ( 8 - x )$$ is shown shaded in Figure 3 below. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-17_419_869_598_587}

\end{figure} All distances are measured in centimetres.
11

Use a single trapezium to find an approximate value of the area of the shaded region, giving your answer in $\mathrm { cm } ^ { 2 }$ to two decimal places.
[0pt] [2 marks]
\section*{Question 11 continues on the next page} 11
Shape $B$ is made from four copies of region $R$ as shown in Figure 4 below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-18_707_711_438_667}
\end{figure} Shape $B$ is cut from metal of thickness 2 mm
The metal has a density of $10.5 \mathrm {~g} / \mathrm { cm } ^ { 3 }$
Use the trapezium rule with six ordinates to calculate an approximate value of the mass of Shape B. Give your answer to the nearest gram.
11
Without further calculation, give one reason why the mass found in part (b) may be:
11
1. an underestimate.
  11
(ii) an overestimate.

AQA Paper 1 2020 June Q12

13 marks Standard +0.3

12 A curve $C$ has equation $$x ^ { 3 } \sin y + \cos y = A x$$ where $A$ is a constant.
$C$ passes through the point $P \left( \sqrt { 3 } , \frac { \pi } { 6 } \right)$
12

Show that $A = 2$
12
1. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - 3 x ^ { 2 } \sin y } { x ^ { 3 } \cos y - \sin y }$
  12
(ii) Hence, find the gradient of the curve at $P$.
12
(iii) The tangent to $C$ at $P$ intersects the $x$-axis at $Q$.
Find the exact $x$-coordinate of $Q$.

AQA Paper 1 2020 June Q13

15 marks Standard +0.3

13 The function f is defined by $$\mathrm { f } ( x ) = \frac { 2 x + 3 } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$ 13

1. Find f-1
  13
(ii) Write down an expression for $\mathrm { ff } ( x )$.
13
The function g is defined by $$g ( x ) = \frac { 2 x ^ { 2 } - 5 x } { 2 } \quad x \in \mathbb { R } , 0 \leq x \leq 4$$ 13
1. Find the range of g .
  13
(ii) Determine whether g has an inverse.
Fully justify your answer.
Show that $$g f ( x ) = \frac { 48 + 29 x - 2 x ^ { 2 } } { 2 x ^ { 2 } - 8 x + 8 }$$ 13
It can be shown that fg is given by $$f g ( x ) = \frac { 4 x ^ { 2 } - 10 x + 6 } { 2 x ^ { 2 } - 5 x - 4 }$$ with domain $\{ x \in \mathbb { R } : 0 \leq x \leq 4 , x \neq a \}$
Find the value of $a$.
Fully justify your answer.

AQA Paper 1 2020 June Q14

9 marks Standard +0.3

14 The function f is defined by $$f ( x ) = 3 ^ { x } \sqrt { x } - 1 \quad \text { where } x \geq 0$$ 14

$\quad \mathrm { f } ( x ) = 0$ has a single solution at the point $x = \alpha$
By considering a suitable change of sign, show that $\alpha$ lies between 0 and 1
14

Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 3 ^ { x } ( 1 + x \ln 9 ) } { 2 \sqrt { x } }$$

(ii) Use the Newton-Raphson method with $x _ { 1 } = 1$ to find $x _ { 3 }$, an approximation for $\alpha$.

Give your answer to five decimal places.

[2 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

(iii) Explain why the Newton-Raphson method fails to find $\alpha$ with $x _ { 1 } = 0$

[2 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

AQA Paper 1 2020 June Q15

10 marks Standard +0.3

15 The region enclosed between the curves $y = \mathrm { e } ^ { x } , y = 6 - \mathrm { e } ^ { \overline { 2 } }$ and the line $x = 0$ is shown shaded in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-28_1155_1009_424_513} Show that the exact area of the shaded region is $$6 \ln 4 - 5$$ Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-30_2491_1736_219_139}

AQA Paper 1 2021 June Q1

1 marks Easy -1.8

1 State the set of values of $x$ which satisfies the inequality $$( x - 3 ) ( 2 x + 7 ) > 0$$ Tick ( $\checkmark$ ) one box. $$\begin{aligned} & \left\{ x : - \frac { 7 } { 2 } < x < 3 \right\} \\ & \left\{ x : x < - 3 \text { or } x > \frac { 7 } { 2 } \right\} \\ & \left\{ x : x < - \frac { 7 } { 2 } \text { or } x > 3 \right\} \\ & \left\{ x : - 3 < x < \frac { 7 } { 2 } \right\} \end{aligned}$$

AQA Paper 1 2021 June Q2

1 marks Easy -1.8

2 Given that $y = \ln ( 5 x )$
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
Circle your answer. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { 5 x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 5 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \ln 5$$


13	A curve, \(C\), has equation
	\(y = \frac { \mathrm { e } ^ { 3 x - 5 } } { x ^ { 2 } }\)

	Show that \(C\) has exactly one stationary point.

	Fully justify your answer.

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