Questions - AQA - A-Level Maths

The distance $O P$ is 16 metres.
Find the value of $a$ that Garry should use in the model.
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AQA Paper 1 2022 June Q14

9 marks Standard +0.8

14 The region bounded by the curve $$y = ( 2 x - 8 ) \ln x$$ and the $x$-axis is shaded in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-26_867_908_543_566} 14

Use the trapezium rule with 5 ordinates to find an estimate for the area of the shaded region. Give your answer correct to three significant figures.
14
Show that the exact area is given by $$32 \ln 2 - \frac { 33 } { 2 }$$ Fully justify your answer.

AQA Paper 1 2022 June Q15

16 marks Challenging +1.2

Given that $$y = \operatorname { cosec } \theta$$ 15
1. Express $y$ in terms of $\sin \theta$. 15
(ii) Hence, prove that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta$$ 15
(iii) Show that $$\frac { \sqrt { y ^ { 2 } - 1 } } { y } = \cos \theta \quad \text { for } 0 < \theta < \frac { \pi } { 2 }$$ 15
1. Use the substitution $$x = 2 \operatorname { cosec } u$$ to show that $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x \quad \text { for } x > 2$$ can be written as $$k \int \sin u \mathrm {~d} u$$ where $k$ is a constant to be found.
  15
(ii) Hence, show $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x = \frac { \sqrt { x ^ { 2 } - 4 } } { 4 x } + c \quad \text { for } x > 2$$ where $c$ is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-32_2492_1721_217_150}
\includegraphics[max width=\textwidth, alt={}]{22ff390e-1360-43bd-8c7f-3d2b58627e91-36_2496_1721_214_148}

AQA Paper 1 2023 June Q1

1 marks Easy -1.2

1 Find the coefficient of $x ^ { 7 }$ in the expansion of $( 2 x - 3 ) ^ { 7 }$
Circle your answer.
-2187-128 2128

AQA Paper 1 2023 June Q2

1 marks Easy -2.5

2 Given that $y = 2 x ^ { 3 }$ find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
Circle your answer.
[0pt] [1 mark]
$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { 2 }$
$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 }$
$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 4 } } { 2 }$
$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 3 }$

AQA Paper 1 2023 June Q3

1 marks Moderate -0.8

3 The curve with equation $y = \ln x$ is transformed by a stretch parallel to the $x$-axis with scale factor 2 Find the equation of the transformed curve.
Circle your answer.
$y = \frac { 1 } { 2 } \ln x \quad y = 2 \ln x \quad y = \ln \frac { x } { 2 } \quad y = \ln 2 x$

AQA Paper 1 2023 June Q4

1 marks Moderate -0.8

4 Given that $\theta$ is a small angle, find an approximation for $\cos 2 \theta$ Circle your answer.
$1 - \frac { \theta ^ { 2 } } { 2 }$
$2 - 2 \theta ^ { 2 }$
$1 - 2 \theta ^ { 2 }$
$1 - \theta ^ { 2 }$

AQA Paper 1 2023 June Q5

4 marks Moderate -0.3

Use the trapezium rule with 6 ordinates ( 5 strips) to find an approximate value for the shaded area. Give your answer to four decimal places.
5
Using your answer to part (a) deduce an estimate for $\int _ { 1 } ^ { 4 } \frac { 20 } { \mathrm { e } ^ { x } - 1 } \mathrm {~d} x$

AQA Paper 1 2023 June Q6

5 marks Standard +0.8

6 Show that the equation
$$\begin{aligned} & \qquad 2 \log _ { 10 } x = \log _ { 10 } 4 + \log _ { 10 } ( x + 8 ) \\ & \text { has exactly one solution. } \\ & \text { Fully justify your answer. } \end{aligned}$$

AQA Paper 1 2023 June Q7

4 marks Moderate -0.3

Given that $n$ is a positive integer, express $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ as a single fraction not involving surds.
7
Hence, deduce that $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ is a rational number for all positive integer values of $n$

AQA Paper 1 2023 June Q8

6 marks Moderate -0.3

8 Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } ( x \sin 4 x ) \mathrm { d } x = - \frac { \pi } { 8 }$$

\includegraphics[max width=\textwidth, alt={}]{6a03a035-ff32-4734-864b-a076aa9cbec0-09_2491_1716_219_153}

AQA Paper 1 2023 June Q9

9 marks Moderate -0.3

9 The points $P$ and $Q$ have coordinates ( $- 6,15$ ) and (12, 19) respectively. 9

1. Find the coordinates of the midpoint of $P Q$ 9
(ii) Find the equation of the perpendicular bisector of $P Q$
Give your answer in the form $a x + b y = c$ where $a , b$ and $c$ are integers.
[0pt] [4 marks]
9
1. A circle passes through the points $P$ and $Q$ The centre of the circle lies on the line with equation $2 x - 5 y = - 30$
  Find the equation of the circle. 9
(ii) The circle intersects the coordinate axes at $n$ points.
State the value of $n$ $$y = \sin x ^ { \circ }$$ for $- 360 \leq x \leq 360$ is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-12_613_1552_532_246}

AQA Paper 1 2023 June Q10

8 marks Moderate -0.3

Point $A$ on the curve has coordinates ( $a , 0.5$ )
10
1. Find the value of $a$
  10
(ii) State the value of $\sin \left( 180 ^ { \circ } - a ^ { \circ } \right)$
10
Point $B$ on the curve has coordinates $\left( b , - \frac { 3 } { 7 } \right)$
10
1. Find the exact value of $\sin \left( b ^ { \circ } - 180 ^ { \circ } \right)$
  10
(ii) Find the exact value of $\cos b ^ { \circ }$

AQA Paper 1 2023 June Q11

9 marks Standard +0.3

11 The $n$th term of a sequence is $u _ { n }$
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + 70$$ where $u _ { 1 } = 400$ and $p$ is a constant.
11

Find an expression, in terms of $p$, for $u _ { 2 }$ 11
It is given that $u _ { 3 } = 382$
11
1. Show that $p$ satisfies the equation $$200 p ^ { 2 } + 35 p - 156 = 0$$ 11
(ii) It is given that the sequence is a decreasing sequence. Find the value of $u _ { 4 }$ and the value of $u _ { 5 }$
11
The limit of $u _ { n }$ as $n$ tends to infinity is $L$
11
1. Write down an equation for $L$
  11
(ii) Find the value of $L$

AQA Paper 1 2023 June Q12

8 marks Easy -1.2

12 One of the rides at a theme park is a room where the floor and ceiling both move up and down for $10 \pi$ seconds. At time $t$ seconds after the ride begins, the distance $f$ metres of the floor above the ground is $$f = 1 - \cos t$$ At time $t$ seconds after the ride begins, the distance $c$ metres of the ceiling above the ground is $$c = 8 - 4 \sin t$$ The ride is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-16_448_766_932_635} 12

Show that the initial distance between the floor and ceiling is 8 metres.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}]{6a03a035-ff32-4734-864b-a076aa9cbec0-17_2500_1721_214_148}

AQA Paper 1 2023 June Q13

9 marks Standard +0.3

13 The function f is defined by $$\mathrm { f } ( x ) = \arccos x \text { for } 0 \leq x \leq a$$ The curve with equation $y = \mathrm { f } ( x )$ is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-18_842_837_550_603} 13

State the value of $a$ 13
1. On the diagram above, sketch the curve with equation $$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ and
  sketch the line with equation $$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ 13
(ii) Explain why the solution to the equation $$x - \cos x = 0$$ must also be a solution to the equation $$\cos x = \arccos x$$ Question 13 continues on the next page 13
Use the Newton-Raphson method with $x _ { 0 } = 0$ to find an approximate solution, $x _ { 3 }$, to the equation $$x - \cos x = 0$$ Give your answer to four decimal places.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-21_2491_1716_219_153}

AQA Paper 1 2023 June Q14

13 marks Standard +0.3

1. Given that $$y = 2 ^ { x }$$ write down $\frac { \mathrm { d } y } { \mathrm {~d} x }$ 14
(ii) Hence find $$\int 2 ^ { x } \mathrm {~d} x$$ 14
The area, $A$, bounded by the curve with equation $y = 2 ^ { x }$, the $x$-axis, the $y$-axis and the line $x = - 4$ is approximated using eight rectangles of equal width as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532} 14
1. Show that the exact area of the largest rectangle is $\frac { \sqrt { 2 } } { 4 }$
  14
(ii) The areas of these rectangles form a geometric sequence with common ratio $\frac { \sqrt { 2 } } { 2 }$
Find the exact value of the total area of the eight rectangles.
Give your answer in the form $k ( 1 + \sqrt { 2 } )$ where $k$ is a rational number.
[0pt] [3 marks]
14
(iii) More accurate approximations for $A$ can be found by increasing the number, $n$, of rectangles used. Find the exact value of the limit of the approximations for $A$ as $n \rightarrow \infty$

AQA Paper 1 2023 June Q15

9 marks Standard +0.8

15 The curve with equation $$x ^ { 2 } + 2 y ^ { 3 } - 4 x y = 0$$ has a single stationary point at $P$ as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-26_656_1138_548_450} 15

Show that the $y$-coordinate of $P$ satisfies the equation $$y ^ { 2 } ( y - 2 ) = 0$$ 15
Hence, find the coordinates of $P$
[0pt] [2 marks]

AQA Paper 1 2023 June Q16

14 marks Moderate -0.8

Given that $$\frac { 1 } { 16 - 9 x ^ { 2 } } \equiv \frac { A } { 4 - 3 x } + \frac { B } { 4 + 3 x }$$ find the values of $A$ and $B$
16
An empty container, in the shape of a cuboid, has length 1.6 metres, width 1.25 metres and depth 0.5 metres, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-29_469_812_404_616} The container has a small hole in the bottom. Water is poured into the container at a rate of 0.16 cubic metres per minute.
At time $t$ minutes after the container starts to be filled, the depth of water is $d$ metres and water leaks out at a rate of $0.36 d ^ { 2 }$ cubic metres per minute. At time $t$ minutes after the container starts to be filled, the volume of water in the container is $V$ cubic metres. 16
1. Show that $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 16 - 9 V ^ { 2 } } { 100 }$$ \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-30_2493_1721_214_150}
  \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-31_2492_1721_217_150} Question number Additional page, if required.
  Write the question numbers in the left-hand margin. Question number Additional page, if required.
  Write the question numbers in the left-hand margin. Question number Additional page, if required.
  Write the question numbers in the left-hand margin.
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