Questions - AQA Paper 1 - A-Level Maths

Year	1980	1985	1990	1995	2000	2005
$\boldsymbol { P }$	75	94	120	156	206	260

It is thought that $P$ can be modelled by $$P = A \times 10 ^ { k t }$$ where $t$ is the number of years after 1980 and $A$ and $k$ are constants.
9

Show algebraically that the graph of $\log _ { 10 } P$ against $t$ should be linear.
9
1. Complete the table below.
  $\boldsymbol { t }$ 0 5 10 15 20 25
  $\boldsymbol { \operatorname { l o g } } _ { \mathbf { 1 0 } } \boldsymbol { P }$ 1.88 1.97 2.08 2.31
  9
(ii) Plot $\log _ { 10 } P$ against $t$, and draw a line of best fit for the data.
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-13_1203_1308_360_367} 9
1. Hence, show that $k$ is approximately 0.02
  9
(ii) Find the value of $A$.
9
Using the model with $k = 0.02$ predict the number of tonnes of annual global production of plastics in 2030. 9
Using the model with $k = 0.02$ predict the year in which $P$ first exceeds 8000
9
Give a reason why it may be inappropriate to use the model to make predictions about future annual global production of plastics.
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-15_2488_1716_219_153}

AQA Paper 1 2021 June Q10

Given that $$y = \tan x$$ use the quotient rule to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x$$ 10
The region enclosed by the curve $y = \tan ^ { 2 } x$ and the horizontal line, which intersects the curve at $x = - \frac { \pi } { 4 }$ and $x = \frac { \pi } { 4 }$, is shaded in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-17_1059_967_461_539} Show that the area of the shaded region is $$\pi - 2$$ Fully justify your answer.
Do not write outside the box
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-19_2488_1716_219_153}

AQA Paper 1 2021 June Q11

11 A curve, $C$, passes through the point with coordinates $( 1,6 )$ The gradient of $C$ is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 6 } ( x y ) ^ { 2 }$$ Show that $C$ intersects the coordinate axes at exactly one point and state the coordinates of this point. Fully justify your answer.

AQA Paper 1 2021 June Q12

2 marks

12 The equation of a curve is $$( x + y ) ^ { 2 } = 4 y + 2 x + 8$$ The curve intersects the positive $x$-axis at the point $P$.
12

Show that the gradient of the curve at $P$ is $- \frac { 3 } { 2 }$

12
Find the equation of the normal to the curve at $P$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.
[2 marks]
$\_\_\_\_$

AQA Paper 1 2021 June Q13

2 marks

Given that $$P ( x ) = 125 x ^ { 3 } + 150 x ^ { 2 } + 55 x + 6$$ use the factor theorem to prove that ( $5 x + 1$ ) is a factor of $\mathrm { P } ( x )$.
[0pt] [2 marks]
13
Factorise $\mathrm { P } ( x )$ completely.
13
Hence, prove that $250 n ^ { 3 } + 300 n ^ { 2 } + 110 n + 12$ is a multiple of 12 when $n$ is a positive whole number.

AQA Paper 1 2021 June Q14

14 The curve $C$ is defined for $t \geq 0$ by the parametric equations $$x = t ^ { 2 } + t \quad \text { and } \quad y = 4 t ^ { 2 } - t ^ { 3 }$$ $C$ is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-26_691_608_541_717} 14

Find the gradient of $C$ at the point where it intersects the positive $x$-axis.
14
1. The area $A$ enclosed between $C$ and the $x$-axis is given by $$A = \int _ { 0 } ^ { b } y \mathrm {~d} x$$ Find the value of $b$.
  14
(ii) Use the substitution $y = 4 t ^ { 2 } - t ^ { 3 }$ to show that $$A = \int _ { 0 } ^ { 4 } \left( 4 t ^ { 2 } + 7 t ^ { 3 } - 2 t ^ { 4 } \right) \mathrm { d } t$$ 14
(iii) Find the value of $A$.

AQA Paper 1 2021 June Q15

Show that $$\sin x - \sin x \cos 2 x \approx 2 x ^ { 3 }$$ for small values of $x$.
15
Hence, show that the area between the graph with equation $$y = \sqrt { 8 ( \sin x - \sin x \cos 2 x ) }$$ the positive $x$-axis and the line $x = 0.25$ can be approximated by $$\text { Area } \approx 2 ^ { m } \times 5 ^ { n }$$ where $m$ and $n$ are integers to be found.
15
1. Explain why $$\int _ { 6.3 } ^ { 6.4 } 2 x ^ { 3 } \mathrm {~d} x$$ is not a suitable approximation for $$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$ Question 15 continues on the next page 15
(ii) Explain how $$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$ may be approximated by $$\int _ { a } ^ { b } 2 x ^ { 3 } \mathrm {~d} x$$ for suitable values of $a$ and $b$.
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-31_2492_1721_217_150}
\includegraphics[max width=\textwidth, alt={}]{042e248a-9efa-4844-957d-f05715900ffc-36_2486_1719_221_150}

AQA Paper 1 2022 June Q1

1 A curve is defined by the parametric equations $$x = \cos \theta \text { and } y = \sin \theta \quad \text { where } 0 \leq \theta \leq 2 \pi$$ Which of the options shown below is a Cartesian equation for this curve?
Circle your answer. $$\frac { y } { x } = \tan \theta \quad x ^ { 2 } + y ^ { 2 } = 1 \quad x ^ { 2 } - y ^ { 2 } = 1 \quad x ^ { 2 } y ^ { 2 } = 1$$

AQA Paper 1 2022 June Q2

2 A periodic sequence is defined by $$U _ { n } = ( - 1 ) ^ { n }$$ State the period of the sequence.
Circle your answer.

AQA Paper 1 2022 June Q3

1 marks

3 The curve $$y = \log _ { 4 } x$$ is transformed by a stretch, scale factor 2 , parallel to the $y$-axis.
State the equation of the curve after it has been transformed.
Circle your answer.
[0pt] [1 mark] $$y = \frac { 1 } { 2 } \log _ { 4 } x \quad y = 2 \log _ { 4 } x \quad y = \log _ { 4 } 2 x \quad y = \log _ { 8 } x$$

\includegraphics[max width=\textwidth, alt={}]{22ff390e-1360-43bd-8c7f-3d2b58627e91-03_2492_1722_217_150}

AQA Paper 1 2022 June Q4

4 The graph of $$y = \mathrm { f } ( x )$$ where $$f ( x ) = a x ^ { 2 } + b x + c$$ is shown in Figure 1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{22ff390e-1360-43bd-8c7f-3d2b58627e91-04_618_634_810_703}

\end{figure} Which of the following shows the graph of $y = \mathrm { f } ^ { \prime } ( x )$ ? Tick $( \checkmark )$ one box.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_2272_437_429_557}
□
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_117_117_1151_1133}
□
□

AQA Paper 1 2022 June Q5

5 Find an equation of the tangent to the curve $$y = ( x - 2 ) ^ { 4 }$$ at the point where $x = 0$

AQA Paper 1 2022 June Q6

Find the first two terms, in ascending powers of $x$, of the binomial expansion of $$\left( 1 - \frac { x } { 2 } \right) ^ { \frac { 1 } { 2 } }$$ 6
Hence, for small values of $x$, show that $$\sin 4 x + \sqrt { \cos x } \approx A + B x + C x ^ { 2 }$$ where $A , B$ and $C$ are constants to be found.

AQA Paper 1 2022 June Q7

3 marks

7 Sketch the graph of $$y = \cot \left( x - \frac { \pi } { 2 } \right)$$ for $0 \leq x \leq 2 \pi$
[0pt] [3 marks]
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-08_1650_1226_587_408}
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-09_2488_1716_219_153}

AQA Paper 1 2022 June Q8

8 The lines $L _ { 1 }$ and $L _ { 2 }$ are parallel.
$L _ { 1 }$ has equation $$5 x + 3 y = 15$$ and $L _ { 2 }$ has equation $$5 x + 3 y = 83$$ $L _ { 1 }$ intersects the $y$-axis at the point $P$.
The point $Q$ is the point on $L _ { 2 }$ closest to $P$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-10_849_917_945_561} 8

1. Find the coordinates of $Q$.
  8
(ii) Hence show that $P Q = k \sqrt { 34 }$, where $k$ is an integer to be found. 8
A circle, $C$, has centre ( $a , - 17$ ).
$L _ { 1 }$ and $L _ { 2 }$ are both tangents to $C$.
8
1. Find $a$.
  8
(ii) Find the equation of $C$.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-13_2493_1732_214_139}

AQA Paper 1 2022 June Q9

4 marks

9 The first three terms of an arithmetic sequence are given by $$2 x + 5 \quad 5 x + 1 \quad 6 x + 7$$ 9

Show that $x = 5$ is the only value which gives an arithmetic sequence.
9
1. Write down the value of the first term of the sequence.
  9
(ii) Find the value of the common difference of the sequence.
9
The sum of the first $N$ terms of the arithmetic sequence is $S _ { N }$ where $$\begin{array} { r } S _ { N } < 100000
S _ { N + 1 } > 100000 \end{array}$$ Find the value of $N$.
[0pt] [4 marks]

AQA Paper 1 2022 June Q10

2 marks

10 The diagram shows a sector of a circle $O A B$.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-16_758_796_360_623} The point $C$ lies on $O B$ such that $A C$ is perpendicular to $O B$.
Angle $A O B$ is $\theta$ radians.
10

Given the area of the triangle $O A C$ is half the area of the sector $O A B$, show that $$\theta = \sin 2 \theta$$ 10

Use a suitable change of sign to show that a solution to the equation $$\theta = \sin 2 \theta$$ lies in the interval given by $\theta \in \left[ \frac { \pi } { 5 } , \frac { 2 \pi } { 5 } \right]$

The Newton-Raphson method is used to find an approximate solution to the equation

$\theta = \sin 2 \theta$

Using $\theta _ { 1 } = \frac { \pi } { 5 }$ as a first approximation for $\theta$ apply the Newton-Raphson method twice

to find the value of $\theta _ { 3 }$ Give your answer to three decimal places.
10

(ii) Explain how a more accurate approximation for $\theta$ can be found using the Newton-Raphson method.
10

(iii) Explain why using $\theta _ { 1 } = \frac { \pi } { 6 }$ as a first approximation in the Newton-Raphson method
[0pt] [2 marks] does not lead to a solution for $\theta$.

AQA Paper 1 2022 June Q11

11 The polynomial $\mathrm { p } ( x )$ is given by $$\mathrm { p } ( x ) = x ^ { 3 } + ( b + 2 ) x ^ { 2 } + 2 ( b + 2 ) x + 8$$ where $b$ is a constant.
11

Use the factor theorem to prove that $( x + 2 )$ is a factor of $\mathrm { p } ( x )$ for all values of $b$.
11
The graph of $y = \mathrm { p } ( x )$ meets the $x$-axis at exactly two points.
11
1. Sketch a possible graph of $y = \mathrm { p } ( x )$
  \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-20_1084_965_1619_532} 11
(ii) Given $\mathrm { p } ( x )$ can be written as $$\mathrm { p } ( x ) = ( x + 2 ) \left( x ^ { 2 } + b x + 4 \right)$$ find the value of $b$. Fully justify your answer.

AQA Paper 1 2022 June Q12

A geometric sequence has first term 1 and common ratio $\frac { 1 } { 2 }$
12
1. Find the sum to infinity of the sequence.
  12
(ii) Hence, or otherwise, evaluate $$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$ 12
Find the smallest positive exact value of $\theta$, in radians, which satisfies the equation $$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$

AQA Paper 1 2022 June Q13

13 Figure 2 shows the approximate shape of the vertical cross section of the entrance to a cave. The cave has a horizontal floor. The entrance to the cave joins the floor at the points $O$ and $P$. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{22ff390e-1360-43bd-8c7f-3d2b58627e91-24_396_991_584_529}

\end{figure} Garry models the shape of the cross section of the entrance to the cave using the equation $$x ^ { 2 } + y ^ { 2 } = a \sqrt { x } - y$$ where $a$ is a constant, and $x$ and $y$ are the horizontal and vertical distances respectively, in metres, measured from $O$. 13

The distance $O P$ is 16 metres.
Find the value of $a$ that Garry should use in the model.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-25_2518_1723_196_148}

AQA Paper 1 2022 June Q14

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

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 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

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

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.
$y = \frac { 1 } { 2 } \ln x \quad y = 2 \ln x \quad y = \ln \frac { x } { 2 } \quad y = \ln 2 x$

\(\boldsymbol { t }\)	0	5	10	15	20	25
\(\boldsymbol { \operatorname { l o g } } _ { \mathbf { 1 0 } } \boldsymbol { P }\)	1.88	1.97	2.08		2.31

12	Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
	[2 marks]
	\(\_\_\_\_\)

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