Questions - AQA AS Paper 1

Find the coordinates of the midpoint of $A B$. 6
Show that the equation of the circle may be written as $$x ^ { 2 } + y ^ { 2 } - 8 x - 2 y = 1$$ 6
$\quad$ The circle has centre $C$ and crosses the $x$-axis at points $D$ and $E$. Find the exact area of triangle $D E C$. 6
The circle has centre $C$ and crosses the $x$-axis at points $D$ and $E$.
The area enclosed between the curve and the $x$-axis is 36 units.
Find the value of $a$.
Fully justify your answer.
[0pt] [6 marks]
$7 \quad$ A curve has equation $y = a ^ { 2 } - x ^ { 2 }$, where $a > 0$

AQA AS Paper 1 2022 June Q8

4 marks

8 A curve has equation $$y = x ^ { 3 } - 6 x + \frac { 9 } { x }$$ 8

Show that the $x$ coordinates of the stationary points of the curve satisfy the equation $$x ^ { 4 } - 2 x ^ { 2 } - 3 = 0$$ 8
Deduce that the curve has exactly two stationary points.
8
Find the coordinates and nature of the two stationary points. Fully justify your answer.
[0pt] [4 marks]
8
Write down the equation of a line which is a tangent to the curve in two places.

AQA AS Paper 1 2022 June Q10

10 Curve $C$ has equation $y = \frac { \sqrt { 2 } } { x ^ { 2 } }$
10

Find an equation of the tangent to $C$ at the point $\left( 2 , \frac { \sqrt { 2 } } { 4 } \right)$
10
Show that the tangent to $C$ at the point $\left( 2 , \frac { \sqrt { 2 } } { 4 } \right)$ is also a normal to the curve at a different point.
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-16_588_978_1969_532}

AQA AS Paper 1 2022 June Q11

1 marks

11 A car, initially at rest, moves with constant acceleration along a straight horizontal road. One of the graphs below shows how the car's velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, changes over time, $t$ seconds. Identify the correct graph.
Tick ( $\checkmark$ ) one box.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_271_296_1219_495}
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_122_140_1290_982}
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_270_298_1583_495} □
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_277_305_1946_488} □
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_280_305_2311_493} □

AQA AS Paper 1 2022 June Q12

12 A horizontal force of 30 N causes a crate to travel with an acceleration of $2 \mathrm {~ms} ^ { - 2 }$, in a straight line, on a smooth horizontal surface. Find the weight of the crate.
Circle your answer. 15 kg $15 g \mathrm {~N} 15 \mathrm {~N} 15 g$ kg

AQA AS Paper 1 2022 June Q13

3 marks

13 Two points $A$ and $B$ lie in a horizontal plane and have coordinates ( $- 2,7$ ) and ( 3,19 ) respectively. A particle moves in a straight line from $A$ to $B$ under the action of a constant resultant force of magnitude 6.5 N Express the resultant force in vector form.
[0pt] [3 marks]

AQA AS Paper 1 2022 June Q14

14 A ball is released from rest from a height $h$ metres above horizontal ground and falls freely downwards. When the ball reaches the ground, its speed is $v \mathrm {~ms} ^ { - 1 }$, where $v \leq 10$
Show that $$h \leq \frac { 50 } { g }$$ \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-21_2488_1728_219_141} \begin{center} \begin{tabular}{|l|l|l|l|} \hline \multicolumn{3}{|l|}{\begin{tabular}{l}

AQA AS Paper 1 2022 June Q15

2 marks

15
15
\end{tabular}} &

Two particles, $P$ and $Q$, are initially at rest at the same point on a horizontal plane.

A force of \(\left[ \begin{array} { l } 4

0 \end{array} \right] \mathrm { N }\) is applied to $P$.

A force of \(\left[ \begin{array} { c } 8

15 \end{array} \right] \mathrm { N }\) is applied to $Q$.

Calculate, to the nearest degree, the acute angle between the two forces.

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

\hline & & &
\hline \end{tabular} \end{center}

\includegraphics[max width=\textwidth, alt={}]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-23_2495_1719_219_150}

AQA AS Paper 1 2022 June Q16

16 Jermaine and his friend Meena are walking in the same direction along a straight path. Meena is walking at a constant speed of $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Jermaine is walking $0.2 \mathrm {~ms} ^ { - 1 }$ more slowly than Meena.
When Jermaine is $d$ metres behind Meena he starts to run with a constant acceleration of $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, for a time of $t$ seconds, until he reaches her. 16

Show that $$d = t ^ { 2 } - 0.2 t$$ 16
When Jermaine's speed is $7.8 \mathrm {~ms} ^ { - 1 }$, he reaches Meena. Given that $u = 1.4$ find the value of $d$.

AQA AS Paper 1 2022 June Q17

3 marks

17
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-26_270_1036_274_552} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t$ seconds, for $0 \leq t < 20$, is given by $$v = 0.5 t + 0.01 t ^ { 2 }$$ 17

Show that when $t = 15$ their acceleration is $0.8 \mathrm {~ms} ^ { - 2 }$
17
The car has a mass of 1500 kg
The caravan has a mass of 850 kg
When $t = 15$ the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N 17
1. Find the total resistance force experienced by the caravan when $t = 15$
  17
(ii) Find the driving force being applied by the car when $t = 15$
[0pt] [3 marks]
17
State one assumption you have made about the tow bar.
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-28_2492_1721_217_150}
\includegraphics[max width=\textwidth, alt={}]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-32_2486_1719_221_150}

AQA AS Paper 1 2023 June Q1

1 At a point $P$ on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at $P$ Circle your answer.
-10
-0.1
0.1
10

AQA AS Paper 1 2023 June Q2

1 marks

2 Identify the expression below which is equivalent to $\left( \frac { 2 x } { 5 } \right) ^ { - 3 }$
Circle your answer.
[0pt] [1 mark]
$\frac { 8 x ^ { 3 } } { 125 }$
$\frac { 125 x ^ { 3 } } { 8 }$
$\frac { 125 } { 8 x ^ { 3 } }$
$\frac { 8 } { 125 x ^ { 3 } }$ Find the two possible values of $a$ The coefficient of $x ^ { 2 }$ in the binomial expansion of $( 1 + a x ) ^ { 6 }$ is $\frac { 20 } { 3 }$

AQA AS Paper 1 2023 June Q3

3 The coefficient of $x ^ { 2 }$ in the binomial expansion of $( 1 + a x ) ^ { 6 }$ is $\frac { 20 } { 3 }$

AQA AS Paper 1 2023 June Q4

Find the possible values of $\tan \theta$, giving your answers in exact form.
4
Hence, or otherwise, solve the equation $$5 \cos ^ { 2 } \theta - 4 \sin ^ { 2 } \theta = 0$$ giving all solutions of $\theta$ to the nearest $0.1 ^ { \circ }$ in the interval $0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$
\begin{center} \begin{tabular}{|l|l|} \hline

AQA AS Paper 1 2023 June Q5

5 (b) & 5 (a) Given that $y = x \sqrt { x }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\hline \end{tabular} \end{center}

AQA AS Paper 1 2023 June Q6

The curve $C _ { 1 }$ has equation $y = 2 x ^ { 2 } - 20 x + 42$ Express the equation of $C _ { 1 }$ in the form $$y = a ( x - b ) ^ { 2 } + c$$ where $a , b$ and $c$ are integers.
6
Write down the coordinates of the minimum point of $C _ { 1 }$
6
The curve $C _ { 1 }$ is mapped onto the curve $C _ { 2 }$ by a stretch in the $y$-direction.
The minimum point of $C _ { 2 }$ is at $( 5 , - 4 )$
Find the equation of $C _ { 2 }$
$7 \quad$ Points $P$ and $Q$ lie on the curve with equation $y = x ^ { 4 }$ The $x$-coordinate of $P$ is $x$
The $x$-coordinate of $Q$ is $x + h$

AQA AS Paper 1 2023 June Q7

2 marks

$\quad$ Expand $( x + h ) ^ { 4 }$
7
Hence, find an expression, in terms of $x$ and $h$, for the gradient of the line $P Q$
7
Explain how to use the answer from part (b) to obtain the gradient function of $y = x ^ { 4 }$
[0pt] [2 marks]

AQA AS Paper 1 2023 June Q8

Show that $$\int _ { 1 } ^ { a } \left( 6 - \frac { 12 } { \sqrt { x } } \right) \mathrm { d } x = 6 a - 24 \sqrt { a } + 18$$ 8
The curve $y = 6 - \frac { 12 } { \sqrt { x } }$, the line $x = 1$ and the line $x = a$ are shown in the diagram below. The shaded region $R _ { 1 }$ is bounded by the curve, the line $x = 1$ and the $x$-axis.
The shaded region $R _ { 2 }$ is bounded by the curve, the line $x = a$ and the $x$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-09_705_931_632_648} It is given that the areas of $R _ { 1 }$ and $R _ { 2 }$ are equal.
Find the value of $a$
Fully justify your answer.

AQA AS Paper 1 2023 June Q9

2 marks

9 A continuous curve has equation $y = \mathrm { f } ( x )$ The curve passes through the points $A ( 2,1 ) , B ( 4,5 )$ and $C ( 6,1 )$
It is given that $f ^ { \prime } ( 4 ) = 0$
Jasmin made two statements about the nature of the curve $y = \mathrm { f } ( x )$ at the point $B$ :
Statement 1: There is a turning point at $B$
Statement 2: There is a maximum point at $B$
9

Draw a sketch of the curve $y = \mathrm { f } ( x )$ such that Statement 1 is correct and Statement 2 is correct.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-10_593_588_1043_817} 9
Draw a sketch of the curve $y = \mathrm { f } ( x )$ such that Statement 1 is correct and Statement 2 is not correct.
\includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-11_607_597_497_813} 9
Draw a sketch of the curve $y = \mathrm { f } ( x )$ such that Statement 1 is not correct and Statement 2 is not correct.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-11_605_597_1541_813} 1 Charlie buys a car for $\pounds 18000$ on 1 January 2016.
The value of the car decreases exponentially.
The car has a value of $\pounds 12000$ on 1 January 2018.

AQA AS Paper 1 2023 June Q10

6 marks

Charlie says:

because the car has lost $\pounds 6000$ after two years, after another two years it will be worth £6000.

Charlie's friend Kaya says:

because the car has lost one third of its value after two years, after another two years it will be worth $\pounds 8000$.

Explain whose statement is correct, justifying the value they have stated.

The value of Charlie's car, $\pounds V , t$ years after 1 January 2016 may be modelled by the equation $V = A \mathrm { e } ^ { - k t }$

where $A$ and $k$ are positive constants.

Find the value of $t$ when the car has a value of $\pounds 10000$, giving your answer to two significant figures.

[5 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

[1 mark] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

AQA AS Paper 1 2023 June Q11

A circle has equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0$$ Find the centre and the radius of the circle.
11
An equilateral triangle has one vertex at the origin, and one side along the line $x = 8$, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-14_697_750_1311_735} 11
1. Show that the vertex at the origin lies inside the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0$
  11
(ii) Prove that the triangle lies completely within the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0$

AQA AS Paper 1 2023 June Q12

12 A particle, initially at rest, starts to move forward in a straight line with constant acceleration, $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ After 6 seconds the particle has a velocity of $3 \mathrm {~ms} ^ { - 1 }$
Find the value of $a$ Circle your answer.
$\begin{array} { l l l l } - 2 & - 0.5 & 0.5 & 2 \end{array}$

AQA AS Paper 1 2023 June Q13

1 marks

13 A resultant force of $\left[ \begin{array} { c } - 2
6 \end{array} \right] \mathrm { N }$ acts on a particle.
The acceleration of the particle is $\left[ \begin{array} { c } - 6
y \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 2 }$
Find the value of $y$
Circle your answer.
[0pt] [1 mark] 231018

\includegraphics[max width=\textwidth, alt={}]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-17_2496_1721_214_148}

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