Questions - AQA AS Paper 1

Find the equation of the tangent to the curve $y = \mathrm { e } ^ { 4 x }$ at the point ( $a , \mathrm { e } ^ { 4 a }$ ).
8
Find the value of $a$ for which this tangent passes through the origin.
8
Hence, find the set of values of $m$ for which the equation
has no real solutions. $$\mathrm { e } ^ { 4 x } = m x$$ has no real solutions.
[0pt] [3 marks]

AQA AS Paper 1 2020 June Q9

9 The diagram below shows a circle and four triangles.
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-12_974_739_356_648}
$A B$ is a diameter of the circle. $C$ is a point on the circumference of the circle.
Triangles $A B K , B C L$ and $C A M$ are equilateral.
Prove that the area of triangle $A B K$ is equal to the sum of the areas of triangle $B C L$ and triangle CAM.

AQA AS Paper 1 2020 June Q10

10 Raj is investigating how the price, $P$ pounds, of a brilliant-cut diamond ring is related to the weight, $C$ carats, of the diamond. He believes that they are connected by a formula $$P = a C ^ { n }$$ where $a$ and $n$ are constants.
10

Express $\ln P$ in terms of $\ln C$.
10
Raj researches the price of three brilliant-cut diamond rings on a website with the following results.
$\boldsymbol { C }$ 0.60 1.15 1.50
$\boldsymbol { P }$ 495 1200 1720
10
1. Plot $\ln P$ against $\ln C$ for the three rings on the grid below.
  \includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-15_1018_1467_751_283} 10
(ii) Explain which feature of the plot suggests that Raj's belief may be correct.
10
(iii) Using the graph on page 15 , estimate the value of $a$ and the value of $n$. 10
Explain the significance of $a$ in this context.
10
Raj wants to buy a ring with a brilliant-cut diamond of weight 2 carats. Estimate the price of such a ring.

AQA AS Paper 1 2020 June Q11

11 A go-kart and driver, of combined mass 55 kg , move forward in a straight line with a constant acceleration of $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ The total driving force is 14 N
Find the total resistance force acting on the go-kart and driver.
Circle your answer. 0 N 3 N 11 N 14 N

AQA AS Paper 1 2020 June Q12

12 One of the following is an expression for the distance between the points represented by position vectors $5 \mathbf { i } - 3 \mathbf { j }$ and $18 \mathbf { i } + 7 \mathbf { j }$ Identify the correct expression.
Tick ( $\checkmark$ ) one box. $$\begin{array} { l l } \sqrt { 13 ^ { 2 } + 4 ^ { 2 } } & \square
\sqrt { 13 ^ { 2 } + 10 ^ { 2 } } & \square
\sqrt { 23 ^ { 2 } + 4 ^ { 2 } } & \square
\sqrt { 23 ^ { 2 } + 10 ^ { 2 } } & \square \end{array}$$

AQA AS Paper 1 2020 June Q13

13 An object is moving in a straight line, with constant acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$, over a time period of $t$ seconds. It has an initial velocity $u$ and final velocity $v$ as shown in the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-19_606_784_493_628} Use the graph to show that $$v = u + a t$$ \includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-21_2488_1728_219_141}

AQA AS Paper 1 2020 June Q15

15 A particle, $P$, is moving in a straight line with acceleration $a \mathrm {~ms} ^ { - 2 }$ at time $t$ seconds, where $$a = 4 - 3 t ^ { 2 }$$ 15

Initially $P$ is stationary.
Find an expression for the velocity of $P$ in terms of $t$.

\includegraphics[max width=\textwidth, alt={}]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-23_2496_1723_214_148}

AQA AS Paper 1 2020 June Q16

1 marks

16 A simple lifting mechanism comprises a light inextensible wire which is passed over a smooth fixed pulley. One end of the wire is attached to a rigid triangular container of mass 2 kg , which rests on horizontal ground. A load of $m \mathrm {~kg}$ is placed in the container.
The other end of the wire is attached to a particle of mass 5 kg , which hangs vertically downwards. The mechanism is initially held at rest as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-24_858_677_943_683} The mechanism is released from rest, and the container begins to move upwards with acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ The wire remains taut throughout the motion.

\includegraphics[max width=\textwidth, alt={}]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-25_2484_1740_219_150}

16 (c) In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ The load reaches a height of 2 metres above the ground 1 second after it is released.
Find the mass of the load.
16 (d) Ignoring air resistance, describe one assumption you have made in your model.
[0pt] [1 mark]

AQA AS Paper 1 2021 June Q1

1 Find the coefficient of the $x$ term in the binomial expansion of $( 3 + x ) ^ { 4 }$ Circle your answer. 122754108

AQA AS Paper 1 2021 June Q2

2 Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x }$ find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$
Circle your answer.
$- \frac { 2 } { x ^ { 2 } }$
$- \frac { 1 } { x ^ { 2 } }$
$\frac { 1 } { x ^ { 2 } }$
$\frac { 2 } { x ^ { 2 } }$

AQA AS Paper 1 2021 June Q3

3 The graph of the equation $y = \frac { 1 } { x }$ is translated by the vector $\left[ \begin{array} { l } 3
0 \end{array} \right]$
3

Write down the equation of the transformed graph. 3
State the equations of the asymptotes of the transformed graph.

AQA AS Paper 1 2021 June Q4

1. Find the equation of the line CD.
  4
(ii) Show that point $D$ has coordinates ( $- 1,2$ )
4
1. Find the sum of the length of $A B$ and the length of $C D$ in simplified surd form.
  4
(ii) Hence, find the area of the trapezium $A B C D$.

AQA AS Paper 1 2021 June Q5

Sketch the curve $$y = ( x - a ) ^ { 2 } ( 3 - x ) \quad \text { where } 0 < a < 3$$ indicating the coordinates of the points where the curve and the axes meet.
\includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-06_1068_1061_543_488} 5
Hence, solve $$( x - a ) ^ { 2 } ( 3 - x ) > 0$$ giving your answer in set notation form.

AQA AS Paper 1 2021 June Q6

6 A curve has the equation $y = \mathrm { e } ^ { - 2 x }$ At point $P$ on the curve the tangent is parallel to the line $x + 8 y = 5$
Find the coordinates of $P$ stating your answer in the form ( $\ln p , q$ ), where $p$ and $q$ are rational.

AQA AS Paper 1 2021 June Q7

2 marks

7 Scientists observed a colony of seabirds over a period of 10 years starting in 2010. They concluded that the number of birds in the colony, its population $P$, could be modelled by a formula of the form $$P = a \left( 10 ^ { b t } \right)$$ where $t$ is the time in years after 2010, and $a$ and $b$ are constants.
7

Explain what the value of $a$ represents.
7
Show that $\log _ { 10 } P = b t + \log _ { 10 } a$
7
The table below contains some data collected by the scientists.
Year 2013 2015
$t$ 3
$P$ 10200 12800
$\log _ { 10 } P$ 4.0086
7
1. Complete the table, giving the $\log _ { 10 } P$ value to 5 significant figures.
  7
(ii) Use the data to calculate the value of $a$ and the value of $b$.
7
(iii) Use the model to estimate the population of the colony in 2024.
7
1. State an assumption that must be made in using the model to estimate the population of the colony in 2024.
  [0pt] [1 mark] 7
(ii) Hence comment, with a reason, on the reliability of your estimate made in part (c)(iii).
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-11_2488_1730_219_141}

AQA AS Paper 1 2021 June Q8

2 marks

1. Show that the equation $$3 \sin \theta \tan \theta = 5 \cos \theta - 2$$ is equivalent to the equation $$( 4 \cos \theta - 3 ) ( 2 \cos \theta + 1 ) = 0$$ 8
(ii) Solve the equation $$3 \sin \theta \tan \theta = 5 \cos \theta - 2$$ for $- 180 ^ { \circ } \leq \theta \leq 180 ^ { \circ }$
8
Hence, deduce all the solutions of the equation $$3 \sin \left( \frac { 1 } { 2 } \theta \right) \tan \left( \frac { 1 } { 2 } \theta \right) = 5 \cos \left( \frac { 1 } { 2 } \theta \right) - 2$$ for $- 180 ^ { \circ } \leq \theta \leq 180 ^ { \circ }$, giving your answers to the nearest degree.
[0pt] [2 marks]

AQA AS Paper 1 2021 June Q9

9 A curve has equation $$y = \frac { a } { \sqrt { x } } + b x ^ { 2 } + \frac { c } { x ^ { 3 } } \quad \text { for } x > 0$$ where $a$, $b$ and $c$ are positive constants.
The curve has a single turning point.
Use the second derivative of $y$ to determine the nature of this turning point.
You do not need to find the coordinates of the turning point.
Fully justify your answer.

AQA AS Paper 1 2021 June Q10

10 Two forces $\left[ \begin{array} { c } 3
- 2 \end{array} \right] \mathrm { N }$ and $\left[ \begin{array} { l } - 7
- 5 \end{array} \right] \mathrm { N }$ act on a particle.
Find the resultant force.
Circle your answer. $$\left[ \begin{array} { c } - 21
10 \end{array} \right] \mathrm { N } \quad \left[ \begin{array} { l } - 4
- 7 \end{array} \right] \mathrm { N } \quad \left[ \begin{array} { l } 4
3 \end{array} \right] \mathrm { N } \quad \left[ \begin{array} { c } 10
7 \end{array} \right] \mathrm { N }$$ Jackie says:
"A person's weight on Earth is directly proportional to their mass."
Tom says:
"A person's weight on Earth is different to their weight on the moon."
Only one of the statements below is correct.
Identify the correct statement.
Tick ( $\checkmark$ ) one box. Jackie and Tom are both wrong. □ Jackie is right but Tom is wrong. □ Jackie is wrong but Tom is right.
\includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-16_118_118_2279_1201} Jackie and Tom are both right. □

AQA AS Paper 1 2021 June Q12

12 A particle $P$ lies at rest on a smooth horizontal table. A constant resultant force, $\mathbf { F }$ newtons, is then applied to P .
As a result $P$ moves in a straight line with constant acceleration $\left[ \begin{array} { l } 8
6 \end{array} \right] \mathrm { ms } ^ { - 2 }$
12

Show that the magnitude of the acceleration of $P$ is $10 \mathrm {~ms} ^ { - 2 }$ 12
Find the speed of $P$ after 3 seconds.
12
Given that $\mathbf { F } = \left[ \begin{array} { c } 2
1.5 \end{array} \right] \mathrm { N }$, find the mass of P .

AQA AS Paper 1 2021 June Q13

2 marks

13 A car, initially at rest, is driven along a straight horizontal road. The graph below is a simple model of how the car's velocity, $v$ metres per second, changes with respect to time, $t$ seconds.
\includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-18_494_1018_500_511} 13

Find the displacement of the car when $t = 45$
13
Shona says:
"This model is too simple. It is unrealistic to assume that the car will instantaneously change its acceleration." On the axes below sketch a graph, for the first 10 seconds of the journey, which would represent a more realistic model.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-19_1769_1175_635_431}

AQA AS Paper 1 2021 June Q14

4 marks

14 A particle, P , is moving along a straight line such that its acceleration $a \mathrm {~ms} ^ { - 2 }$, at any time, $t$ seconds, may be modelled by $$a = 3 + 0.2 t$$ When $t = 2$, the velocity of P is $k \mathrm {~m} \mathrm {~s} ^ { - 1 }$
14

Show that the initial velocity of P is given by the expression $( k - 6.4 ) \mathrm { ms } ^ { - 1 }$
[0pt] [4 marks]
14
The initial velocity of P is one fifth of the velocity when $t = 2$ Find the value of $k$.

AQA AS Paper 1 2021 June Q15

1 marks

15 In this question, use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A box, B, of mass 4 kg lies at rest on a fixed rough horizontal shelf.
One end of a light string is connected to B .
The string passes over a smooth peg, attached to the end of the shelf.
The other end of the string is connected to particle, P , of mass 1 kg , which hangs freely below the shelf as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-22_778_910_760_566} B is initially held at rest with the string taut.
B is then released.
B and P both move with constant acceleration $a \mathrm {~ms} ^ { - 2 }$ As B moves across the shelf it experiences a total resistance force of 5 N
15

State one type of force that would be included in the total resistance force. 15
Show that $a = 1$
15
When B has moved forward exactly 20 cm the string breaks.
Find how much further B travels before coming to rest.
15
State one assumption you have made when finding your solutions in parts (b) or (c). [1 mark]

AQA AS Paper 1 2022 June Q1

1 marks

1 Express as a single logarithm $$\log _ { 10 } 2 - \log _ { 10 } x$$ Circle your answer.
[0pt] [1 mark]
$\log _ { 10 } ( 2 + x ) \quad \log _ { 10 } ( 2 - x ) \quad \log _ { 10 } ( 2 x ) \quad \log _ { 10 } \left( \frac { 2 } { x } \right)$

AQA AS Paper 1 2022 June Q2

1 marks

2 The graph of the function $y = \cos \frac { 1 } { 2 } x$ for $0 ^ { \circ } \leq x \leq 360 ^ { \circ }$ is one of the graphs shown below. Identify the correct graph.
Tick ( $\checkmark$ ) one box.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_373_634_671_502}
□
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_387_634_1133_502}
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_113_111_1265_1306}
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_366_629_1610_502}
□
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_368_629_2074_502}
□

AQA AS Paper 1 2022 June Q3

3 Find the coefficient of the $x ^ { 3 }$ term in the expansion of $\left( 3 x + \frac { 1 } { 2 } \right) ^ { 4 }$

\(\boldsymbol { C }\)	0.60	1.15	1.50
\(\boldsymbol { P }\)	495	1200	1720

Year	2013	2015
\(t\)	3
\(P\)	10200	12800
\(\log _ { 10 } P\)	4.0086

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