Questions Paper 2 - A-Level Maths

1. Find the binomial expansion of $( 1 + 3 x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$
  9
(ii) Show that the first three terms in the binomial expansion of $$\frac { 1 } { 2 - 3 x }$$ form a geometric sequence and state the common ratio.
9
It is given that $$\frac { 36 x } { ( 1 + 3 x ) ( 2 - 3 x ) } \equiv \frac { P } { ( 2 - 3 x ) } + \frac { Q } { ( 1 + 3 x ) }$$ where $P$ and $Q$ are integers. Find the value of $P$ and the value of $Q$
9
1. Using your answers to parts (a) and (b), find the binomial expansion of $$\frac { 12 x } { ( 1 + 3 x ) ( 2 - 3 x ) }$$ up to and including the term in $x ^ { 2 }$
  [0pt] [2 marks]
  9
(ii) Find the range of values of $x$ for which the binomial expansion of $$\frac { 12 x } { ( 1 + 3 x ) ( 2 - 3 x ) }$$ is valid.

AQA Paper 2 2024 June Q10

10 The function f is defined by $$f ( x ) = x ^ { 2 } + 2 \cos x \text { for } - \pi \leq x \leq \pi$$ Determine whether the curve with equation $y = \mathrm { f } ( x )$ has a point of inflection at the point where $x = 0$ Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-19_2491_1757_173_121}

AQA Paper 2 2024 June Q11

A student states that 3 is the smallest value of $k$ in the interval $3 < k < 4$ Explain the error in the student's statement.
11
The student's teacher says there is no smallest value of $k$ in the interval $3 < k < 4$ The teacher gives the following correct proof: Step 1: Assume there is a smallest number in the interval $3 < k < 4$ and let this smallest number be $x$ Step 2: let $y = \frac { 3 + x } { 2 }$
Step 3: $3 < y < x$ which is a contradiction.
Step 4: Therefore, there is no smallest number in interval $3 < k < 4$ 11
1. Explain the contradiction stated in Step 3
  11
(ii) Prove that there is no largest value of $k$ in the interval $3 < k < 4$
\section*{END OF SECTION A TURN OVER FOR SECTION B}

AQA Paper 2 2024 June Q12

12 Two constant forces act on a particle, of mass 2 kilograms, so that it moves forward in a straight line. The two forces are:

a forward driving force of 10 newtons
a resistance force of 4 newtons.

Find the acceleration of the particle.
Circle your answer.
$2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$12 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

AQA Paper 2 2024 June Q13

13 A car starting from rest moves forward in a straight line. The motion of the car is modelled by the velocity-time graph below:
\includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-23_476_738_459_715} One of the following assumptions about the motion of the car is implied by the graph. Identify this assumption. Tick ( ✓ ) one box. The car never accelerates. □ The acceleration of the car is always positive. □ The acceleration of the car can change instantaneously. □ The acceleration of the car is never constant. □

AQA Paper 2 2024 June Q14

3 marks

14 The displacement, $r$ metres, of a particle at time $t$ seconds is $$r = 6 t - 2 t ^ { 2 }$$ 14

Find the value of $r$ when $t = 4$
[0pt] [1 mark] 14
Determine the range of values of $t$ for which the displacement is positive.
[0pt] [2 marks]

AQA Paper 2 2024 June Q15

15 Two forces, $\mathbf { F } _ { \mathbf { 1 } }$ and $\mathbf { F } _ { \mathbf { 2 } }$, are acting on a particle of mass 3 kilograms. It is given that $$\mathbf { F } _ { \mathbf { 1 } } = \left[ \begin{array} { c } a
23 \end{array} \right] \text { newtons and } \mathbf { F } _ { \mathbf { 2 } } = \left[ \begin{array} { l } 4
b \end{array} \right] \text { newtons }$$ where $a$ and $b$ are constants. The particle has an acceleration of $\left[ \begin{array} { c } 4 b
a \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 2 }$ Find the value of $a$ and the value of $b$

AQA Paper 2 2024 June Q16

16 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ An apple tree stands on horizontal ground.
An apple hangs, at rest, from a branch of the tree.
A second apple also hangs, at rest, from a different branch of the tree.
The vertical distance between the two apples is $d$ centimetres.
At the same instant both apples begin to fall freely under gravity.
The first apple hits the ground after 0.5 seconds.
The second apple hits the ground 0.1 seconds later.
Show that $d$ is approximately 54

AQA Paper 2 2024 June Q17

17 A uniform rod is resting on two fixed supports at points $A$ and $B$.
$A$ lies at a distance $x$ metres from one end of the rod.
$B$ lies at a distance $( x + 0.1 )$ metres from the other end of the rod.
The rod has length $2 L$ metres and mass $m$ kilograms.
The rod lies horizontally in equilibrium as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-28_332_880_726_644} The reaction force of the support on the rod at $B$ is twice the reaction force of the support on the rod at $A$. Show that $$L - x = k$$ where $k$ is a constant to be found.

AQA Paper 2 2024 June Q18

18 A particle is moving in a straight line through the origin $O$ The displacement of the particle, $r$ metres, from $O$, at time $t$ seconds is given by $$r = p + 2 t - q \mathrm { e } ^ { - 0.2 t }$$ where $p$ and $q$ are constants.
When $t = 3$, the acceleration of the particle is $- 1.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
18

Show that $q \approx 82$
18
The particle has an initial displacement of 5 metres. Find the value of $p$ Give your answer to two significant figures.
Turn over for the next question

AQA Paper 2 2024 June Q19

19 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A toy shoots balls upwards with an initial velocity of $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
The advertisement for this toy claims the balls can reach a maximum height of 2.5 metres from the ground. 19

Suppose that the toy shoots the balls vertically upwards.
19
1. Verify the claim in the advertisement.
  19
(ii) State two modelling assumptions you have made in verifying this claim.
19
In fact the toy shoots the balls anywhere between 0 and 11 degrees from the vertical. The range of maximum heights, $h$ metres, above the ground which can be reached by the balls may be expressed as $$k < h \leq 2.5$$ Find the value of $k$

AQA Paper 2 2024 June Q20

1 marks

20 Two particles $P$ and $Q$ are moving in separate straight lines across a smooth horizontal surface.
$P$ moves with constant velocity $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
$Q$ moves from position vector $( 5 \mathbf { i } - 7 \mathbf { j } )$ metres to position vector $( 14 \mathbf { i } + 5 \mathbf { j } )$ metres during a 3 second period. 20

Show that $P$ and $Q$ move along parallel lines.
20
Stevie says
Q is also moving with a constant velocity of $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
Explain why Stevie may be incorrect.
[0pt] [1 mark] Question 20 continues on the next page 20
A third particle $R$ is moving with a constant speed of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, in a straight line, across the same surface.
$P$ and $R$ move along lines that intersect at a fixed point $X$
It is given that:
- $P$ passes through $X$ exactly 2 seconds after $R$ passes through $X$
- $P$ and $R$ are exactly 13 metres apart 3 seconds after $R$ passes through $X$
Show that $P$ and $R$ move along perpendicular lines.

AQA Paper 2 Specimen Q1

1 State the values of $| x |$ for which the binomial expansion of $( 3 + 2 x ) ^ { - 4 }$ is valid. Circle your answer. $$| x | < \frac { 2 } { 3 } \quad | x | < 1 \quad | x | < \frac { 3 } { 2 } \quad | x | < 3$$

AQA Paper 2 Specimen Q2

2 A zoologist is investigating the growth of a population of red squirrels in a forest.
She uses the equation $N = \frac { 200 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 5 } } }$ as a model to predict the number of squirrels, $N$, in the population $t$ weeks after the start of the investigation. What is the size of the squirrel population at the start of the investigation?
Circle your answer.
5
20
40
200

AQA Paper 2 Specimen Q3

4 marks

3 A curve is defined by the parametric equations $$x = t ^ { 3 } + 2 , \quad y = t ^ { 2 } - 1$$ 3

Find the gradient of the curve at the point where $t = - 2$
[0pt] [4 marks]
3
Find a Cartesian equation of the curve.

AQA Paper 2 Specimen Q5

8 marks

5
20
40
200 3 A curve is defined by the parametric equations $$x = t ^ { 3 } + 2 , \quad y = t ^ { 2 } - 1$$ 3

Find the gradient of the curve at the point where $t = - 2$
[0pt] [4 marks]
3
Find a Cartesian equation of the curve.
4 The equation $x ^ { 3 } - 3 x + 1 = 0$ has three real roots. 4
Show that one of the roots lies between - 2 and - 1
4
Taking $x _ { 1 } = - 2$ as the first approximation to one of the roots, use the Newton-Raphson method to find $x _ { 2 }$, the second approximation.
[0pt] [3 marks]
4
Explain why the Newton-Raphson method fails in the case when the first approximation is $x _ { 1 } = - 1$
[0pt] [1 mark]

AQA Paper 2 Specimen Q6

5 marks

6 A curve $C$, has equation $y = x ^ { 2 } - 4 x + k$, where $k$ is a constant.
It crosses the $x$-axis at the points $( 2 + \sqrt { 5 } , 0 )$ and $( 2 - \sqrt { 5 } , 0 )$
6

Find the value of $k$.
[0pt] [2 marks] 6
Sketch the curve $C$, labelling the exact values of all intersections with the axes.
[0pt] [3 marks]

AQA Paper 2 Specimen Q7

3 marks

7 A student notices that when he adds two consecutive odd numbers together the answer always seems to be the difference between two square numbers. He claims that this will always be true.
He attempts to prove his claim as follows: Step 1: Check first few cases
$3 + 5 = 8$ and $8 = 3 ^ { 2 } - 1 ^ { 2 }$
$5 + 7 = 12$ and $12 = 4 ^ { 2 } - 2 ^ { 2 }$
$7 + 9 = 16$ and $16 = 5 ^ { 2 } - 3 ^ { 2 }$ Step 2: Use pattern to predict and check a large example
$101 + 103 = 204$
subtract 1 and divide by 2 for the first number
Add 1 and divide by two for the second number
$52 ^ { 2 } - 50 ^ { 2 } = 204$ it works! \section*{Step 3: Conclusion} The first few cases work and there is a pattern, which can be used to predict larger numbers. Therefore, it must be true for all consecutive odd numbers. 7

Explain what is wrong with the student's "proof". 7
Prove that the student's claim is correct.
[0pt] [3 marks]
Turn over for the next question

AQA Paper 2 Specimen Q8

8 marks

8 A curve has equation $y = 2 x \cos 3 x + \left( 3 x ^ { 2 } - 4 \right) \sin 3 x$ 8

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer in the form $\left( m x ^ { 2 } + n \right) \cos 3 x$, where $m$ and $n$ are integers.
[0pt] [4 marks] 8
Show that the $x$-coordinates of the points of inflection of the curve satisfy the equation $$\cot 3 x = \frac { 9 x ^ { 2 } - 10 } { 6 x }$$ [4 marks]

AQA Paper 2 Specimen Q9

10 marks

Three consecutive terms in an arithmetic sequence are $3 \mathrm { e } ^ { - p } , 5,3 \mathrm { e } ^ { p }$
Find the possible values of $p$. Give your answers in an exact form.
[0pt] [6 marks]
9
Prove that there is no possible value of $q$ for which $3 \mathrm { e } ^ { - q } , 5,3 \mathrm { e } ^ { q }$ are consecutive terms of a geometric sequence.
[0pt] [4 marks]

AQA Paper 2 Specimen Q11

2 marks

11 A uniform rod, $A B$, has length 3 metres and mass 24 kg .
A particle of mass $M \mathrm {~kg}$ is attached to the rod at $A$.
The rod is balanced in equilibrium on a support at $C$, which is 0.8 metres from $A$.
\includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-17_275_1308_735_424} Find the value of $M$.
[0pt] [2 marks]

AQA Paper 2 Specimen Q12

4 marks

12 A particle moves on a straight line with a constant acceleration, $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
The initial velocity of the particle is $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
After $T$ seconds the particle has velocity $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
This information is shown on the velocity-time graph.
\includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-18_602_1065_813_541} The displacement, $S$ metres, of the particle from its initial position at time $T$ seconds is given by the formula $$S = \frac { 1 } { 2 } ( U + V ) T$$ 12

By considering the gradient of the graph, or otherwise, write down a formula for $a$ in terms of $U , V$ and $T$.
[0pt] [1 mark] 12
Hence show that $V ^ { 2 } = U ^ { 2 } + 2 a S$
[0pt] [3 marks]

AQA Paper 2 Specimen Q13

5 marks

13 The three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ are acting on a particle. $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 25 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }
& \mathbf { F } _ { 2 } = ( - 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { N }
& \mathbf { F } _ { 3 } = ( 15 \mathbf { i } - 28 \mathbf { j } ) \mathrm { N } \end{aligned}$$ The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are horizontal and vertical respectively.
The resultant of these three forces is $\mathbf { F }$ newtons. 13

1. Find the magnitude of F, giving your answer to three significant figures.
  [0pt] [2 marks] 13
(ii) Find the acute angle that $\mathbf { F }$ makes with the horizontal, giving your answer to the nearest $0.1 ^ { \circ }$
[0pt] [2 marks]
13
The fourth force, $F _ { 4 }$, is applied to the particle so that the four forces are in equilibrium. Find $\mathbf { F } _ { 4 }$, giving your answer in terms of $\mathbf { i }$ and $\mathbf { j }$.
[0pt] [1 mark]
Turn over for the next question

AQA Paper 2 Specimen Q14

3 marks

14 The graph below models the velocity of a small train as it moves on a straight track for 20 seconds. The front of the train is at the point $A$ when $t = 0$
The mass of the train is 800 kg .
\includegraphics[max width=\textwidth, alt={}, center]{a57b0526-cf9c-44d6-a349-cac392f85a70-22_645_1374_699_479} 14

Find the total distance travelled in the 20 seconds.
14
Find the distance of the front of the train from the point $A$ at the end of the 20 seconds.
[0pt] [1 mark]
14
Find the maximum magnitude of the resultant force acting on the train.
[0pt] [2 marks]
14
Explain why, in reality, the graph may not be an accurate model of the motion of the train.

AQA Paper 2 Specimen Q15

8 marks

15 At time $t = 0$, a parachutist jumps out of an airplane that is travelling horizontally.
The velocity, $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$, of the parachutist at time $t$ seconds is given by: $$\mathbf { v } = \left( 40 \mathrm { e } ^ { - 0.2 t } \right) \mathbf { i } + 50 \left( \mathrm { e } ^ { - 0.2 t } - 1 \right) \mathbf { j }$$ The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are horizontal and vertical respectively.
Assume that the parachutist is at the origin when $t = 0$
Model the parachutist as a particle. 15

Find an expression for the position vector of the parachutist at time $t$.
[0pt] [4 marks] 15
The parachutist opens her parachute when she has travelled 100 metres horizontally.
Find the vertical displacement of the parachutist from the origin when she opens her parachute.
[0pt] [4 marks]
15
Carefully, explaining the steps that you take, deduce the value of $g$ used in the formulation of this model.

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