Questions - AQA - A-Level Maths

Given that $$\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { A } { ( r + 1 ) ( r + 2 ) } + \frac { B } { ( r + 2 ) ( r + 3 ) }$$ find the values of the integers $A$ and $B$
3
Use the method of differences to show clearly that $$\sum _ { r = 9 } ^ { 97 } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 89 } { 19800 }$$

AQA Further Paper 1 Specimen Q5

4 marks

5
- 1
3 \end{array} \right] \quad \left[ \begin{array} { l } 2
1
1 \end{array} \right]$$ 2 Use the definitions of $\cosh x$ and $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$ to show that $\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1$
[0pt] [2 marks]
3

Given that $$\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } \equiv \frac { A } { ( r + 1 ) ( r + 2 ) } + \frac { B } { ( r + 2 ) ( r + 3 ) }$$ find the values of the integers $A$ and $B$
3
Use the method of differences to show clearly that $$\sum _ { r = 9 } ^ { 97 } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 89 } { 19800 }$$ 4 A student states that $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \cos x + \sin x } { \cos x - \sin x } \mathrm {~d} x$ is not an improper integral because $\frac { \cos x + \sin x } { \cos x - \sin x }$ is defined at both $x = 0$ and $x = \frac { \pi } { 2 }$ Assess the validity of the student's argument.
[0pt] [2 marks]
$5 \quad \mathrm { p } ( z ) = z ^ { 4 } + 3 z ^ { 2 } + a z + b , a \in \mathbb { R } , b \in \mathbb { R }$
$2 - 3 \mathrm { i }$ is a root of the equation $\mathrm { p } ( \mathrm { z } ) = 0$ 5
Express $\mathrm { p } ( z )$ as a product of quadratic factors with real coefficients.
5
Solve the equation $\mathrm { p } ( z ) = 0$.

AQA Further Paper 1 Specimen Q6

7 marks

Obtain the general solution of the differential equation $$\tan x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \sin x \tan x$$ where $0 < x < \frac { \pi } { 2 }$
[0pt] [5 marks]
6
Hence find the particular solution of this differential equation, given that $y = \frac { 1 } { 2 \sqrt { 2 } }$
when $x = \frac { \pi } { 4 }$
[0pt] [2 marks]

AQA Further Paper 1 Specimen Q7

5 marks

7 Three planes have equations, $$\begin{gathered} x - y + k z = 3
k x - 3 y + 5 z = - 1
x - 2 y + 3 z = - 4 \end{gathered}$$ Where $k$ is a real constant. The planes do not meet at a unique point. 7

Find the possible values of $k$ 7
There are two possible geometric configurations of the given planes. Identify each possible configurations, stating the corresponding value of $k$ Fully justify your answer.
[0pt] [5 marks]
7
Given further that the equations of the planes form a consistent system, find the solution of the system of equations.

AQA Further Paper 1 Specimen Q8

1 marks

8 A curve has equation $$y = \frac { 5 - 4 x } { 1 + x }$$ 8

Sketch the curve.
\includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-10_1205_1219_886_360} 8
Hence sketch the graph of $y = \left| \frac { 5 - 4 x } { 1 + x } \right|$.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-11_1203_1202_641_331}

AQA Further Paper 1 Specimen Q9

10 marks

9 A line has Cartesian equations $x - p = \frac { y + 2 } { q } = 3 - z$ and a plane has
equation r. $\left[ \begin{array} { r } 1
- 1
- 2 \end{array} \right] = - 3$ 9

In the case where the plane fully contains the line, find the values of $p$ and $q$.
[0pt] [3 marks]
9
In the case where the line intersects the plane at a single point, find the range of values of $p$ and $q$.
[0pt] [3 marks]
9
In the case where the angle $\theta$ between the line and the plane satisfies $\sin \theta = \frac { 1 } { \sqrt { 6 } }$ and the line intersects the plane at $z = 0$ 9
1. Find the value of $q$.
  [0pt] [4 marks]
  9
(ii) Find the value of $p$.

AQA Further Paper 1 Specimen Q10

9 marks

10 The curve, $C$, has equation $y = \frac { x } { \cosh x }$
10

Show that the $x$-coordinates of any stationary points of $C$ satisfy the equation $\tanh x = \frac { 1 } { x }$
[0pt] [3 marks] 10
1. Sketch the graphs of $y = \tanh x$ and $y = \frac { 1 } { x }$ on the axes below.
  [0pt] [2 marks]
  \includegraphics[max width=\textwidth, alt={}, center]{a155b39a-6835-4d62-a481-41ef822bbd5f-14_1151_1226_1461_358} 10
(ii) Hence determine the number of stationary points of the curve $C$. 10
Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 0$ at each of the stationary points of the curve $C$.
[0pt] [4 marks]

AQA Further Paper 1 Specimen Q11

6 marks

Prove that $\frac { \sinh \theta } { 1 + \cosh \theta } + \frac { 1 + \cosh \theta } { \sinh \theta } \equiv 2 \operatorname { coth } \theta$ Explicitly state any hyperbolic identities that you use within your proof.
[0pt] [4 marks] LL
LL
LL
LL
LL
LL
LL
L
11
Solve $\frac { \sinh \theta } { 1 + \cosh \theta } + \frac { 1 + \cosh \theta } { \sinh \theta } = 4$ giving your answer in an exact form.
[0pt] [2 marks]

AQA Further Paper 1 Specimen Q12

3 marks

12 The function $\mathrm { f } ( x ) = \cosh ( \mathrm { i } x )$ is defined over the domain $\{ x \in \mathbb { R } : - a \pi \leq x \leq a \pi \}$, where $a$ is a positive integer. By considering the graph of $y = [ f ( x ) ] ^ { n }$, find the mean value of $[ f ( x ) ] ^ { n }$, when $n$ is an odd positive integer. Fully justify your answer.
[0pt] [3 marks]

AQA Further Paper 1 Specimen Q13

5 marks

13 Given that $\mathbf { M } = \left[ \begin{array} { l l l } 1 & 1 & 1
1 & 1 & 1
1 & 1 & 1 \end{array} \right]$, prove that $\mathbf { M } ^ { n } = \left[ \begin{array} { l l l } 3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 }
3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 }
3 ^ { n - 1 } & 3 ^ { n - 1 } & 3 ^ { n - 1 } \end{array} \right]$ for all $n \in \mathbb { N }$
[0pt] [5 marks] LL LL L
LL
LL
LL L
L

AQA Further Paper 1 Specimen Q14

12 marks

14 A particle, $P$, of mass $M$ is released from rest and moves along a horizontal straight line through a point $O$. When $P$ is at a displacement of $x$ metres from $O$, moving with a speed $v \mathrm {~ms} ^ { - 1 }$, a force of magnitude $| 8 M x |$ acts on the particle directed towards $O$. A resistive force, of magnitude $4 M v$, also acts on $P$. 14

Initially $P$ is held at rest at a displacement of 1 metre from $O$. Describe completely the motion of $P$ after it is released. Fully justify your answer.
[0pt] [8 marks]
14
It is decided to alter the resistive force so that the motion of $P$ is critically damped. Determine the magnitude of the resistive force that will produce critically damped motion.
[0pt] [4 marks]

AQA Further Paper 1 Specimen Q15

11 marks

15 An isolated island is populated by rabbits and foxes. At time $t$ the number of rabbits is $x$ and the number of foxes is $y$. It is assumed that:

The number of foxes increases at a rate proportional to the number of rabbits. When there are 200 rabbits the number of foxes is increasing at a rate of 20 foxes per unit period of time.
If there were no foxes present, the number of rabbits would increase by $120 \%$ in a unit period of time.
When both foxes and rabbits are present the foxes kill rabbits at a rate that is equal to $110 \%$ of the current number of foxes.
At time $t = 0$, the number of foxes is 20 and the number of rabbits is 80 .

1. Construct a mathematical model for the number of rabbits.
  [0pt] [9 marks]
  15
(ii) Use this model to show that the number of rabbits has doubled after approximately 0.7 units of time.
[0pt] [1 mark] 15
Suggest one way in which the model that you have used for the number of rabbits could be refined.
[0pt] [1 mark]

AQA Further Paper 2 Specimen Q1

1 marks

1 Given that $z _ { 1 } = 4 e ^ { \mathrm { i } \frac { \pi } { 3 } }$ and $z _ { 2 } = 2 e ^ { \mathrm { i } \frac { \pi } { 4 } }$
state the value of $\arg \left( \frac { z _ { 1 } } { z _ { 2 } } \right)$
Circle your answer.
[0pt] [1 mark]
$\frac { \pi } { 12 }$
$\frac { 4 } { 3 }$
$\frac { 7 \pi } { 12 }$
2

AQA Further Paper 2 Specimen Q2

3 marks

2 Given that $z$ is a complex number and that $z ^ { * }$ is the complex conjugate of $z$
prove that $z z ^ { * } - | z | ^ { 2 } = 0$
[0pt] [3 marks] LL

AQA Further Paper 2 Specimen Q3

3 marks

3 The transformation T is defined by the matrix $\mathbf { M }$. The transformation S is defined by the matrix $\mathbf { M } ^ { - 1 }$. Given that the point $( x , y )$ is invariant under transformation T , prove that $( x , y )$ is also an invariant point under transformation S .
[0pt] [3 marks]

AQA Further Paper 2 Specimen Q4

4 marks

4 Solve the equation $z ^ { 3 } = i$, giving your answers in the form $e ^ { i \theta }$, where $- \pi < \theta \leq \pi$
[0pt] [4 marks]

AQA Further Paper 2 Specimen Q5

4 marks

5 Find the smallest value $\theta$ of for which $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { 5 } = \frac { 1 } { \sqrt { 2 } } ( 1 - \mathrm { i } ) \{ \theta \in \mathbb { R } : \theta > 0 \}$$ [4 marks]

AQA Further Paper 2 Specimen Q6

5 marks

6 Prove that $8 ^ { n } - 7 n + 6$ is divisible by 7 for all integers $n \geq 0$
[0pt] [5 marks]

AQA Further Paper 2 Specimen Q7

2 marks

7 A small, hollow, plastic ball, of mass $m \mathrm {~kg}$ is at rest at a point $O$ on a polished horizontal surface. The ball is attached to two identical springs. The other ends of the springs are attached to the points $P$ and $Q$ which are 1.8 metres apart on a straight line through $O$. The ball is struck so that it moves away from $O$, towards $P$ with a speed of $0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. As the ball moves, its displacement from $O$ is $x$ metres at time $t$ seconds after the motion starts. The force that each of the springs applies to the ball is $12.5 m x$ newtons towards $O$. The ball is to be modelled as a particle. The surface is assumed to be smooth and it is assumed that the forces applied to the ball by the springs are the only horizontal forces acting on the ball. 7

Find the minimum distance of the ball from $P$, in the subsequent motion. 7
In practice the minimum distance predicted by the model is incorrect.
Is the minimum distance predicted by the model likely to be too big or too small?
Explain your answer with reference to the model.
[0pt] [2 marks]

AQA Further Paper 2 Specimen Q8

5 marks

8 Given that $I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x \quad n \geq 0$
show that $n I _ { n } = ( n - 1 ) I _ { n - 2 } \quad n \geq 2$
[0pt] [5 marks]

AQA Further Paper 2 Specimen Q9

6 marks

9 A student claims:
"Given any two non-zero square matrices, $\mathbf { A }$ and $\mathbf { B }$, then $( \mathbf { A B } ) ^ { - 1 } = \mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 }$ " 9

Explain why the student's claim is incorrect giving a counter example.
[0pt] [2 marks]
9
Refine the student's claim to make it fully correct.
[0pt] [1 mark]
9
Prove that your answer to part (b) is correct.
[0pt] [3 marks]

AQA Further Paper 2 Specimen Q10

8 marks

10 Evaluate the improper integral $\int _ { 0 } ^ { \infty } \frac { 4 x - 30 } { \left( x ^ { 2 } + 5 \right) ( 3 x + 2 ) } \mathrm { d } x$, showing the limiting process used. Give your answer as a single term.
[0pt] [8 marks]

AQA Further Paper 2 Specimen Q11

4 marks

11 The diagram shows a sketch of a curve $C$, the pole $O$ and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{21084ed7-43f8-47c6-80c2-930ccf340d37-14_622_978_374_571} The polar equation of $C$ is $r = 4 + 2 \cos \theta , \quad - \pi \leq \theta \leq \pi$ 11

Show that the area of the region bounded by the curve $C$ is $18 \pi$
11
Points $A$ and $B$ lie on the curve $C$ such that $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ and $A O B$ is an equilateral triangle. Find the polar equation of the line segment $A B$
[0pt] [4 marks]
$12 \quad \mathbf { M } = \left[ \begin{array} { r r r } - 1 & 2 & - 1
2 & 2 & - 2
- 1 & - 2 & - 1 \end{array} \right]$

AQA Further Paper 2 Specimen Q12

18 marks

Given that 4 is an eigenvalue of $\mathbf { M }$, find a corresponding eigenvector.
[0pt] [3 marks] 12
Given that $\mathbf { M U } = \mathbf { U D }$, where $\mathbf { D }$ is a diagonal matrix, find possible matrices for $\mathbf { D }$ and $\mathbf { U }$. [8 marks]
$13 \quad \mathbf { S }$ is a singular matrix such that $$\operatorname { det } \mathbf { S } = \left| \begin{array} { c c c } a & a & x
x - b & a - b & x + 1
x ^ { 2 } & a ^ { 2 } & a x \end{array} \right|$$ Express the possible values of $x$ in terms of $a$ and $b$.
[0pt] [7 marks]

AQA Further Paper 2 Specimen Q14

9 marks

14 Given that the vectors $\mathbf { a }$ and $\mathbf { b }$ are perpendicular, prove that
$| ( \mathbf { a } + 5 \mathbf { b } ) \times ( \mathbf { a } - 4 \mathbf { b } ) | = k | \mathbf { a } | | \mathbf { b } |$, where $k$ is an integer to be found. Explicitly state any properties of the vector product that you use within your proof.
[0pt] [9 marks] LL

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