Questions - AQA - A-Level Maths

AQA Further Paper 2 2020 June Q8

$\quad$ Factorise $\left| \begin{array} { c c c } 2 a + b + x & x + b & x ^ { 2 } + b ^ { 2 }
0 & a & - a ^ { 2 }
a + b & b & b ^ { 2 } \end{array} \right|$ as fully as possible.
8
The matrix $\mathbf { M }$ is defined by $$\mathbf { M } = \left[ \begin{array} { c c c } 13 + x & x + 3 & x ^ { 2 } + 9
0 & 5 & - 25
8 & 3 & 9 \end{array} \right]$$ Under the transformation represented by $\mathbf { M }$, a solid of volume $0.625 \mathrm {~m} ^ { 3 }$ becomes a solid of volume $300 \mathrm {~m} ^ { 3 }$ Use your answer to part (a) to find the possible values of $x$.
Use $\mathbf { C }$ to show that $\cos \frac { \pi } { 12 }$ can be written in the form $\frac { \sqrt { \sqrt { m } + n } } { 2 }$, where $m$ and $n$ are integers.

AQA Further Paper 2 2020 June Q10

10 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { 1 } = 0 \quad u _ { n + 1 } = \frac { 5 } { 6 - u _ { n } }$$ Prove by induction that, for all integers $n \geq 1$, $$u _ { n } = \frac { 5 ^ { n } - 5 } { 5 ^ { n } - 1 }$$

AQA Further Paper 2 2020 June Q11

Starting from the series given in the formulae booklet, show that the general term of the Maclaurin series for $$\frac { \sin x } { x } - \cos x$$ is $$( - 1 ) ^ { r + 1 } \frac { 2 r } { ( 2 r + 1 ) ! } x ^ { 2 r }$$ 11
Show that $$\lim _ { x \rightarrow 0 } \left[ \frac { \frac { \sin x } { x } - \cos x } { 1 - \cos x } \right] = \frac { 2 } { 3 }$$

AQA Further Paper 2 2020 June Q12

6 marks

Given that $I = \int _ { a } ^ { b } \mathrm { e } ^ { 2 t } \sin t \mathrm {~d} t$, show that $$I = \left[ q \mathrm { e } ^ { 2 t } \sin t + r \mathrm { e } ^ { 2 t } \cos t \right] _ { a } ^ { b }$$ where $q$ and $r$ are rational numbers to be found.
[0pt] [6 marks]
12

A small object is initially at rest. The subsequent motion of the object is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } + v = 5 \mathrm { e } ^ { t } \sin t$$ where $v$ is the velocity at time $t$.
Find the speed of the object when $t = 2 \pi$, giving your answer in exact form.

13	Charlotte is trying to solve this mathematical problem:
	Find the general solution of the differential equation $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 10 \mathrm { e } ^ { - 2 x }$

Charlotte's solution starts as follows:

Particular integral: $y = \lambda \mathrm { e } ^ { - 2 x }$

so $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \lambda \mathrm { e } ^ { - 2 x }$

and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 \lambda \mathrm { e } ^ { - 2 x }$

AQA Further Paper 2 2020 June Q13

Show that Charlotte's method will fail to find a particular integral for the differential equation.
13
Explain how Charlotte should have started her solution differently and find the general solution of the differential equation.

AQA Further Paper 2 2020 June Q14

14 The diagram shows the polar curve $C _ { 1 }$ with equation $r = 2 \sin \theta$ The diagram also shows part of the polar curve $C _ { 2 }$ with equation $r = 1 + \cos 2 \theta$
\includegraphics[max width=\textwidth, alt={}, center]{b4ba8a08-333d-4efc-a0ed-14fef2d99410-20_378_897_456_954} 14

On the diagram above, complete the sketch of $C _ { 2 }$ 14
Show that the area of the region shaded in the diagram is equal to $$k \pi + m \alpha - \sin 2 \alpha + q \sin 4 \alpha$$ where $\alpha = \sin ^ { - 1 } \left( \frac { \sqrt { 5 } - 1 } { 2 } \right)$, and $k , m$ and $q$ are rational numbers.

AQA Further Paper 2 2020 June Q15

15 The points $A ( 7,2,8 ) , B ( 7 , - 4,0 )$ and $C ( 3,3.2,9.6 )$ all lie in the plane $\Pi$. 15

Find a Cartesian equation of the plane $\Pi$.
15
The line $L _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { c } 5
- 0.4
4.8 \end{array} \right] + \mu \left[ \begin{array} { c } 15
3
4 \end{array} \right]$ 15
1. Show that $L _ { 1 }$ lies in the plane $\Pi$.
  15
(ii) Show that every point on $L _ { 1 }$ is equidistant from $B$ and $C$.
15
The line $L _ { 2 }$ lies in the plane $\Pi$, and every point on $L _ { 2 }$ is equidistant from $A$ and $B$.
15
The points $A , B$ and $C$ all lie on a circle $G$. The point $D$ is the centre of circle $G$. Find the coordinates of $D$.
\includegraphics[max width=\textwidth, alt={}, center]{b4ba8a08-333d-4efc-a0ed-14fef2d99410-26_2488_1719_219_150}

AQA Further Paper 2 2021 June Q1

1 Which of the following matrices is singular?
Circle your answer.
$\left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right]$
$\left[ \begin{array} { l l } 1 & 1
2 & 2 \end{array} \right]$
$\left[ \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right]$
$\left[ \begin{array} { c c } 1 & - 2
1 & 2 \end{array} \right]$

AQA Further Paper 2 2021 June Q2

1 marks

2 Find arg ( $- 4 - 7 \mathrm { i }$ ) to the nearest degree.
Circle your answer.
[0pt] [1 mark]
$- 120 ^ { \circ }$
$- 60 ^ { \circ }$
$30 ^ { \circ }$
$60 ^ { \circ }$

AQA Further Paper 2 2021 June Q3

AQA Further Paper 2 2021 June Q4

3 marks

Show that $$( r + 1 ) ^ { 2 } - r ^ { 2 } = 2 r + 1$$ 4
Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } ( 2 r + 1 ) = n ^ { 2 } + 2 n$$ 4
Verify that using the formula for $\sum _ { r = 1 } ^ { n } r$ gives the same result as that given in part (b).
[0pt] [3 marks]

AQA Further Paper 2 2021 June Q5

5 The equation $$z ^ { 3 } + 2 z ^ { 2 } - 5 z - 3 = 0$$ has roots $\alpha , \beta$ and $\gamma$
Find a cubic equation with roots $$\frac { 1 } { 2 } \alpha - 1 , \frac { 1 } { 2 } \beta - 1 \text { and } \frac { 1 } { 2 } \gamma - 1$$

AQA Further Paper 2 2021 June Q6

6 The ellipse $E _ { 1 }$ has equation $$x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$$ $E _ { 1 }$ is translated by the vector $\left[ \begin{array} { l } 3
0 \end{array} \right]$ to give the ellipse $E _ { 2 }$
6

Write down the equation of $E _ { 2 }$ 6
The ellipse $E _ { 3 }$ has equation $$\frac { x ^ { 2 } } { 4 } + ( y - 3 ) ^ { 2 } = 1$$ Describe the transformation that maps $E _ { 2 }$ to $E _ { 3 }$ 6
Each of the lines $L _ { A }$ and $L _ { B }$ is a tangent to both $E _ { 2 }$ and $E _ { 3 }$
$L _ { A }$ is closer to the origin than $L _ { B }$
$E _ { 2 }$ and $E _ { 3 }$ both lie between $L _ { A }$ and $L _ { B }$
Sketch and label $E _ { 2 } , E _ { 3 } , L _ { A }$ and $L _ { B }$ on the axes below.
You do not need to show the values of the axis intercepts for $L _ { A }$ and $L _ { B }$
\includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-09_1095_1095_726_475} 6
Explain, without doing any calculations, why $L _ { A }$ has an equation of the form $$x + y = c$$ where $c$ is a constant.

AQA Further Paper 2 2021 June Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-10_854_1027_264_520} The diagram shows a curve known as an astroid.
The curve has parametric equations $$\begin{aligned} & x = 4 \cos ^ { 3 } t
& y = 4 \sin ^ { 3 } t
& ( 0 \leq t < 2 \pi ) \end{aligned}$$ The section of the curve from $t = 0$ to $t = \frac { \pi } { 2 }$ is rotated through $2 \pi$ radians about the $x$-axis. Show that the curved surface area of the shape formed is equal to $\frac { b \pi } { c }$, where $b$ and $c$ are integers.

AQA Further Paper 2 2021 June Q8

6 marks

8 The complex number $z$ satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where $a$ is real.
Show that $a$ must lie in the interval $[ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]$, where $s$ and $t$ are prime numbers.
[0pt] [6 marks]

AQA Further Paper 2 2021 June Q9

7 marks

The line $L$ has polar equation $$r = \frac { 7 } { 4 } \sec \theta \quad \left( - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } \right)$$ Show that $L$ is perpendicular to the initial line.
9
The curve $C$ has polar equation $$r = 3 + \cos \theta \quad ( - \pi < \theta \leq \pi )$$ Find the polar coordinates of the points of intersection of $L$ and $C$ Fully justify your answer.
9
The region $R$ is the set of points such that
and $$r > \frac { 7 } { 4 } \sec \theta$$ Find the exact area of $R$ $$r < 3 + \cos \theta$$ Find the exact area of $R$
[0pt] [7 marks]

AQA Further Paper 2 2021 June Q10

10 In a colony of seabirds, there are $y$ birds at time $t$ years. 10

The rate of reduction in the number of birds due to birds dying or leaving the colony is proportional to the number of birds. In one year the reduction in the number of birds due to birds dying or leaving the colony is equal to $16 \%$ of the number of birds at the start of the year. If no birds are born or join the colony, find the constant $k$ such that $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k y$$ Give your answer to three significant figures.
10
A wildlife protection group takes measures to support the colony.
The rate of reduction in the number of birds due to birds dying or leaving the colony is the same as in part (a), but in addition:
- The rate of increase in the number of birds due to births is $20 t$ per year.
- The wildlife protection group brings 45 birds into the colony each year.
Write down a first-order differential equation for $y$ and $t$
10
The initial number of birds is 340 Solve your differential equation from part (b) to find $y$ in terms of $t$
10
Describe two limitations of the model you have used. Limitation 1 $\_\_\_\_$
Limitation 2 $\_\_\_\_$

AQA Further Paper 2 2021 June Q11

11 The Cartesian equation of the line $L _ { 1 }$ is $$\frac { x + 1 } { 3 } = \frac { - y + 5 } { 2 } = \frac { 2 z + 5 } { 3 }$$ The Cartesian equation of the line $L _ { 2 }$ is $$\frac { 2 x - 1 } { 2 } = \frac { y - 14 } { m } = \frac { z + 12 } { p }$$ The non-singular matrix $\mathbf { N } = \left[ \begin{array} { c c c } - 0.5 & 1 & 2
1 & b & 4
- 3 & - 2 & c \end{array} \right]$ maps the line $L _ { 1 }$ onto the line $L _ { 2 }$
Calculate the values of the constants $b , c , m$ and $p$
Fully justify your answers.

AQA Further Paper 2 2021 June Q12

12 The integral $S _ { n }$ is defined by $$S _ { n } = \int _ { 0 } ^ { a } x ^ { n } \sinh x \mathrm {~d} x \quad ( n \geq 0 )$$ 12

Show that for $n \geq 2$ $$S _ { n } = n ( n - 1 ) S _ { n - 2 } + a ^ { n } \cosh a - n a ^ { n - 1 } \sinh a$$
12
Hence show that $\int _ { 0 } ^ { 1 } x ^ { 4 } \sinh x d x = \frac { 9 } { 2 } e + \frac { 65 } { 2 } e ^ { - 1 } - 24$

AQA Further Paper 2 2021 June Q13

4 marks

Two of the solutions to the equation $\cos 6 \theta = 0$ are $\theta = \frac { \pi } { 4 }$ and $\theta = \frac { 3 \pi } { 4 }$
Find the other solutions to the equation $\cos 6 \theta = 0$ for $0 \leq \theta \leq \pi$ 13
Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ 13
Use the fact that $\theta = \frac { \pi } { 4 }$ and $\theta = \frac { 3 \pi } { 4 }$ are solutions to the equation $\cos 6 \theta = 0$ to find a factor of $32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$ in the form ( $a \cos ^ { 2 } \theta + b$ ), where $a$ and $b$ are integers.
[0pt] [4 marks]
Hence show that $$\cos \left( \frac { 11 \pi } { 12 } \right) = - \sqrt { \frac { 2 + \sqrt { 3 } } { 4 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-25_2492_1721_217_150}

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