Questions - AQA Further Paper 2

12	Prove that Bonnie is right.

13(a) Explain why $\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x$ is an improper integral.

[1 mark]

13(b) Evaluate $\int _ { 3 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - 2 x } \mathrm {~d} x$

Show the limiting process.

[9 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

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AQA Further Paper 2 2019 June Q14

Use the method of differences to show that $$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$ where $a , b$ and $c$ are integers.
Question 14 continues on the next page 14
Show that, for any number $k$ greater than $\frac { 12 } { 5 }$, if the difference between $\frac { 5 } { 12 }$ and $S _ { n }$ is less than $\frac { 1 } { k }$, then $$n > \frac { k - 5 + \sqrt { k ^ { 2 } + 1 } } { 2 }$$

AQA Further Paper 2 2019 June Q15

9 marks

Find the value of $r$. 15
Show that $\mu = 3$ 15
Solve the coupled differential equations to find both $x$ and $y$ in terms of $t$.
[0pt] [9 marks]

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Additional page, if required.
Write the question numbers in the left-hand margin.

AQA Further Paper 2 2020 June Q1

1 Three of the four expressions below are equivalent to each other.
Which of the four expressions is not equivalent to any of the others? Circle your answer.
$\mathbf { a } \times ( \mathbf { a } + \mathbf { b } )$
$( \mathbf { a } + \mathbf { b } ) \times \mathbf { b }$
$( \mathbf { a } - \mathbf { b } ) \times \mathbf { b }$
$\mathbf { a } \times ( \mathbf { a } - \mathbf { b } )$

AQA Further Paper 2 2020 June Q2

2 Given that arg $( a + b \mathrm { i } ) = \varphi$, where $a$ and $b$ are positive real numbers and $0 < \varphi < \frac { \pi } { 2 }$, three of the following four statements are correct. Which statement is not correct? Tick $( \checkmark )$ one box. $$\begin{aligned} & \arg ( - a - b \mathrm { i } ) = \pi - \varphi
& \arg ( a - b \mathrm { i } ) = - \varphi
& \arg ( b + a \mathrm { i } ) = \frac { \pi } { 2 } - \varphi
& \arg ( b - a \mathrm { i } ) = \varphi - \frac { \pi } { 2 } \end{aligned}$$

AQA Further Paper 2 2020 June Q3

3 Find the gradient of the tangent to the curve $$y = \sin ^ { - 1 } x$$ at the point where $x = \frac { 1 } { 5 }$
Circle your answer.

$\frac { 5 \sqrt { 6 } } { 12 }$

$\frac { 2 \sqrt { 6 } } { 5 }$

$\frac { 4 \sqrt { 3 } } { 25 }$

$\frac { 25 } { 24 }$

AQA Further Paper 2 2020 June Q4

4 The matrices A and B are defined as follows: $$\begin{aligned} & \mathbf { A } = \left[ \begin{array} { l l } x + 1 & 2
x + 2 & - 3 \end{array} \right]
& \mathbf { B } = \left[ \begin{array} { c c } x - 4 & x - 2
0 & - 2 \end{array} \right] \end{aligned}$$ Show that there is a value of $x$ for which $\mathbf { A B } = k \mathbf { I }$, where $\mathbf { I }$ is the $2 \times 2$ identity matrix and $k$ is an integer to be found.

AQA Further Paper 2 2020 June Q5

5 Solve the inequality $$\frac { 2 x + 3 } { x - 1 } \leq x + 5$$

AQA Further Paper 2 2020 June Q6

6 Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers.

AQA Further Paper 2 2020 June Q7

7 The diagram shows part of the graph of $y = \cos ^ { - 1 } x$ The diagram shows part of the graph of $y = \cos ^ { - 1 } x$
\includegraphics[max width=\textwidth, alt={}, center]{b4ba8a08-333d-4efc-a0ed-14fef2d99410-07_689_958_358_539} The finite region enclosed by the graph of $y = \cos ^ { - 1 } x$, the $y$-axis, the $x$-axis and the line $x = 0.8$ is rotated by $2 \pi$ radians about the $x$-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places.

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]

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