Questions Further Paper 2

By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < \frac { 2 n - 1 } { n }$$
Use a similar method to find, in terms of $n$, a low er $\mathbf { H }$
- $\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }$.

CAIE Further Paper 2 2020 Specimen Q5

5 Th cn e $C$ has parametric equations $$x = \mathrm { e } ^ { t } - 4 t + 3 \quad y = 8 \mathrm { e } ^ { \frac { 1 } { 2 } t } , \quad \text { f } \mathbf { D } \quad 0 \leqslant t \leqslant 2$$

Find, in terms of e, the length of $C$.
Find, in terms of $\pi$ and $e$, the area of the surface generated when $C$ is rotated through $2 \pi$ radians ab the $x$-ax s.
[0pt] [\$

CAIE Further Paper 2 2020 Specimen Q6

Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$
Hen esh th the eq tion $x ^ { 2 } - 4 x + 5 = 0$ s ro $\operatorname { stan } ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)$ ad $\operatorname { an } ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)$.

CAIE Further Paper 2 2020 Specimen Q7

Starting from the definition of tanh in terms of exponentials, prove that $\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$. [ $\beta$
Given that $y = \operatorname { tah } ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right) , \mathrm { s } \quad$ th $\mathrm { t } ( 2 x + 1 ) \frac { \mathrm { dy } } { \mathrm { dx } } + 1 = 0$
Hence find the first three terms in the Maclaurin's series for $\tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)$ in the form $$a \ln 3 + b x + c x ^ { 2 }$$ wh re $a , b$ ad $c$ are constants to be determined.

CAIE Further Paper 2 2020 Specimen Q8

1. Fid bet basb le s a for which the system of equations $$\begin{array} { r l } x - 2 y - 2 z + z & 0
  2 x + ( a - 9 y - 0 z + 1 E & 0
  3 x - 6 y + 2 a z + 9 & 0 \end{array}$$ h san q sbtu in
2. Given that $a = - 3$, show that the system of equations in part (i) $\mathbf { b } \mathbf { s } \mathbf { n }$ sb t in In erp et th s situation geometrically.
The matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 1 & 2
0 & 2 & 2
- 1 & 1 & 3 \end{array} \right)$$
1. Find b eig le so A.
2. Use th ch racteristic eq tiw $\mathbf { A }$ tof id $\mathbf { A } ^ { - 1 }$. If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE Further Paper 2 2024 November Q7

Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is $\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }$ .
\includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-15_2723_33_99_22}
Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which $y = 6$ when $x = 3$.

CAIE Further Paper 2 2024 November Q8

By considering the binomial expansion of $\left( z + \frac { 1 } { z } \right) ^ { 7 }$ ,where $z = \cos \theta + \mathrm { i } \sin \theta$ ,use de Moivre's theorem to show that $$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$ where $a , b , c$ and $d$ are constants to be determined.
Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta$.
Show that $$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-18_2718_42_107_2007}
Using the results given in parts (a) and (b), find the exact value of $I _ { 9 }$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 2 2024 November Q7

Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is $\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }$ .
\includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-15_2723_33_99_22}
Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which $y = 6$ when $x = 3$.

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

AQA Further Paper 2 Specimen Q15

8 marks

Show that $\left( 1 - \frac { 1 } { 4 } \mathrm { e } ^ { 2 \mathrm { i } \theta } \right) \left( 1 - \frac { 1 } { 4 } \mathrm { e } ^ { - 2 \mathrm { i } \theta } \right) = \frac { 1 } { 16 } ( 17 - 8 \cos 2 \theta )$
[0pt] [3 marks]
15
Given that the series $\mathrm { e } ^ { 2 \mathrm { i } \theta } + \frac { 1 } { 4 } \mathrm { e } ^ { 4 \mathrm { i } \theta } + \frac { 1 } { 16 } \mathrm { e } ^ { 6 \mathrm { i } \theta } + \frac { 1 } { 64 } \mathrm { e } ^ { 8 \mathrm { i } \theta } + \ldots$. has a sum to infinity, express this sum to infinity in terms of $\mathrm { e } ^ { 2 \mathrm { i } \theta }$
15
Hence show that $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 4 ^ { n - 1 } } \cos 2 n \theta = \frac { 16 \cos 2 \theta - 4 } { 17 - 8 \cos 2 \theta }$
[0pt] [4 marks]
15
Deduce a similar expression for $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 4 ^ { n - 1 } } \sin 2 n \theta$
[0pt] [1 mark]

AQA Further Paper 2 Specimen Q16

9 marks

16 A designer is using a computer aided design system to design part of a building. He models part of a roof as a triangular prism $A B C D E F$ with parallel triangular ends $A B C$ and $D E F$, and a rectangular base $A C F D$. He uses the metre as the unit of length.
\includegraphics[max width=\textwidth, alt={}, center]{21084ed7-43f8-47c6-80c2-930ccf340d37-22_510_766_484_776} The coordinates of $B , C$ and $D$ are ( $3,1,11$ ), ( $9,3,4$ ) and ( $- 4,12,4$ ) respectively.
He uses the equation $x - 3 y = 0$ for the plane $A B C$.
He uses $\left[ \mathbf { r } - \left( \begin{array} { c } - 4
12
4 \end{array} \right) \right] \times \left( \begin{array} { c } 4
- 12
0 \end{array} \right) = \left( \begin{array} { l } 0
0
0 \end{array} \right)$ for the equation of the line $A D$.
Find the volume of the space enclosed inside this section of the roof.
[0pt] [9 marks]

CAIE Further Paper 2 2020 June Q4

By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } x ^ { 2 } d x < \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
Use a similar method to find, in terms of $n$, a lower bound for $\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm {~d} x$.

CAIE Further Paper 2 2020 November Q4

By considering the sum of the areas of the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) d x \leqslant \frac { 3 n ^ { 2 } + 2 n - 1 } { 4 n ^ { 2 } }$$
Use a similar method to find, in terms of $n$, a lower bound for $\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) \mathrm { dx }$.

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