Questions - AQA Further Paper 2

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]

AQA Further Paper 2 2019 June Q1

1 marks

1 Given that $z$ is a complex number, and that $z ^ { * }$ is the complex conjugate of $z$, which of the following statements is not always true? Circle your answer.
[0pt] [1 mark] $$\left( z ^ { * } \right) ^ { * } = z \quad z z ^ { * } = | z | ^ { 2 } \quad ( - z ) ^ { * } = - \left( z ^ { * } \right) \quad z - z ^ { * } = z ^ { * } - z$$

AQA Further Paper 2 2019 June Q2

1 marks

2 Which of the straight lines given below is an asymptote to the curve $$y = \frac { a x ^ { 2 } } { x - 1 }$$ where $a$ is a non-zero constant? Circle your answer.
[0pt] [1 mark]
$y = a x + a$
$y = a x$
$y = a x - a$
$y = a$

AQA Further Paper 2 2019 June Q3

1 marks

3 The set $\mathcal { A }$ is defined by $\mathcal { A } = \{ x : - \sqrt { } 2 < x < 0 \} \cup \{ x : 0 < x < \sqrt { } 2 \}$
Which of the inequalities given below has $\mathcal { A }$ as its solution?
Circle your answer.
[0pt] [1 mark]
$\left| x ^ { 2 } - 1 \right| > 1$
$\left| x ^ { 2 } - 1 \right| \geq 1$
$\left| x ^ { 2 } - 1 \right| < 1$
$\left| x ^ { 2 } - 1 \right| \leq 1$

AQA Further Paper 2 2019 June Q4

4 The positive integer $k$ is such that $$\sum _ { r = 1 } ^ { k } ( 3 r - k ) = 90$$ Find the value of $k$.

AQA Further Paper 2 2019 June Q5

5 A curve has equation $y = \cosh x$ Show that the arc length of the curve from $x = a$ to $x = b$, where $0 < a < b$, is equal to
$\sinh b - \sinh a$
$6 \quad$ A circle $C$ in the complex plane has equation $| z - 2 - 5 \mathrm { i } | = a$ The point $z _ { 1 }$ on $C$ has the least argument of any point on $C$, and $\arg \left( z _ { 1 } \right) = \frac { \pi } { 4 }$ Prove that $a = \frac { 3 \sqrt { } 2 } { 2 }$

AQA Further Paper 2 2019 June Q7

4 marks

7 The points $A , B$ and $C$ have coordinates $A ( 4,5,2 ) , B ( - 3,2 , - 4 )$ and $C ( 2,6,1 )$ 7

Use a vector product to show that the area of triangle $A B C$ is $\frac { 5 \sqrt { 11 } } { 2 }$
[0pt] [4 marks]
7
The points $A , B$ and $C$ lie in a plane.
Find a vector equation of the plane in the form r.n $= k$
7
Hence find the exact distance of the plane from the origin.

AQA Further Paper 2 2019 June Q8

The line $y = m x$ is a tangent to $P _ { 2 }$
Prove that $m = \pm \sqrt { \frac { a } { b } }$
Solutions using differentiation will be given no marks.
8
The line $y = \sqrt { \frac { a } { b } } x$ meets $P _ { 2 }$ at the point $D$.
The finite region $R$ is bounded by the $x$-axis, $P _ { 2 }$ and a line through $D$ perpendicular to the $x$-axis. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid.
Find, in terms of $a$ and $b$, the volume of this solid.
Fully justify your answer.
Find the eigenvalues and corresponding eigenvectors of the matrix

AQA Further Paper 2 2019 June Q9

4 marks

Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { 5 } & \frac { 2 } { 5 }
\frac { - 3 } { 5 } & \frac { 13 } { 10 } \end{array} \right]$$ 9
Find matrices $\mathbf { U }$ and $\mathbf { D }$ such that $\mathbf { D }$ is a diagonal matrix and $\mathbf { M } = \mathbf { U D U } ^ { - 1 }$
9
Given that $\mathbf { M } ^ { n } \rightarrow \mathbf { L }$ as $n \rightarrow \infty$, find the matrix $\mathbf { L }$.
[0pt] [4 marks]
9
The transformation represented by $\mathbf { L }$ maps all points onto a line. Find the equation of this line.
\begin{center} \begin{tabular}{ | l | }

AQA Further Paper 2 2019 June Q10

7 marks

10
- $\begin{array} { c } \text { Prove by induction that } \mathrm { f } ( n ) = n ^ { 3 } + 3 n ^ { 2 } + 8 n \text { is divisible by } 6 \text { for all integers } n \geq 1
\text { [7 marks] }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - }
\text { - } \end{array}$
-
\end{tabular} \end{center}

\includegraphics[max width=\textwidth, alt={}]{f1ec515d-184a-4462-a6d2-5876d3e19117-13_2488_1716_219_153}

AQA Further Paper 2 2019 June Q11

11 The line $L _ { 1 }$ has equation $$\frac { x - 2 } { 3 } = \frac { y + 4 } { 8 } = \frac { 4 z - 5 } { 5 }$$ The line $L _ { 2 }$ has equation $$\left( \mathbf { r } - \left[ \begin{array} { c } - 2
0
3 \end{array} \right] \right) \times \left[ \begin{array} { l } 2
1
3 \end{array} \right] = \mathbf { 0 }$$ Find the shortest distance between the two lines, giving your answer to three significant figures.

Questions — AQA Further Paper 2 (101 questions)

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AQA Further Paper 2 2019 June Q1

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