AQA Further Paper 2 (Further Paper 2) 2022 June

1 Find the imaginary part of $$\frac { 5 + \mathrm { i } } { 1 - \mathrm { i } }$$ Circle your answer.
-3
-2

2
3 2 Find the mean value of the function \(\mathrm { f } ( x ) = 10 x ^ { 4 }\) between \(x = 0\) and \(x = a\) Circle your answer.
[0pt] [1 mark]
\(10 a ^ { 3 }\)
\(40 a ^ { 3 }\)
\(2 a ^ { 4 }\)
\(4 a ^ { 5 }\)

3 The roots of the equation \(x ^ { 2 } - p x - 6 = 0\) are \(\alpha\) and \(\beta\) Find \(\alpha ^ { 2 } + \beta ^ { 2 }\) in terms of \(p\)
Circle your answer.
\(p ^ { 2 } - 6\)
\(p ^ { 2 } + 6\)
\(p ^ { 2 } - 12\)
\(p ^ { 2 } + 12\)

4 Which of the following graphs intersects the graph of \(y = \sinh x\) at exactly one point? Circle your answer.
\(y = \operatorname { cosech } x\)
\(y = \cosh x\)
\(y = \operatorname { coth } x\)
\(y = \operatorname { sech } x\)

5 Prove by induction that, for all integers \(n \geq 1\), $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \left\{ \frac { 1 } { 2 } n ( n + 1 ) \right\} ^ { 2 }$$ [4 marks]

6 The diagram below shows part of the graph of \(y = \mathrm { f } ( x )\) The line \(T P Q\) is a tangent to the graph of \(y = \mathrm { f } ( x )\) at the point \(P \left( \frac { a + b } { 2 } , \mathrm { f } \left( \frac { a + b } { 2 } \right) \right)\)
The points \(S ( a , 0 )\) and \(T\) lie on the line \(x = a\)
The points \(Q\) and \(R ( b , 0 )\) lie on the line \(x = b\)
\includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-05_748_696_669_671} Sharon uses the mid-ordinate rule with one strip to estimate the value of the integral \(\int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { d } x\) By considering the area of the trapezium QRST, state, giving reasons, whether you would expect Sharon's estimate to be an under-estimate or an over-estimate.

7 The function f is defined by $$\mathrm { f } ( x ) = \frac { a x - 5 } { 2 x + b } \quad x \in \mathbb { R } , x \neq \frac { 9 } { 2 }$$ where \(a\) and \(b\) are integers.
The graph of \(y = \mathrm { f } ( x )\) has asymptotes \(x = \frac { 9 } { 2 }\) and \(y = 3\)
7

1. Find the value of \(a\) and the value of \(b\)
   7
2. Solve the inequality $$\mathrm { f } ( x ) \leq x + 2$$ Fully justify your answer.

8

1. The function f is defined as \(\mathrm { f } ( x ) = \sec x\) 8
2. 1. Show that \(\mathrm { f } ^ { ( 4 ) } ( 0 ) = 5\)
      8
3. (ii) Hence find the first three non-zero terms of the Maclaurin series for \(\mathrm { f } ( x ) = \sec x\)
   8
4. Prove that $$\lim _ { x \rightarrow 0 } \left( \frac { \sec x - \cosh x } { x ^ { 4 } } \right) = \frac { 1 } { 6 }$$

9

1. A curve passes through the point (5, 12.3) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( x ^ { 2 } - 9 \right) ^ { \frac { 1 } { 2 } } + \frac { 2 x y } { x ^ { 2 } - 9 } \quad x > 3$$ Use Euler's step by step method once, and then the midpoint formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right) , \quad x _ { r + 1 } = x _ { r } + h$$ once, each with a step length of 0.1 , to estimate the value of \(y\) when \(x = 5.2\)
   Give your answer to six significant figures.
   9
2. 1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \left( x ^ { 2 } - 9 \right) ^ { \frac { 1 } { 2 } } + \frac { 2 x y } { x ^ { 2 } - 9 } \quad ( x > 3 )$$ 9
3. (ii) Given that \(y\) satisfies the differential equation in part (b)(i) and that \(y = 12.3\) when \(x = 5\), find the value of \(y\) when \(x = 5.2\) Give your answer to six significant figures.
   [0pt] [3 marks]
   9
4. Comment on the accuracy of your answer to part (a).

10 The curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1$$ The curve \(C _ { 2 }\) has equation $$x ^ { 2 } - 25 y ^ { 2 } - 6 x - 200 y - 416 = 0$$ 10

1. Find a sequence of transformations that maps the graph of \(C _ { 1 }\) onto the graph of \(C _ { 2 }\) [4 marks]
   10
2. Find the equations of the asymptotes to \(C _ { 2 }\)
   Give your answers in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

11

1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 5 } { 2 } & - \frac { 3 } { 2 }
   - \frac { 3 } { 2 } & \frac { 13 } { 2 } \end{array} \right]$$ 11
2. 1. Describe how the directions of the invariant lines of the transformation represented by \(\mathbf { M }\) are related to each other. Fully justify your answer.
      [0pt] [2 marks]
      11
3. (ii) Describe fully the transformation represented by \(\mathbf { M }\)

12 The shaded region shown in the diagram below is bounded by the \(x\)-axis, the curve \(y = \mathrm { f } ( x )\), and the lines \(x = a\) and \(x = b\)
\includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-16_661_721_406_662} The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid.
12

1. Show that the volume of this solid is $$\pi \int _ { a } ^ { b } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x$$ 12
2. In the case where \(a = 1 , b = 2\) and $$f ( x ) = \frac { x + 3 } { ( x + 1 ) \sqrt { x } }$$ show that the volume of the solid is $$\pi \left( \ln \left( \frac { 2 ^ { m } } { 3 ^ { n } } \right) - \frac { 2 } { 3 } \right)$$ where \(m\) and \(n\) are integers.

13

1. The matrix A represents a reflection in the line \(y = m x\), where \(m\) is a constant. Show that \(\mathbf { A } = \left( \frac { 1 } { m ^ { 2 } + 1 } \right) \left[ \begin{array} { c c } 1 - m ^ { 2 } & 2 m
   2 m & m ^ { 2 } - 1 \end{array} \right]\)
   You may use the result in the formulae booklet. 13
2. \(\quad\) The matrix \(\mathbf { B }\) is defined as \(\mathbf { B } = \left[ \begin{array} { l l } 3 & 0
   0 & 3 \end{array} \right]\)
   Show that \(( \mathbf { B A } ) ^ { 2 } = k \mathbf { I }\)
   where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix and \(k\) is an integer.
   13
3. 1. The diagram below shows a point \(P\) and the line \(y = m x\) Draw four lines on the diagram to demonstrate the result proved in part (b).
      Label as \(P ^ { \prime }\) the image of \(P\) under the transformation represented by (BA) \({ } ^ { 2 }\)
      \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-20_579_1068_584_488} 13
4. (ii) Explain how your completed diagram shows the result proved in part (b).
   13
5. The matrix \(\mathbf { C }\) is defined as \(\mathbf { C } = \left[ \begin{array} { c c } \frac { 12 } { 5 } & \frac { 9 } { 5 }
   \frac { 9 } { 5 } & - \frac { 12 } { 5 } \end{array} \right]\)
   Find the value of \(m\) such that \(\mathbf { C } = \mathbf { B A }\) Fully justify your answer.
   [0pt] [4 marks]

14 On an isolated island some rabbits have been accidently introduced. In order to eliminate them, conservationists have introduced some birds of prey.
At time \(t\) years \(( t \geq 0 )\) there are \(x\) rabbits and \(y\) birds of prey.
At time \(t = 0\) there are 1755 rabbits and 30 birds of prey.
When \(t > 0\) it is assumed that:

• the rabbits will reproduce at a rate of \(a \%\) per year
• each bird of prey will kill, on average, \(b\) rabbits per year
• the death rate of the birds of prey is \(c\) birds per year
• the number of birds of prey will increase at a rate of \(d \%\) of the rabbit population per year.

This system is represented by the coupled differential equations: $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4 x - 13 y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.01 x - 1.95 \end{aligned}$$ 14

1. State the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\)
   [0pt] [2 marks]
   14
2. Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\)