AQA Further Paper 2 (Further Paper 2) 2021 June

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

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

3 The line \(L\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 3
2
0 \end{array} \right] + \lambda \left[ \begin{array} { c } - 1
- 2
5 \end{array} \right]\) Which of the following lines is perpendicular to the line \(L\) ?
Tick \(( \checkmark )\) one box. $$\begin{aligned} & \mathbf { r } = \left[ \begin{array} { c } 2
- 3
4 \end{array} \right] + \mu \left[ \begin{array} { c } 1
2
- 5 \end{array} \right]
& \mathbf { r } = \left[ \begin{array} { l } 1
0
1 \end{array} \right] + \mu \left[ \begin{array} { c } 2
- 3
1 \end{array} \right]
& \mathbf { r } = \left[ \begin{array} { l } 1
2
1 \end{array} \right] + \mu \left[ \begin{array} { l } 1
1
2 \end{array} \right]
& \mathbf { r } = \left[ \begin{array} { l } 0
3
2 \end{array} \right] + \mu \left[ \begin{array} { l } 4
3
2 \end{array} \right] \end{aligned}$$ □
□
□
□

4

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

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

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

1. Write down the equation of \(E _ { 2 }\) 6
2. 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
3. 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
4. Explain, without doing any calculations, why \(L _ { A }\) has an equation of the form $$x + y = c$$ where \(c\) is a constant.

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.

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]

9

1. 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
2. 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
3. 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]

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

1. 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
2. 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
3. The initial number of birds is 340 Solve your differential equation from part (b) to find \(y\) in terms of \(t\)
   10
4. Describe two limitations of the model you have used. Limitation 1 \(\_\_\_\_\)
   Limitation 2 \(\_\_\_\_\)

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.

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

1. 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 \(\int _ { 0 } ^ { 1 } x ^ { 4 } \sinh x d x = \frac { 9 } { 2 } e + \frac { 65 } { 2 } e ^ { - 1 } - 24\)

13

1. 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
2. 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
3. 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]
4. 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}