Questions — AQA Further Paper 2 (101 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA Further Paper 2 2021 June Q9
7 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]
AQA Further Paper 2 2021 June Q10
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 \(\_\_\_\_\)
AQA Further Paper 2 2021 June Q11
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.
AQA Further Paper 2 2021 June Q12
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
  		2. Hence show that \(\int _ { 0 } ^ { 1 } x ^ { 4 } \sinh x d x = \frac { 9 } { 2 } e + \frac { 65 } { 2 } e ^ { - 1 } - 24\)
AQA Further Paper 2 2021 June Q13
4 marks
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}
AQA Further Paper 2 2022 June Q1
1 Find the imaginary part of $$\frac { 5 + \mathrm { i } } { 1 - \mathrm { i } }$$ Circle your answer.
-3
-2
AQA Further Paper 2 2022 June Q2
1 marks
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]
\(10 a ^ { 3 }\)
\(40 a ^ { 3 }\)
\(2 a ^ { 4 }\)
\(4 a ^ { 5 }\)
AQA Further Paper 2 2022 June Q3
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.
\(p ^ { 2 } - 6\)
\(p ^ { 2 } + 6\)
\(p ^ { 2 } - 12\)
\(p ^ { 2 } + 12\)
AQA Further Paper 2 2022 June Q4
4 Which of the following graphs intersects the graph of \(y = \sinh x\) at exactly one point? Circle your answer.
\(y = \operatorname { cosech } x\)
\(y = \cosh x\)
\(y = \operatorname { coth } x\)
\(y = \operatorname { sech } x\)
AQA Further Paper 2 2022 June Q5
4 marks
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]
AQA Further Paper 2 2022 June Q6
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.
AQA Further Paper 2 2022 June Q7
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.
AQA Further Paper 2 2022 June Q8
8
  1. The function f is defined as \(\mathrm { f } ( x ) = \sec x\) 8
    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 $$\lim _ { x \rightarrow 0 } \left( \frac { \sec x - \cosh x } { x ^ { 4 } } \right) = \frac { 1 } { 6 }$$
AQA Further Paper 2 2022 June Q9
3 marks
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
    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).
AQA Further Paper 2 2022 June Q10
4 marks
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.
AQA Further Paper 2 2022 June Q11
2 marks
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
    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 \(\mathbf { M }\)
AQA Further Paper 2 2022 June Q12
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.
AQA Further Paper 2 2022 June Q13
4 marks
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
    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]
AQA Further Paper 2 2022 June Q14
5 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 \(t\)
AQA Further Paper 2 2023 June Q1
1 Given that \(y = \sin x + \sinh x\), find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y\)
Circle your answer. $$\begin{array} { l l l l } 2 \sin x & - 2 \sin x & 2 \sinh x & - 2 \sinh x \end{array}$$
AQA Further Paper 2 2023 June Q2
1 marks
2 Which one of the expressions below is not equal to zero?
Circle your answer.
[0pt] [1 mark]
\(\lim _ { x \rightarrow \infty } \left( x ^ { 2 } \mathrm { e } ^ { - x } \right)\)
\(\lim _ { x \rightarrow 0 } \left( x ^ { 5 } \ln x \right)\)
\(\lim _ { x \rightarrow \infty } \left( \frac { \mathrm { e } ^ { x } } { x ^ { 5 } } \right)\)
\(\lim _ { x \rightarrow 0 } \left( x ^ { 3 } \mathrm { e } ^ { x } \right)\)
AQA Further Paper 2 2023 June Q3
3 The determinant \(A = \left| \begin{array} { l l l } 1 & 1 & 1
2 & 0 & 2
3 & 2 & 1 \end{array} \right|\)
Which one of the determinants below has a value which is not equal to the value of \(A\) ?
Tick ( \(\checkmark\) ) one box.
\(\left| \begin{array} { l l l } 313
202
321 \end{array} \right|\)\(\square\)
\(\left| \begin{array} { l l l } 123
102
121 \end{array} \right|\)\(\square\)
\(\left| \begin{array} { l l l } 222
101
321 \end{array} \right|\)\(\square\)
\(\left| \begin{array} { l l l } 111
321
202 \end{array} \right|\)\(\square\)
AQA Further Paper 2 2023 June Q4
1 marks
4 It is given that \(\mathrm { f } ( x ) = \cosh ^ { - 1 } ( x - 3 )\)
Which of the sets listed below is the greatest possible domain of the function \(f\) ?
Circle your answer.
[0pt] [1 mark] $$\{ x : x \geq 4 \} \quad \{ x : x \geq 3 \} \quad \{ x : x \geq 1 \} \quad \{ x : x \geq 0 \}$$
AQA Further Paper 2 2023 June Q5
5 Josh and Zoe are solving the following mathematics problem: The curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The matrix \(\mathbf { M } = \left[ \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right]\) maps \(C _ { 1 }\) onto \(C _ { 2 }\)
Find the equations of the asymptotes of \(C _ { 2 }\) Josh says that to solve this problem you must first carry out the transformation on \(C _ { 1 }\) to find \(C _ { 2 }\), and then find the asymptotes of \(C _ { 2 }\) Zoe says that you will get the same answer if you first find the asymptotes of \(C _ { 1 }\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C _ { 2 }\) Show that Zoe is correct.
AQA Further Paper 2 2023 June Q6
6
  1. Express \(- 5 - 5 \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leq \pi\)
    6
  2. The point on an Argand diagram that represents \(- 5 - 5 \mathrm { i }\) is one of the vertices of an equilateral triangle whose centre is at the origin. Find the complex numbers represented by the other two vertices of the triangle.
    Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leq \pi\)