Questions Further Paper 2 (305 questions)

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AQA Further Paper 2 2022 June Q1
1 marks Easy -1.8
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 Easy -1.2
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] \(10 a ^ { 3 }\) \(40 a ^ { 3 }\) \(2 a ^ { 4 }\) \(4 a ^ { 5 }\)
AQA Further Paper 2 2022 June Q3
1 marks Moderate -0.8
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
1 marks Moderate -0.5
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 Standard +0.3
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
3 marks Challenging +1.2
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
8 marks Standard +0.8
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
10 marks Challenging +1.2
8
  1. The function f is defined as \(\mathrm { f } ( x ) = \sec x\) 8
    1. (i) Show that \(\mathrm { f } ^ { ( 4 ) } ( 0 ) = 5\) 8
    2. (ii) Hence find the first three non-zero terms of the Maclaurin series for \(\mathrm { f } ( x ) = \sec x\) 8
    3. 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
14 marks Standard +0.8
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. (i) 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 (b) (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
  3. Comment on the accuracy of your answer to part (a).
AQA Further Paper 2 2022 June Q10
7 marks Challenging +1.2
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
9 marks Standard +0.3
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. (i) 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 (b) (ii) Describe fully the transformation represented by \(\mathbf { M }\)
AQA Further Paper 2 2022 June Q12
11 marks Standard +0.3
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
16 marks Challenging +1.2
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. (i) 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 (c) (ii) Explain how your completed diagram shows the result proved in part (b).
    13
  4. 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
14 marks Challenging +1.8
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 \(t\)
CAIE Further Paper 2 2020 June Q1
6 marks Standard +0.3
Find the general solution of the differential equation $$\frac{d^2x}{dt^2} - 8\frac{dx}{dt} - 9x = 9e^{8t}.$$ [6]
CAIE Further Paper 2 2020 June Q2
6 marks Challenging +1.2
Let \(I_n = \int_0^1 (1+3x)^n e^{-3x} dx\), where \(n\) is an integer.
  1. Show that \(3I_n = 1 - 4^n e^{-3} + 3nI_{n-1}\). [3]
  2. Find the exact value of \(I_2\). [3]
CAIE Further Paper 2 2020 June Q3
8 marks Challenging +1.2
The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} 5 & -1 & 7 \\ 0 & 6 & 0 \\ 7 & 7 & 5 \end{pmatrix}.$$
  1. Find the eigenvalues of \(\mathbf{A}\). [4]
  2. Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\). [4]
CAIE Further Paper 2 2020 June Q4
8 marks Challenging +1.2
\includegraphics{figure_4} The diagram shows the curve with equation \(y = \ln x\) for \(x \geqslant 1\), together with a set of \((N-1)\) rectangles of unit width.
  1. By considering the sum of the areas of these rectangles, show that $$\ln N! > N \ln N - N + 1.$$ [5]
  2. Use a similar method to find, in terms of \(N\), an upper bound for \(\ln N!\). [3]
CAIE Further Paper 2 2020 June Q5
9 marks Challenging +1.3
The curve \(C\) has parametric equations $$x = \frac{1}{2}t^2 - \ln t, \quad y = 2t + 1, \quad \text{for } \frac{1}{2} \leqslant t \leqslant 2.$$
  1. Find the exact length of \(C\). [5]
  2. Find \(\frac{d^2y}{dx^2}\) in terms of \(t\), simplifying your answer. [4]
CAIE Further Paper 2 2020 June Q6
12 marks Standard +0.8
  1. Starting from the definitions of \(\tanh\) and \(\sech\) in terms of exponentials, prove that $$1 - \tanh^2 \theta = \sech^2 \theta.$$ [3]
The variables \(x\) and \(y\) are such that \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\), for \(-\frac{1}{4}\pi < x < \frac{3}{4}\pi\).
  1. By differentiating the equation \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\) with respect to \(x\), show that $$\frac{dy}{dx} = -\operatorname{cosec}\left(x + \frac{1}{4}\pi\right).$$ [4]
  2. Hence find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\cos\left(x + \frac{1}{4}\pi\right)\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [5]
CAIE Further Paper 2 2020 June Q7
11 marks Challenging +1.3
  1. Show that an appropriate integrating factor for $$(x^2 + 1)\frac{dy}{dx} + y\sqrt{x^2 + 1} = x^2 - x\sqrt{x^2 + 1}$$ is \(x + \sqrt{x^2 + 1}\). [4]
  2. Hence find the solution of the differential equation $$(x^2 + 1)\frac{dy}{dx} + y\sqrt{x^2 + 1} = x^2 - x\sqrt{x^2 + 1}$$ for which \(y = \ln 2\) when \(x = 0\). Give your answer in the form \(y = f(x)\). [7]
CAIE Further Paper 2 2020 June Q8
15 marks Challenging +1.8
  1. Use de Moivre's theorem to show that \(\sin^6 \theta = -\frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10)\). [6]
It is given that \(\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6\cos 4\theta + 15\cos 2\theta + 10)\).
  1. Find the exact value of \(\int_0^{\frac{1}{4}\pi}\left(\cos^6\left(\frac{1}{4}x\right) + \sin^6\left(\frac{1}{4}x\right)\right)dx\). [4]
  2. Express each root of the equation \(16c^6 + 16\left(1-c^2\right)^3 - 13 = 0\) in the form \(\cos k\pi\), where \(k\) is a rational number. [5]
CAIE Further Paper 2 2021 November Q1
5 marks Standard +0.3
It is given that \(y = \sinh(x^2) + \cosh(x^2)\).
  1. Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x^4\). [2]
  2. Deduce the value of \(\frac{d^4y}{dx^4}\) when \(x = 0\). [1]
  3. Use your answer to part (a) to find an approximation to \(\int_0^{\frac{1}{2}} y \, dx\), giving your answer as a rational fraction in its lowest terms. [2]
CAIE Further Paper 2 2021 November Q2
7 marks Standard +0.3
Find the solution of the differential equation $$\frac{dy}{dx} + \frac{4x^3y}{x^4 + 5} = 6x$$ for which \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f(x)\). [7]
CAIE Further Paper 2 2021 November Q3
8 marks Challenging +1.2
\includegraphics{figure_3} The diagram shows the curve with equation \(y = 1 - x^2\) for \(0 \leq x \leq 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).
  1. By considering the sum of the areas of the rectangles, show that $$\int_0^1 (1 - x^2) \, dx < \frac{4n^2 + 3n - 1}{6n^2}.$$ [4]
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int_0^1 (1 - x^2) \, dx\). [4]