Questions — CAIE Further Paper 2 (195 questions)

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CAIE Further Paper 2 2020 Specimen Q3
8 marks Standard +0.8
3 Find the solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = \frac { \sin x } { x }$$ for which \(y = 0\) when \(x = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( x )\).
CAIE Further Paper 2 2020 Specimen Q4
8 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{b5503355-3952-47dc-91f4-80a674349b4a-06_538_949_269_557} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } }\) for \(x > 0\), together with a set of \(( n - 1 )\) rectangles of unit width.
  1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < \frac { 2 n - 1 } { n } .$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }\).
CAIE Further Paper 2 2020 Specimen Q5
10 marks Challenging +1.2
5 The curve \(C\) has parametric equations $$x = \mathrm { e } ^ { t } - 4 t + 3 , \quad y = 8 \mathrm { e } ^ { \frac { 1 } { 2 } t } , \quad \text { for } 0 \leqslant t \leqslant 2 .$$
  1. Find, in terms of e , the length of \(C\).
  2. Find, in terms of \(\pi\) and e , the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE Further Paper 2 2020 Specimen Q6
10 marks Challenging +1.8
6
  1. Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
  2. Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
CAIE Further Paper 2 2020 Specimen Q7
12 marks Challenging +1.8
7
  1. Starting from the definition of tanh in terms of exponentials, prove that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\). [3]
  2. Given that \(y = \tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\), show that \(( 2 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 1 = 0\).
  3. Hence find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\) in the form $$a \ln 3 + b x + c x ^ { 2 } ,$$ where \(a , b\) and \(c\) are constants to be determined.
CAIE Further Paper 2 2020 Specimen Q8
15 marks Standard +0.3
8
    1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { r } x - 2 y - 2 z + 7 = 0
CAIE Further Paper 2 2024 November Q7
10 marks Challenging +1.8
7
  1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) . \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-15_2723_33_99_22}
  2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).
CAIE Further Paper 2 2024 November Q8
14 marks Hard +2.3
8
  1. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 7 }\) ,where \(z = \cos \theta + \mathrm { i } \sin \theta\) ,use de Moivre's theorem to show that $$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$ where \(a , b , c\) and \(d\) are constants to be determined.
    Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta\).
  2. Show that $$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-18_2718_42_107_2007}
  3. Using the results given in parts (a) and (b), find the exact value of \(I _ { 9 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2024 November Q7
10 marks Challenging +1.8
7
  1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) . \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-15_2723_33_99_22}
  2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).
CAIE Further Paper 2 2020 June Q4
8 marks Challenging +1.2
  1. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } x ^ { 2 } d x < \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm {~d} x\).
CAIE Further Paper 2 2020 November Q4
8 marks Challenging +1.2
  1. By considering the sum of the areas of the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) d x \leqslant \frac { 3 n ^ { 2 } + 2 n - 1 } { 4 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) \mathrm { dx }\).
CAIE Further Paper 2 2021 November Q4
10 marks Challenging +1.8
  1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { \ln r } { r ^ { 2 } } < \frac { 2 + 3 \ln 2 } { 4 } - \frac { 1 + \ln N } { N }$$
  2. Use a similar method to find, in terms of \(N\), a lower bound for \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { N } } \frac { \ln \mathrm { r } } { \mathrm { r } ^ { 2 } }\).
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]
CAIE Further Paper 2 2021 November Q4
10 marks Challenging +1.8
  1. Write down all the roots of the equation \(x^5 - 1 = 0\). [2]
  2. Use de Moivre's theorem to show that \(\cos 4\theta = 8\cos^4 \theta - 8\cos^2 \theta + 1\). [4]
  3. Use the results of parts (a) and (b) to express each real root of the equation $$8x^9 - 8x^7 + x^5 - 8x^4 + 8x^2 - 1 = 0$$ in the form \(\cos k\pi\), where \(k\) is a rational number. [4]
CAIE Further Paper 2 2021 November Q5
10 marks Standard +0.3
The curve \(C\) has parametric equations $$x = 3t + 2t^{-1} + at^3, \quad y = 4t - \frac{3}{2}t^{-1} + bt^3, \quad \text{for } 1 \leq t \leq 2,$$ where \(a\) and \(b\) are constants.
  1. It is given that \(a = \frac{2}{3}\) and \(b = -\frac{1}{2}\). Show that \(\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = \frac{25}{4}(t^2 + t^{-2})^2\) and find the exact length of \(C\). [6]
  2. It is given instead that \(a = b = 0\). Find the value of \(\frac{d^2y}{dx^2}\) when \(t = 1\). [4]