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CAIE Further Paper 2 2020 June Q1
6 marks Standard +0.8
1 Find the general solution of the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } - 8 \frac { d x } { d t } - 9 x = 9 e ^ { 8 t }$$
CAIE Further Paper 2 2020 June Q2
6 marks Challenging +1.3
2 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } ( 1 + 3 \mathrm { x } ) ^ { \mathrm { n } } \mathrm { e } ^ { - 3 \mathrm { x } } \mathrm { dx }\), where \(n\) is an integer.
  1. Show that \(3 \mathrm { I } _ { \mathrm { n } } = 1 - 4 ^ { \mathrm { n } } \mathrm { e } ^ { - 3 } + 3 \mathrm { nl } _ { \mathrm { n } - 1 }\).
  2. Find the exact value of \(I _ { 2 }\).
CAIE Further Paper 2 2020 June Q3
8 marks Standard +0.3
3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 1 & 7 \\ 0 & 6 & 0 \\ 7 & 7 & 5 \end{array} \right) .$$
  1. Find the eigenvalues of \(\mathbf { A }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-06_568_1614_294_262} The diagram shows the curve with equation \(\mathrm { y } = \ln \mathrm { x }\) for \(x \geqslant 1\), together with a set of ( \(N - 1\) ) rectangles of unit width.
  3. By considering the sum of the areas of these rectangles, show that $$\ln N ! > N \ln N - N + 1 .$$
  4. Use a similar method to find, in terms of \(N\), an upper bound for \(\operatorname { In } N\) !.
CAIE Further Paper 2 2020 June Q5
9 marks Challenging +1.2
5 The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 1 } { 2 } \mathrm { t } ^ { 2 } - \ln \mathrm { t } , \quad \mathrm { y } = 2 \mathrm { t } + 1 , \quad \text { for } \frac { 1 } { 2 } \leqslant t \leqslant 2$$
  1. Find the exact length of \(C\).
  2. Find \(\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } }\) in terms of \(t\), simplifying your answer.
CAIE Further Paper 2 2020 June Q6
12 marks Challenging +1.2
6
  1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh ^ { 2 } \theta = \operatorname { sech } ^ { 2 } \theta$$ \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1552_374_347} \includegraphics[max width=\textwidth, alt={}]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_67_1569_466_328} ......................................................................................................................................... ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1573_735_324} \includegraphics[max width=\textwidth, alt={}]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1573_826_324} .......................................................................................................................................... . ........................................................................................................................................ . The variables \(x\) and \(y\) are such that \(\tanh y = \cos \left( x + \frac { 1 } { 4 } \pi \right)\), for \(- \frac { 1 } { 4 } \pi < x < \frac { 3 } { 4 } \pi\).
  2. By differentiating the equation \(\tanh y = \cos \left( x + \frac { 1 } { 4 } \pi \right)\) with respect to \(x\), show that $$\frac { \mathrm { dy } } { \mathrm { dx } } = - \operatorname { cosec } \left( \mathrm { x } + \frac { 1 } { 4 } \pi \right)$$
  3. 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 \(\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }\), giving the exact values of the constants \(a , b\) and \(c\).
CAIE Further Paper 2 2020 June Q7
11 marks Challenging +1.8
7
  1. Show that an appropriate integrating factor for $$\left( x ^ { 2 } + 1 \right) \frac { d y } { d x } + y \sqrt { x ^ { 2 } + 1 } = x ^ { 2 } - x \sqrt { x ^ { 2 } + 1 }$$ is \(x + \sqrt { x ^ { 2 } + 1 }\).
  2. Hence find the solution of the differential equation $$\left( x ^ { 2 } + 1 \right) \frac { d y } { d x } + y \sqrt { x ^ { 2 } + 1 } = x ^ { 2 } - x \sqrt { x ^ { 2 } + 1 }$$ for which \(y = \ln 2\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2020 June Q8
15 marks Challenging +1.8
8
  1. Use de Moivre's theorem to show that \(\sin ^ { 6 } \theta = - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )\).
    It is given that \(\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 6 } \left( \frac { 1 } { 4 } x \right) + \sin ^ { 6 } \left( \frac { 1 } { 4 } x \right) \right) \mathrm { d } x\).
  3. Express each root of the equation \(16 c ^ { 6 } + 16 \left( 1 - c ^ { 2 } \right) ^ { 3 } - 13 = 0\) in the form \(\cos k \pi\), where \(k\) is a rational number.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 2 2021 June Q1
5 marks Standard +0.3
1
  1. Given that \(a\) is an integer, show that the system of equations $$\begin{aligned} a x + 3 y + z & = 14 \\ 2 x + y + 3 z & = 0 \\ - x + 2 y - 5 z & = 17 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
  2. Find the value of \(a\) for which \(x = 1 , y = 4 , z = - 2\) is the solution to the system of equations in part (a).
CAIE Further Paper 2 2021 June Q2
7 marks Standard +0.3
2 The variables \(x\) and \(y\) are related by the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 \frac { d y } { d x } + 2 y = 2 x + 1$$
  1. Find the general solution for \(y\) in terms of \(x\).
  2. State an approximate solution for large positive values of \(x\).
CAIE Further Paper 2 2021 June Q3
10 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{fa2213b3-480c-44cb-8ba0-ebd2b94d3d90-04_851_805_251_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
  1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x < U _ { n }\), where $$\mathrm { U } _ { \mathrm { n } } = \left( \frac { \mathrm { n } + 1 } { 2 \mathrm { n } } \right) ^ { 2 }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x\).
  3. Find the least value of \(n\) such that \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } < 10 ^ { - 3 }\).
CAIE Further Paper 2 2021 June Q4
9 marks Challenging +1.2
4 Find the solution of the differential equation $$\sin \theta \frac { d y } { d \theta } + y = \tan \frac { 1 } { 2 } \theta$$ where \(0 < \theta < \pi\), given that \(y = 1\) when \(\theta = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( \theta )\). [You may use without proof the result that \(\int \operatorname { cosec } \theta d \theta = \ln \tan \frac { 1 } { 2 } \theta\).]
CAIE Further Paper 2 2021 June Q5
10 marks Challenging +1.2
5
  1. State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\).
  2. Given that \(z\) is an \(n\)th root of unity and \(z \neq 1\), deduce that \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = 0\).
  3. Given instead that \(z = \frac { 1 } { 3 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to show that $$\sum _ { m = 1 } ^ { \infty } 3 ^ { - m } \cos m \theta = \frac { 3 \cos \theta - 1 } { 10 - 6 \cos \theta }$$
CAIE Further Paper 2 2021 June Q6
11 marks Challenging +1.2
6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 5 & - \frac { 22 } { 3 } & 8 \\ 0 & - 6 & 0 \\ 0 & 0 & 1 \end{array} \right)$$
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 3 }\).
CAIE Further Paper 2 2021 June Q7
10 marks Challenging +1.2
7
  1. It is given that \(\mathrm { y } = \operatorname { sech } ^ { - 1 } \left( \mathrm { x } + \frac { 1 } { 2 } \right)\).
    Express cosh \(y\) in terms of \(x\) and hence show that \(\sinh y \frac { d y } { d x } = - \frac { 1 } { \left( x + \frac { 1 } { 2 } \right) ^ { 2 } }\).
  2. Find the first three terms in the Maclaurin's series for \(\operatorname { sech } ^ { - 1 } \left( x + \frac { 1 } { 2 } \right)\) in the form $$\ln a + b x + c x ^ { 2 }$$ where \(a\), \(b\) and \(c\) are constants to be determined.
CAIE Further Paper 2 2021 June Q8
13 marks Challenging +1.8
8 The curve \(C\) has parametric equations $$\mathbf { x } = 2 \cosh t , \quad \mathbf { y } = \frac { 3 } { 2 } \mathbf { t } - \frac { 1 } { 4 } \sinh 2 \mathbf { t } , \text { for } 0 \leqslant t \leqslant 1$$
  1. Find \(\frac { \mathrm { dx } } { \mathrm { dt } }\) and show that \(\frac { \mathrm { dy } } { \mathrm { dt } } = 1 - \sinh ^ { 2 } \mathrm { t }\).
    The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
    1. Show that \(\mathrm { A } = \pi \int _ { 0 } ^ { 1 } \left( \frac { 3 } { 2 } \mathrm { t } - \frac { 1 } { 4 } \sinh 2 \mathrm { t } \right) ( 1 + \cosh 2 \mathrm { t } ) \mathrm { dt }\).
    2. Hence find \(A\) in terms of \(\pi , \sinh 2\) and \(\cosh 2\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 2 2021 June Q3
10 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{fd247a71-4680-45d8-89d2-fef17ed3a5e9-04_851_805_251_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
  1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x < U _ { n }\), where $$\mathrm { U } _ { \mathrm { n } } = \left( \frac { \mathrm { n } + 1 } { 2 \mathrm { n } } \right) ^ { 2 }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x\).
  3. Find the least value of \(n\) such that \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } < 10 ^ { - 3 }\).
CAIE Further Paper 2 2021 June Q1
7 marks Standard +0.8
1
  1. Find \(a\) and \(b\) such that $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = \left( z ^ { 5 } - a \right) \left( z ^ { 3 } - b \right) .$$
  2. Hence find the roots of $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
CAIE Further Paper 2 2021 June Q2
7 marks Challenging +1.2
2 Find the Maclaurin's series for \(\ln \cosh x\) up to and including the term in \(x ^ { 4 }\).
CAIE Further Paper 2 2021 June Q3
10 marks Challenging +1.8
3 \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-04_540_1511_276_274} The diagram shows the curve \(\mathrm { y } = \frac { \mathrm { x } } { 2 \mathrm { x } ^ { 2 } - 1 }\) for \(x \geqslant 1\), together with a set of \(N - 1\) rectangles of unit
width. width.
  1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { r } { 2 r ^ { 2 } - 1 } < \frac { 1 } { 4 } \ln \left( 2 N ^ { 2 } - 1 \right) + 1$$
  2. Use a similar method to find, in terms of \(N\), a lower bound for \(\sum _ { r = 1 } ^ { N } \frac { r } { 2 r ^ { 2 } - 1 }\).
CAIE Further Paper 2 2021 June Q4
7 marks Challenging +1.2
4 By considering the binomial expansions of \(\left( z + \frac { 1 } { z } \right) ^ { 5 }\) and \(\left( z - \frac { 1 } { z } \right) ^ { 5 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that $$\tan ^ { 5 } \theta = \frac { \sin 5 \theta - \mathrm { a } \sin 3 \theta + \mathrm { b } \sin \theta } { \cos 5 \theta + \mathrm { a } \cos 3 \theta + \mathrm { b } \cos \theta }$$ where \(a\) and \(b\) are integers to be determined.
CAIE Further Paper 2 2021 June Q5
10 marks Standard +0.8
5 The variables \(x\) and \(y\) are related by the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } - 3 y = 4 e ^ { - x }$$
  1. Find the value of the constant \(k\) such that \(\mathrm { y } = \mathrm { kxe } ^ { - \mathrm { x } }\) is a particular integral of the differential equation.
  2. Find the solution of the differential equation for which \(\mathrm { y } = \frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 1 } { 2 }\) when \(x = 0\).
CAIE Further Paper 2 2021 June Q6
10 marks Challenging +1.2
6
  1. Starting from the definitions of sinh and cosh in terms of exponentials, prove that $$2 \sinh ^ { 2 } x = \cosh 2 x - 1$$ \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-10_67_1550_374_347} \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-10_65_1569_468_328} \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-10_67_1573_557_324} \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-10_70_1573_646_324} \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-10_72_1573_735_324} \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-10_72_1570_826_324} \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-10_74_1570_916_324} \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-10_69_1570_1007_324}
  2. Find the solution to the differential equation $$\frac { d y } { d x } + y \operatorname { coth } x = 4 \sinh x$$ for which \(y = 1\) when \(x = \ln 3\).
CAIE Further Paper 2 2021 June Q7
11 marks Challenging +1.8
7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 3 } { 2 } } \left( 4 + \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).
  1. Find the exact value of \(I _ { 1 }\), expressing your answer in logarithmic form.
  2. By considering \(\frac { d } { d x } \left( x \left( 4 + x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } n } \right)\), or otherwise, show that $$4 n l _ { n + 2 } = \frac { 3 } { 2 } \left( \frac { 2 } { 5 } \right) ^ { n } + ( n - 1 ) l _ { n } .$$
  3. Find the value of \(I _ { 5 }\).
CAIE Further Paper 2 2021 June Q8
13 marks Standard +0.8
8
  1. Find the value of \(a\) for which the system of equations $$\begin{array} { r } 13 x + 18 y - 28 z = 0 \\ - 4 x - a y + 8 z = 0 \\ 2 x + 6 y - 5 z = 0 \end{array}$$ does not have a unique solution.
    The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 13 & 18 & - 28 \\ - 4 & - 1 & 8 \\ 2 & 6 & - 5 \end{array} \right)$$
  2. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\).
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
  4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\) in terms of \(\mathbf { A }\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 2 2022 June Q1
5 marks
1 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { \frac { 3 } { 4 } \theta }\) for \(0 \leqslant \theta \leqslant \alpha\).
Given that the length of \(C\) is \(s\), find \(\alpha\) in terms of \(s\).