Questions — CAIE Further Paper 2 (186 questions)

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CAIE Further Paper 2 2021 June Q1
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
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
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
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
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
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
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
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
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 \(s\).
CAIE Further Paper 2 2022 June Q2
2
  1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh 2 x = 2 \sinh ^ { 2 } x + 1$$
  2. Find the set of values of \(k\) for which \(\cosh 2 \mathrm { x } = \mathrm { ksinh } \mathrm { x }\) has two distinct real roots.
CAIE Further Paper 2 2022 June Q3
3 The variables \(t\) and \(x\) are related by the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + \frac { d x } { d t } + x = t ^ { 2 } + 1$$
  1. Find the general solution for \(x\) in terms of \(t\).
  2. Deduce an approximate value of \(\frac { \mathrm { d } ^ { 2 } \mathrm { x } } { \mathrm { dt } ^ { 2 } }\) for large positive values of \(t\).
CAIE Further Paper 2 2022 June Q4
4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\).
\includegraphics[max width=\textwidth, alt={}, center]{114ece0d-558d-4c02-8a77-034b3681cff9-06_824_1161_376_450}
  1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
  2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
  3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).
CAIE Further Paper 2 2022 June Q5
5 The variables \(x\) and \(y\) are such that \(y = 0\) when \(x = 0\) and $$( x + 1 ) y + ( x + y + 1 ) ^ { 3 } = 1$$
  1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 3 } { 4 }\) when \(x = 0\).
  2. Find the Maclaurin's series for \(y\) up to and including the term in \(x ^ { 2 }\).
CAIE Further Paper 2 2022 June Q6
10 marks
6 Use the substitution \(y = v x\) to find the solution of the differential equation $$x \frac { d y } { d x } = y + \sqrt { 9 x ^ { 2 } + y ^ { 2 } }$$ for which \(y = 0\) when \(x = 1\). Give your answer in the form \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), where \(\mathrm { f } ( x )\) is a polynomial in \(x\). [10]
\includegraphics[max width=\textwidth, alt={}, center]{114ece0d-558d-4c02-8a77-034b3681cff9-10_51_1648_527_246}
CAIE Further Paper 2 2022 June Q7
7
  1. Use de Moivre's theorem to show that $$\operatorname { cosec } 7 \theta = \frac { \operatorname { cosec } ^ { 7 } \theta } { 7 \operatorname { cosec } ^ { 6 } \theta - 56 \operatorname { cosec } ^ { 4 } \theta + 112 \operatorname { cosec } ^ { 2 } \theta - 64 }$$
  2. Hence obtain the roots of the equation $$x ^ { 7 } - 14 x ^ { 6 } + 112 x ^ { 4 } - 224 x ^ { 2 } + 128 = 0$$ in the form \(\operatorname { cosec } q \pi\), where \(q\) is rational.
CAIE Further Paper 2 2022 June Q8
8
  1. Find the value of \(a\) for which the system of equations $$\begin{gathered} 3 x + a y = 0
    5 x - y = 0
    x + 3 y + 2 z = 0 \end{gathered}$$ does not have a unique solution.
    The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 0 & 0
    5 & - 1 & 0
    1 & 3 & 2 \end{array} \right)$$
  2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
  3. Use the characteristic equation of \(\mathbf { A }\) to show that $$( \mathbf { A } + 6 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ where \(b\) is an integer to be determined.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2022 June Q4
4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\).
\includegraphics[max width=\textwidth, alt={}, center]{69c540e1-1dad-45a1-9809-7629d16260e0-06_824_1161_376_450}
  1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
  2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
  3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).
CAIE Further Paper 2 2022 June Q6
10 marks
6 Use the substitution \(y = v x\) to find the solution of the differential equation $$x \frac { d y } { d x } = y + \sqrt { 9 x ^ { 2 } + y ^ { 2 } }$$ for which \(y = 0\) when \(x = 1\). Give your answer in the form \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), where \(\mathrm { f } ( x )\) is a polynomial in \(x\). [10]
\includegraphics[max width=\textwidth, alt={}, center]{69c540e1-1dad-45a1-9809-7629d16260e0-10_51_1648_527_246}
CAIE Further Paper 2 2022 June Q1
1 Find the roots of the equation \(z ^ { 3 } = 7 \sqrt { 3 } - 7 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi \leqslant \theta < \pi\).
CAIE Further Paper 2 2022 June Q2
2
  1. Find the coefficient of \(x ^ { 2 }\) in the Maclaurin's series for \(- \ln \cos x\).
  2. Find the length of the arc of the curve with equation \(\mathrm { y } = - \operatorname { Incos } \mathrm { x }\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 4 } \pi\).
CAIE Further Paper 2 2022 June Q3
3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 6 & - 9 & 5
5 & - 8 & 5
1 & - 1 & 2 \end{array} \right)$$
  1. Find the eigenvalues of \(\mathbf { A }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }\), where \(p\) and \(q\) are constants to be determined.
CAIE Further Paper 2 2022 June Q4
4 It is given that $$x = - t + \tan ^ { - 1 } t \quad \text { and } \quad y = t + \sinh ^ { - 1 } t$$
  1. Show that \(\frac { d y } { d x } = - \frac { t ^ { 2 } + 1 + \sqrt { t ^ { 2 } + 1 } } { t ^ { 2 } }\).
  2. Find the value of \(\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } }\) when \(t = \frac { 3 } { 4 }\).
CAIE Further Paper 2 2022 June Q5
5 Find the solution of the differential equation $$x ( x + 7 ) \frac { d y } { d x } + 7 y = x$$ for which \(y = 7\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2022 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{23b06b1c-997f-425d-ae3d-bd4cc1295605-10_771_1146_260_497} The diagram shows the curve with equation \(\mathrm { y } = \ln ( 1 + \mathrm { x } )\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles each of width \(\frac { 1 } { n }\).
  1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \ln ( 1 + x ) d x < U _ { n }\), where $$U _ { n } = \frac { 1 } { n } \ln \frac { ( 2 n ) ! } { n ! } - \ln n$$
  2. Use a similar method to find, in terms of \(n\), a lower bound \(\mathrm { L } _ { \mathrm { n } }\) for \(\int _ { 0 } ^ { 1 } \ln ( 1 + x ) \mathrm { d } x\).
  3. By simplifying \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } }\), show that \(\lim _ { \mathrm { n } \rightarrow \infty } \left( \mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).
CAIE Further Paper 2 2022 June Q7
7 The variables \(x\) and \(y\) are related by the differential equation $$4 \frac { d ^ { 2 } y } { d x ^ { 2 } } - y = 3$$ It is given that, when \(x = 0 , y = - 3\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = 2\).
  1. Find \(y\) in terms of \(x\).
  2. Deduce the exact value of \(x\) for which \(y = 0\). Give your answer in logarithmic form.