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CAIE Further Paper 2 2023 November Q1
4 marks Standard +0.8
1 Show that the system of equations $$\begin{aligned} 14 x - 4 y + 6 z & = 5 \\ x + y + k z & = 3 \\ - 21 x + 6 y - 9 z & = 14 \end{aligned}$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically.
CAIE Further Paper 2 2023 November Q2
5 marks Standard +0.8
2 Find the roots of the equation \(( z + 5 i ) ^ { 3 } = 4 + 4 \sqrt { 3 } i\), giving your answers in the form \(r \cos \theta + i ( r \sin \theta - 5 )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).
CAIE Further Paper 2 2023 November Q3
6 marks Challenging +1.2
3 Find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 } { 2 } e ^ { x } \right)\) 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 2023 November Q4
10 marks Standard +0.8
4 Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + 3 y = 27 x ^ { 2 }$$ given that, when \(x = 0 , y = 2\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = - 8\).
CAIE Further Paper 2 2023 November Q5
10 marks Standard +0.8
5 The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 2 } { 3 } \mathrm { t } ^ { \frac { 3 } { 2 } } - 2 \mathrm { t } ^ { \frac { 1 } { 2 } } , \quad \mathrm { y } = 2 \mathrm { t } + 5 , \quad \text { for } 0 < t \leqslant 3$$
  1. Find the exact length of \(C\).
  2. Find the set of values of \(t\) for which \(\frac { d ^ { 2 } y } { d x ^ { 2 } } > 0\).
CAIE Further Paper 2 2023 November Q6
14 marks Standard +0.3
6
  1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\sinh 2 x = 2 \sinh x \cosh x$$ \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_67_1550_374_347} \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_58_1569_475_328} \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_58_1569_566_328} \includegraphics[max width=\textwidth, alt={}]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_54_1566_657_328} ....................................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_54_1570_840_324} \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_53_1570_932_324} \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-10_53_1570_1023_324}
  2. Using the substitution \(\mathrm { u } = \sinh \mathrm { x }\), find \(\int \sinh ^ { 2 } 2 x \cosh x \mathrm { dx }\).
  3. Find the particular solution of the differential equation $$\frac { d y } { d x } + y \tanh x = \sinh ^ { 2 } 2 x$$ given that \(y = 4\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2023 November Q7
11 marks
7 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } - 6 & 2 & 13 \\ 0 & - 2 & 5 \\ 0 & 0 & 8 \end{array} \right) .$$
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
CAIE Further Paper 2 2023 November Q8
15 marks Challenging +1.8
8
  1. State the sum of the series \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 }\), for \(z \neq 1\).
  2. By letting \(z = \cos \theta + i \sin \theta\), where \(\cos \theta \neq 1\), show that $$1 + \cos \theta + \cos 2 \theta + \ldots + \cos ( n - 1 ) \theta = \frac { 1 } { 2 } \left( 1 - \cos n \theta + \frac { \sin n \theta \sin \theta } { 1 - \cos \theta } \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-15_833_785_214_680} The diagram shows the curve with equation \(\mathrm { y } = \cos \mathrm { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
  3. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } \cos x d x < \frac { 1 } { 2 n } \left( 1 - \cos 1 + \frac { \sin 1 \sin \frac { 1 } { n } } { 1 - \cos \frac { 1 } { n } } \right)$$
  4. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \cos x d x\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2023 November Q1
5 marks Standard +0.3
1 Find the Maclaurin's series for \(\ln ( x + 2 ) + \ln \left( x ^ { 2 } + 5 \right)\) up to and including the term in \(x ^ { 2 }\).
CAIE Further Paper 2 2023 November Q2
7 marks Standard +0.8
2 It is given that $$x = 1 + \frac { 1 } { t } \quad \text { and } \quad y = t e ^ { t }$$
  1. Show that \(\frac { d y } { d x } = - e ^ { t } \left( t ^ { 3 } + t ^ { 2 } \right)\).
  2. Find \(\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } }\) in terms of \(t\).
CAIE Further Paper 2 2023 November Q3
8 marks Challenging +1.8
3
  1. Use de Moivre's theorem to show that $$\cos 5 \theta = 16 \cos ^ { 5 } \theta - 20 \cos ^ { 3 } \theta + 5 \cos \theta$$
  2. Hence obtain the roots of the equation $$32 x ^ { 5 } - 40 x ^ { 3 } + 10 x - \sqrt { 2 } = 0$$ in the form \(\cos ( q \pi )\), where \(q\) is a rational number.
CAIE Further Paper 2 2023 November Q4
9 marks Standard +0.3
4 Find the solution of the differential equation $$\frac { d y } { d x } + 3 y = \sin x$$ for which \(y = 1\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2023 November Q5
10 marks Hard +2.3
5 \includegraphics[max width=\textwidth, alt={}, center]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-08_773_1161_278_443} The diagram shows part of the curve \(\mathrm { y } = \mathrm { xsech } ^ { 2 } \mathrm { x }\) and its maximum point \(M\).
  1. Show that, at \(M\), $$2 x \tanh x - 1 = 0$$ and verify that this equation has a root between 0.7 and 0.8 .
  2. By considering a suitable set of rectangles, use the diagram to show that \(\sum _ { r = 2 } ^ { n } r \operatorname { sech } ^ { 2 } r < n \tanh n + \operatorname { lnsechn } - \tanh 1 - \operatorname { lnsech } 1\).
CAIE Further Paper 2 2023 November Q6
10 marks Challenging +1.2
6 The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & - 1 \end{array} \right) .$$
  1. State the eigenvalues of \(\mathbf { P }\).
  2. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
    The \(3 \times 3\) matrix \(\mathbf { A }\) has distinct non-zero eigenvalues \(a , \frac { 1 } { 2 } , 2\) with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 2 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } 1 \\ 1 \\ - 1 \end{array} \right) ,$$ respectively.
  3. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
CAIE Further Paper 2 2023 November Q7
12 marks Challenging +1.8
7
  1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$2 \sinh ^ { 2 } A = \cosh 2 A - 1$$ \includegraphics[max width=\textwidth, alt={}, center]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-12_79_1556_358_347} \includegraphics[max width=\textwidth, alt={}]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-12_69_1575_466_328} ....................................................................................................................................... ........................................................................................................................................
  2. A curve has equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).
    Use the substitution \(\mathrm { X } = \frac { 1 } { 2 } \operatorname { sinhu }\) to show that \(S = \frac { 1 } { 32 } \pi \left( \frac { 820 } { 81 } - \ln 3 \right)\).
CAIE Further Paper 2 2023 November Q8
14 marks Challenging +1.8
8 It is given that \(\mathbf { v } = y ^ { 4 }\) and $$y ^ { 3 } \frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 y ^ { 2 } \left( \frac { d y } { d x } \right) ^ { 2 } + y ^ { 3 } \frac { d y } { d x } + y ^ { 4 } = e ^ { - 2 x }$$
  1. Show that $$\frac { d ^ { 2 } v } { d x ^ { 2 } } + \frac { d v } { d x } + 4 v = 4 e ^ { - 2 x }$$
  2. Find \(y\) in terms of \(x\), given that, when \(x = 0 , y = 1\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 3 } { 8 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2023 November Q4
10 marks Standard +0.3
4 Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + 3 y = 27 x ^ { 2 }$$ given that, when \(x = 0 , y = 2\) and \(\frac { d y } { d x } = - 8\).
CAIE Further Paper 2 2023 November Q6
14 marks Standard +0.3
6
  1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\sinh 2 x = 2 \sinh x \cosh x$$ \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_67_1550_374_347} \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_58_1569_475_328} \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_58_1569_566_328} \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_59_1566_657_328} \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_58_1570_749_324} \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_54_1570_840_324} \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_63_1570_922_324} \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-10_67_1570_1009_324}
  2. Using the substitution \(\mathrm { u } = \sinh \mathrm { x }\), find \(\int \sinh ^ { 2 } 2 x \cosh x \mathrm {~d} x\).
  3. Find the particular solution of the differential equation $$\frac { d y } { d x } + y \tanh x = \sinh ^ { 2 } 2 x$$ given that \(y = 4\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2023 November Q8
15 marks Challenging +1.8
8
  1. State the sum of the series \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 }\), for \(z \neq 1\).
  2. By letting \(z = \cos \theta + i \sin \theta\), where \(\cos \theta \neq 1\), show that $$1 + \cos \theta + \cos 2 \theta + \ldots + \cos ( n - 1 ) \theta = \frac { 1 } { 2 } \left( 1 - \cos n \theta + \frac { \sin n \theta \sin \theta } { 1 - \cos \theta } \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-15_833_785_214_680} The diagram shows the curve with equation \(\mathrm { y } = \cos \mathrm { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
  3. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } \cos x d x < \frac { 1 } { 2 n } \left( 1 - \cos 1 + \frac { \sin 1 \sin \frac { 1 } { n } } { 1 - \cos \frac { 1 } { n } } \right)$$
  4. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \cos x d x\).
    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 Q1
4 marks Standard +0.8
1 Find the set of values of \(k\) for which the system of equations $$\begin{array} { r } x + 5 y + 6 z = 1 \\ k x + 2 y + 2 z = 2 \\ - 3 x + 4 y + 8 z = 3 \end{array}$$ has a unique solution and interpret this situation geometrically.
CAIE Further Paper 2 2024 November Q2
6 marks Standard +0.8
2 It is given that $$x = 1 + \frac { 1 } { t } \quad \text { and } \quad y = \cos ^ { - 1 } t \quad \text { for } 0 < t < 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ^ { 2 } } { \sqrt { 1 - t ^ { 2 } } }\). \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-05_2723_33_99_22}
  2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - t ^ { a } \left( 1 - t ^ { 2 } \right) ^ { b } \left( 2 - t ^ { 2 } \right)\) ,where \(a\) and \(b\) are constants to be determined.
CAIE Further Paper 2 2024 November Q3
12 marks Challenging +1.8
3 A curve has equation \(y = \mathrm { e } ^ { x }\) for \(\ln \frac { 4 } { 3 } \leqslant x \leqslant \ln \frac { 12 } { 5 }\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
  1. Use the substitution \(u = \mathrm { e } ^ { x }\) to show that $$A = 2 \pi \int _ { \frac { 4 } { 3 } } ^ { \frac { 12 } { 5 } } \sqrt { 1 + u ^ { 2 } } \mathrm {~d} u$$
  2. Use the substitution \(u = \sinh v\) to show that $$A = \pi \left( \frac { 904 } { 225 } + \ln \frac { 5 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-06_2716_38_109_2012} \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-07_2726_35_97_20}
CAIE Further Paper 2 2024 November Q4
9 marks Standard +0.3
4 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 11 & 1 & 8 \\ 0 & - 2 & 0 \\ - 16 & 1 & 13 \end{array} \right)$$
  1. Show that \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and state the corresponding eigenvalue.
  2. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 19 \lambda - 30 = 0\) and hence find the other eigenvalues of \(\mathbf { A }\). \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-08_2717_35_106_2015} \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-09_2726_33_97_22}
  3. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
CAIE Further Paper 2 2024 November Q5
10 marks Standard +0.3
5 Find the particular solution of the differential equation $$6 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = t ^ { 2 } + t + 1$$ given that, when \(t = 0 , x = 12\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 6\).
[0pt] [10] \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-10_2715_40_110_2007} \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-11_2726_35_97_20}
CAIE Further Paper 2 2024 November Q6
10 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-12_533_1532_278_264} The diagram shows the curve with equation \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) 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 \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x > L _ { N }\), where $$L _ { N } = \frac { 1 } { 2 N \left( 2 ^ { \frac { 1 } { N } } - 1 \right) }$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-12_2717_38_109_2009}
  2. Use a similar method to find, in terms of \(N\), an upper bound \(U _ { N }\) for \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x\).
  3. Find the least value of \(N\) such that \(U _ { N } - L _ { N } \leqslant 10 ^ { - 3 }\).
  4. Given that \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x = \frac { 1 } { 2 \ln 2 }\) ,use the value of \(N\) found in part(c)to find upper and lower bounds for \(\ln 2\) .