Questions — CAIE Further Paper 2 (186 questions)

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CAIE Further Paper 2 2023 November Q4
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
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
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
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
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
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
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
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
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 \(\ln 2\) .
CAIE Further Paper 2 2024 November Q1
1 Find the value of \(\int _ { 6 } ^ { 7 } \frac { 1 } { \sqrt { ( x - 5 ) ^ { 2 } - 1 } } \mathrm {~d} x\), giving your answer in the form \(\ln ( a + \sqrt { b } )\), where \(a\) and \(b\) are integers to be determined.
CAIE Further Paper 2 2024 November Q2
2 The curve \(C\) has equation $$4 y ^ { 2 } + 4 \ln ( x y ) = 1 .$$
  1. Show that, at the point \(\left( 2 , \frac { 1 } { 2 } \right)\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - \frac { 1 } { 6 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-04_2718_35_107_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-05_2725_35_99_20}
  2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(\left( 2 , \frac { 1 } { 2 } \right)\).
CAIE Further Paper 2 2024 November Q3
3 The curve \(C\) has parametric equations $$x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 t } - \frac { 1 } { 3 } t ^ { 3 } - \frac { 1 } { 2 } , \quad y = 2 \mathrm { e } ^ { t } ( t - 1 ) , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the exact length of \(C\) .
\includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-07_2726_35_97_20}
CAIE Further Paper 2 2024 November Q4
4
  1. Use de Moivre's theorem to show that $$\cot 6 \theta = \frac { \cot ^ { 6 } \theta - 15 \cot ^ { 4 } \theta + 15 \cot ^ { 2 } \theta - 1 } { 6 \cot ^ { 5 } \theta - 20 \cot ^ { 3 } \theta + 6 \cot \theta } .$$ \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-08_2718_35_107_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-09_2723_33_99_22}
  2. Hence obtain the roots of the equation $$x ^ { 6 } - 6 x ^ { 5 } - 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } - 6 x - 1 = 0$$ in the form \(\cot ( q \pi )\), where \(q\) is a rational number.
CAIE Further Paper 2 2024 November Q5
5 Find the particular solution of the differential equation $$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x ^ { 2 }$$ given that, when \(x = 0 , y = \frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
\includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-11_2726_35_97_20}
\includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-12_869_636_260_715}
\includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-12_2720_38_109_2009}
(b) Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { 1 - x } \mathrm {~d} x\).
(c) Show that \(\lim _ { n \rightarrow \infty } \left( U _ { n } - L _ { n } \right) = 0\).
(d) Use the Maclaurin's series for \(\mathrm { e } ^ { x }\) given in the list of formulae (MF19) to find the first three terms of the series expansion of \(z \left( 1 - \mathrm { e } ^ { - \frac { 1 } { z } } \right)\), in ascending powers of \(\frac { 1 } { z }\), and deduce the value of \(\lim _ { n \rightarrow \infty } \left( U _ { n } \right)\).
CAIE Further Paper 2 2024 November Q7
7
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \ln ( \tanh x ) ) = 2 \operatorname { cosech } 2 x\).
    \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-14_2717_35_106_2015}
    \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-15_2723_33_99_22}
  2. Find the solution of the differential equation $$\sinh 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = \sinh 2 x$$ for which \(y = 5\) when \(x = \ln 2\). Give your answer in an exact form.
CAIE Further Paper 2 2024 November Q8
8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 0 & 0
0 & 7 & 9
4 & 1 & 7 \end{array} \right)$$
  1. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 12 \lambda ^ { 2 } + 12 \lambda + 80 = 0\) and find the eigenvalues of A.
    \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-16_2718_38_106_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-17_2723_33_99_22}
  2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { A } + r \mathbf { I } ,$$ where \(p , q\) and \(r\) are integers to be determined.
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } - 3 \mathbf { I } ) ^ { 4 } = \mathbf { P D P } ^ { - 1 }\) .
    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 Q2
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]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-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
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]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-06_2716_38_109_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-07_2726_35_97_20}
CAIE Further Paper 2 2024 November Q4
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]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-08_2715_44_110_2006}
    \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-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
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]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-10_2715_40_110_2007}
\includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-11_2726_35_97_20}
CAIE Further Paper 2 2024 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-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]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-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 \(\ln 2\) .
CAIE Further Paper 2 2024 November Q8
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]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-18_2716_40_109_2009}
  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 2020 Specimen Q1
1 Fid b g a ral sb t iord to d fferen ial eq tion $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 4 x = 72 \quad t ^ { 2 }$$
CAIE Further Paper 2 2020 Specimen Q2
2 Fid \(\mathbf { b }\) ex ct le \(\mathbf { 6 } \int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 3 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\).
CAIE Further Paper 2 2020 Specimen Q3
3 Fid b sb utiw the id fferen ial eq tin $$x \frac { \mathrm { dy } } { \mathrm { dx } } + 3 y = \frac { \sin x } { x }$$ fo wh ch \(y = O _ { N }\) b \(\mathrm { n } x = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( x )\).