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CAIE Further Paper 2 2024 November Q1
4 marks Standard +0.3
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
7 marks Standard +0.8
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
7 marks Challenging +1.2
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
10 marks Challenging +1.2
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
10 marks Standard +0.3
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
10 marks Challenging +1.2
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
14 marks Challenging +1.8
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
6 marks Challenging +1.2
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
12 marks Challenging +1.3
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
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]{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 Standard +0.8
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
10 marks Challenging +1.2
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
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]{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
6 marks Standard +0.8
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
6 marks Standard +0.8
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
8 marks Standard +0.8
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 \(y = \mathrm { f } ( x )\).
CAIE Further Paper 2 2020 Specimen Q4
8 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{6ff1b572-4cd8-433d-ba16-ffc8cda44476-06_545_958_264_552} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } }\) fo \(x > 0\) tg th rwith a set \(6 ( n - 1 )\) rectab es 6 in t witd h
  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 low er \(\mathbf { H }\)
    • \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }\).
CAIE Further Paper 2 2020 Specimen Q5
10 marks Challenging +1.2
5 Th cn e \(C\) has parametric equations $$x = \mathrm { e } ^ { t } - 4 t + 3 \quad y = 8 \mathrm { e } ^ { \frac { 1 } { 2 } t } , \quad \text { f } \mathbf { D } \quad 0 \leqslant t \leqslant 2$$
  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 ab the \(x\)-ax s.
    [0pt] [\$
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. Hen esh th the eq tion \(x ^ { 2 } - 4 x + 5 = 0\) s ro \(\operatorname { stan } ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) ad \(\operatorname { an } ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
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 \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\). [ \(\beta\)
  2. Given that \(y = \operatorname { tah } ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right) , \mathrm { s } \quad\) th \(\mathrm { t } ( 2 x + 1 ) \frac { \mathrm { dy } } { \mathrm { dx } } + 1 = 0\)
  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 $$a \ln 3 + b x + c x ^ { 2 }$$ wh re \(a , b\) ad \(c\) are constants to be determined.
CAIE Further Paper 2 2020 Specimen Q8
15 marks Standard +0.3
8
    1. Fid bet basb le s a for which the system of equations $$\begin{array} { r l } x - 2 y - 2 z + z & 0 \\ 2 x + ( a - 9 y - 0 z + 1 E & 0 \\ 3 x - 6 y + 2 a z + 9 & 0 \end{array}$$ h san q sbtu in
    2. Given that \(a = - 3\), show that the system of equations in part (i) \(\mathbf { b } \mathbf { s } \mathbf { n }\) sb t in In erp et th s situation geometrically.
  2. The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 1 & 2 \\ 0 & 2 & 2 \\ - 1 & 1 & 3 \end{array} \right)$$
    1. Find b eig le so A.
    2. Use th ch racteristic eq tiw \(\mathbf { A }\) tof id \(\mathbf { A } ^ { - 1 }\). 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 FP2 2014 June Q1
Easy -1.8
1 Tmall mt ad a ual ad ad a ma ad ctly Ty a mg a tagt \(l\) tam dc \(\quad t\) a mt tal tabl \(T d\) ad td - clld dctly \(t\) T cct ttut
bt t -
  1. tat t d at t cll $$\begin{array} { c c } \pi & \theta \\ \hline \pi & \theta \end{array}$$
  2. G tat \(t\) magtud \(t\) mul xcd by dug \(t\) cll - \(d t\) alu
CAIE FP2 2014 June Q2
2 Aatcl ma kgmaacaccltct adadu mt At tm tatcl at tt At tm cd agl tat tadal cmt \(t\) acclat at tm cd a magtud \(\quad \pi \quad \theta m\) Fd
  1. talu ta cmt t acclat t ual t
  2. \(t\) magtud \(t\) ultat \(c\) actg
CAIE FP2 2014 June Q3
Standard +0.8
3 A atcl attacd \(t d\) a lgt xtbl tg lgt \(\quad T t d\) t tg attacd t a xd t t atcl agg ulbum t \(t\) tg tcal \(t\) ga tal mul magtud t tg mak a agl t t dad tcal at \(\quad\) t \(t\)
  1. tat - \(\pi \quad c \quad \theta\)
  2. It g tat \(\quad { } _ { \pi } ^ { \alpha } \quad \theta \quad\) a t ctat Jut \(\quad\) yg yu a dcb tmt ac tllg ca
    (a)
    (b)
CAIE FP2 2014 June Q4
Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{e2ff0097-b2a1-4901-b880-4ef4505c9cbe-3_529_606_260_767}
A um dad lgtT d t ulbum a mt g\(t\)
\(t d \quad\) tg \(a u g t\)al la \(T\) dtacad tagl btad
t tal- A atcl ma- attacd t t d atdagam
Fd t mal act atad dduc tat-
\(T\) cct ct bt td ad t latat