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CAIE Further Paper 2 2023 June Q7
11 marks Challenging +1.8
7
  1. Use the substitution \(\mathrm { u } = \mathrm { x } ^ { 2 } - 1\) to find \(\int \frac { x } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x\). \includegraphics[max width=\textwidth, alt={}, center]{482b2236-1f1b-4c53-a1bc-0277cf63dc62-12_778_1548_1007_296} The diagram shows the curve with equation \(\mathrm { y } = \cosh ^ { - 1 } \mathrm { x }\) together with a set of \(( N - 1 )\) rectangles of unit width.
  2. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 2 } ^ { N } \ln \left( r + \sqrt { r ^ { 2 } - 1 } \right) > N \ln \left( N + \sqrt { N ^ { 2 } - 1 } \right) - \sqrt { N ^ { 2 } - 1 }$$
  3. Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { \mathrm { r } = 2 } ^ { \mathrm { N } } \ln \left( \mathrm { r } + \sqrt { \mathrm { r } ^ { 2 } - 1 } \right)\).
CAIE Further Paper 2 2023 June Q8
14 marks Standard +0.8
8
  1. Starting from the definitions of sech and tanh in terms of exponentials, prove that $$1 - \operatorname { sech } ^ { 2 } t = \tanh ^ { 2 } t$$ \includegraphics[max width=\textwidth, alt={}]{482b2236-1f1b-4c53-a1bc-0277cf63dc62-14_77_1547_360_347} ......................................................................................................................................... ......................................................................................................................................... . ........................................................................................................................................ ........................................................................................................................................ ....................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{482b2236-1f1b-4c53-a1bc-0277cf63dc62-14_72_1573_911_324} \includegraphics[max width=\textwidth, alt={}, center]{482b2236-1f1b-4c53-a1bc-0277cf63dc62-14_67_1573_1005_324} The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 1 } { 2 } \tanh ^ { 2 } \mathrm { t } + \text { Insecht } , \quad \mathrm { y } = 1 + \tanh ^ { 4 } \mathrm { t } , \quad \text { for } t > 0$$
  2. Show that \(\frac { d y } { d x } = - 4 \operatorname { sech } ^ { 2 } t\).
  3. Find the coordinates of the point on \(C\) with \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = - \frac { 9 } { 2 }\), giving your answer in the form \(( a + \ln b , c )\) where \(a , b\) and \(c\) are rational numbers.
    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 June Q1
6 marks Standard +0.8
1
  1. Find the Maclaurin series for \(\sin ^ { - 1 } x\) up to and including the term in \(x ^ { 3 }\).
  2. Deduce an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 5 } } \frac { 1 } { \sqrt { 1 - u ^ { 2 } } } \mathrm {~d} u\), giving your answer as a fraction.
CAIE Further Paper 2 2023 June Q2
7 marks Standard +0.8
2 The variables \(x\) and \(y\) are related by the differential equation $$6 \frac { d ^ { 2 } x } { d t ^ { 2 } } + 5 \frac { d x } { d t } + x = t ^ { 2 } + 10 t + 13$$
  1. Find the general solution for \(x\) in terms of \(t\).
  2. State an approximate solution for large positive values of \(t\).
CAIE Further Paper 2 2023 June Q3
7 marks Challenging +1.2
3 By considering the binomial expansions of \(\left( z + \frac { 1 } { z } \right) ^ { 4 }\) and \(\left( z - \frac { 1 } { z } \right) ^ { 4 }\), where \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to show that $$\cot ^ { 4 } \theta = \frac { \cos 4 \theta + a \cos 2 \theta + b } { \cos 4 \theta - a \cos 2 \theta + b }$$ where \(a\) and \(b\) are integers to be determined.
CAIE Further Paper 2 2023 June Q4
8 marks Standard +0.8
4 The curve \(C\) has equation $$4 y ^ { 3 } + ( x + y ) ^ { 6 } = 109 .$$
  1. Show that, at the point \(( - 4,3 )\) on \(C , \frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 1 } { 17 }\).
  2. Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( - 4,3 )\).
CAIE Further Paper 2 2023 June Q5
11 marks Challenging +1.2
5
  1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$2 \cosh ^ { 2 } x = \cosh 2 x + 1$$ \includegraphics[max width=\textwidth, alt={}, center]{d421652f-576d-4843-abbf-54404e225fec-08_67_1550_374_347}
  2. Find the solution of the differential equation $$\frac { d y } { d x } + 2 y \tanh x = 1$$ for which \(y = 1\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2023 June Q6
11 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{d421652f-576d-4843-abbf-54404e225fec-10_1015_988_260_577} The diagram shows the curve with equation \(\mathrm { y } = ( 1 - \mathrm { x } ) ^ { 2 }\) 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 \(\int _ { 0 } ^ { 1 } ( 1 - x ) ^ { 2 } d x < U _ { n }\), where $$U _ { n } = \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } ( 1 - x ) ^ { 2 } d x\).
  3. Show that \(\lim _ { n \rightarrow \infty } \left( U _ { n } - L _ { n } \right) = 0\).
CAIE Further Paper 2 2023 June Q7
11 marks Challenging +1.8
7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 4 } { 3 } } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).
  1. Find the exact value of \(I _ { - 1 }\) giving your answer in the form \(\ln a\), where \(a\) is an integer to be determined.
  2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm { n } \right)\), or otherwise, show that $$( \mathrm { n } + 1 ) \mathrm { I } _ { \mathrm { n } } = \mathrm { nl } _ { \mathrm { n } - 2 } + \frac { 4 } { 3 } \left( \frac { 5 } { 3 } \right) ^ { \mathrm { n } }$$
  3. A curve has equation \(y = x ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The arc length of the curve is denoted by \(s\). Use the substitution \(\mathrm { u } = 2 \mathrm { x }\) to show that \(\mathrm { s } = \frac { 1 } { 2 } \mathrm { l } _ { 1 }\) and find the exact value of \(s\).
CAIE Further Paper 2 2023 June Q8
14 marks Challenging +1.2
8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & - 6 a & 2 a + 2 \\ 0 & 1 - a & 0 \\ 0 & 2 - a & - 1 \end{array} \right)$$ where \(a\) is a constant with \(a \neq 0\) and \(a \neq 1\).
  1. Show that the equation \(\mathbf { A } \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } 1 \\ 2 \\ 3 \end{array} \right)\) has a unique solution and interpret this situation geometrically.
  2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , 1 - a\) and - 1 .
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 4 } = \mathbf { P D P } ^ { - 1 }\).
  4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 4 }\) in terms of \(\mathbf { A }\) and \(a\).
    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 June Q1
5 marks Standard +0.3
1 Find the roots of the equation \(z ^ { 3 } = - 108 \sqrt { 3 } + 108\) i, giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).
CAIE Further Paper 2 2024 June Q2
4 marks Moderate -0.3
2 Find the Maclaurin's series for \(\mathrm { e } ^ { 1 + x ^ { 2 } } + \mathrm { e } ^ { 1 - x }\) up to and including the term in \(x ^ { 2 }\).
CAIE Further Paper 2 2024 June Q3
7 marks Challenging +1.2
3 It is given that $$\mathrm { x } = \sin ^ { - 1 } \mathrm { t } \quad \text { and } \quad \mathrm { y } = \mathrm { tcos } ^ { - 1 } \mathrm { t } , \quad \text { for } 0 \leqslant t < 1 .$$
  1. Show that \(\frac { d y } { d x } = - t + \sqrt { 1 - t ^ { 2 } } \cos ^ { - 1 } t\).
  2. Find \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) in terms of \(t\).
CAIE Further Paper 2 2024 June Q4
8 marks Challenging +1.8
4 It is given that, for \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \ln 3 } \operatorname { sech } ^ { \mathrm { n } } \mathrm { xdx }\).
  1. Show that, for \(n \geqslant 2\), $$( n - 1 ) \mathrm { I } _ { n } = \left( \frac { 3 } { 5 } \right) ^ { n - 2 } \left( \frac { 4 } { 5 } \right) + ( n - 2 ) \mathrm { I } _ { n - 2 }$$ [You may use the result that \(\frac { \mathrm { d } } { \mathrm { dx } } ( \operatorname { sech } x ) = - \tanh x \operatorname { sech } x\).]
  2. Find the value of \(I _ { 4 }\).
CAIE Further Paper 2 2024 June Q5
11 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{bca7281b-a6a9-4b4c-94e5-3da2a561ad86-08_663_1152_260_452} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } - \mathrm { x } ^ { 2 }\) 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 \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x < U _ { n }\), where $$U _ { n } = \left( 1 + \frac { 1 } { n } \right) \left( \frac { 2 } { 3 } - \frac { 1 } { 6 n } \right) .$$
  2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x\).
  3. Show that \(\lim _ { n \rightarrow \infty } \left( \mathrm { U } _ { n } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).
CAIE Further Paper 2 2024 June Q6
12 marks Challenging +1.2
6
  1. Show that \(( \cosh x + \sinh x ) ^ { \frac { 1 } { 2 } } = \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
  2. Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } + 3 y = 5 ( \cosh x + \sinh x ) ^ { \frac { 1 } { 2 } }$$ given that, when \(x = 0 , y = 1\) and \(\frac { d y } { d x } = \frac { 4 } { 3 }\).
CAIE Further Paper 2 2024 June Q7
12 marks Challenging +1.2
7
  1. Use the substitution \(\mathrm { u } = 1 + \mathrm { x } ^ { 2 }\) to find $$\int \frac { x } { \sqrt { 1 + x ^ { 2 } } } d x$$
  2. Find the solution of the differential equation $$x \frac { d y } { d x } - y = x ^ { 2 } \sinh ^ { - 1 } x$$ given that \(y = 1\) when \(x = 1\). Give your answer in the form \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\).
CAIE Further Paper 2 2024 June Q8
16 marks Challenging +1.8
8
  1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { c l } 6 x + a y & = 3 \\ 2 x - y & = 1 \\ x + 5 y + 4 z & = 2 \end{array}$$ has a unique solution.
  2. Show that the system of equations in part (a) is consistent for all values of \(a\).
    The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 0 & 0 \\ 2 & - 1 & 0 \\ 1 & 5 & 4 \end{array} \right)$$
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
  4. Use the characteristic equation of \(\mathbf { A }\) to show that $$( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ 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 2024 June Q5
11 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{114be67d-a57f-4c36-8f1c-974a2719c1f1-08_663_1152_260_452} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } - \mathrm { x } ^ { 2 }\) 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 \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x < U _ { n }\), where $$U _ { n } = \left( 1 + \frac { 1 } { n } \right) \left( \frac { 2 } { 3 } - \frac { 1 } { 6 n } \right) .$$
  2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x\).
  3. Show that \(\lim _ { n \rightarrow \infty } \left( \mathrm { U } _ { n } - \mathrm { L } _ { \mathrm { n } } \right) = 0\).
CAIE Further Paper 2 2024 June Q1
5 marks Standard +0.8
1 Find the exact value of \(\int _ { 2 } ^ { \frac { 7 } { 2 } } \frac { 1 } { \sqrt { 4 x - x ^ { 2 } - 1 } } \mathrm {~d} x\).
CAIE Further Paper 2 2024 June Q2
9 marks Challenging +1.2
2 The curve \(C\) has parametric equations $$x = \cosh t , \quad y = \sinh t , \quad \text { for } 0 < t \leqslant \frac { 3 } { 5 }$$ The length of \(C\) is denoted by \(s\).
  1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 5 } } \sqrt { \cosh 2 t } \mathrm {~d} t\). \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-04_2714_37_143_2008}
  2. By finding the Maclaurin's series for \(\sqrt { \cosh 2 t }\) up to and including the term in \(t ^ { 2 }\) ,deduce an approximation to \(s\) .
CAIE Further Paper 2 2024 June Q3
8 marks Standard +0.8
3 The curve \(C\) has equation $$x ^ { 3 } + 2 x y + 8 y ^ { 3 } = - 12$$
  1. Show that, at the point \(( - 2 , - 1 )\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - \frac { 1 } { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-06_2714_37_143_2008}
  2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( - 2 , - 1 )\).
CAIE Further Paper 2 2024 June Q4
10 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_408_1433_296_315} The diagram shows the curve with equation \(y = x ^ { - 2 }\) for \(2 \leqslant x \leqslant N\) together with a set of ( \(N - 2\) ) rectangles of unit width.
  1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } } > \frac { 3 } { 2 } - \frac { 1 } { N } + \frac { 1 } { N ^ { 2 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_2718_35_141_2012}
  2. Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } }\).
  3. Deduce lower and upper bounds for \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\).
CAIE Further Paper 2 2024 June Q5
10 marks Challenging +1.2
5
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 10 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 338 \sin t$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-10_2715_35_143_2012}
  2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx R \sin ( t - \phi ) ,$$ where the constants \(R\) and \(\phi\) are to be determined.
CAIE Further Paper 2 2024 June Q6
7 marks Challenging +1.2
6
  1. Show that \(\sum _ { r = 1 } ^ { n } z ^ { 4 r } = \frac { z ^ { 4 n + 2 } - z ^ { 2 } } { z ^ { 2 } - z ^ { - 2 } }\), for \(z ^ { 2 } \neq z ^ { - 2 }\).
  2. By letting \(z = \cos \theta + \mathrm { i } \sin \theta\), show that, if \(\sin 2 \theta \neq 0\), $$\sum _ { r = 1 } ^ { n } \sin ( 4 r \theta ) = \frac { \cos 2 \theta - \cos ( 4 n + 2 ) \theta } { 2 \sin 2 \theta }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-12_2718_35_143_2012}