Questions - CAIE Further Paper 2

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CAIE Further Paper 2 2024 June Q7

Use the substitution $\mathrm { u } = 1 + \mathrm { x } ^ { 2 }$ to find $$\int \frac { x } { \sqrt { 1 + x ^ { 2 } } } d x$$
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

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.
Show that the system of equations in part (a) is consistent for all values of $a$.
The matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 0 & 0
2 & - 1 & 0
1 & 5 & 4 \end{array} \right)$$
Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { P D P } ^ { - 1 }$.
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

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 $\frac { 1 } { n }$.

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 $$U _ { n } = \left( 1 + \frac { 1 } { n } \right) \left( \frac { 2 } { 3 } - \frac { 1 } { 6 n } \right) .$$
Use a similar method to find, in terms of $n$, a lower bound $L _ { n }$ for $\int _ { 0 } ^ { 1 } \left( 2 x - x ^ { 2 } \right) d x$.
Show that $\lim _ { n \rightarrow \infty } \left( \mathrm { U } _ { n } - \mathrm { L } _ { \mathrm { n } } \right) = 0$.

CAIE Further Paper 2 2024 June Q1

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

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 $s$.

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}
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

3 The curve $C$ has equation $$x ^ { 3 } + 2 x y + 8 y ^ { 3 } = - 12$$

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}
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

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.

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}
Use a similar method to find, in terms of $N$, an upper bound for $\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } }$.
Deduce lower and upper bounds for $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }$.

CAIE Further Paper 2 2024 June Q5

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}
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

Show that $\sum _ { r = 1 } ^ { n } z ^ { 4 r } = \frac { z ^ { 4 n + 2 } - z ^ { 2 } } { z ^ { 2 } - z ^ { - 2 } }$, for $z ^ { 2 } \neq z ^ { - 2 }$.
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}

CAIE Further Paper 2 2024 June Q7

Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \frac { x } { 2 } \sqrt { x ^ { 2 } - 9 } - \frac { 9 } { 2 } \cosh ^ { - 1 } \frac { x } { 3 } \right) = \sqrt { x ^ { 2 } - 9 }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_67_1579_413_324}
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_77_1581_497_322}
Find the solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = x ^ { 2 } \sqrt { x ^ { 2 } - 9 }$$ given that $y = 1$ when $x = 3$. Give your answer in the form $y = \mathrm { f } ( x )$.
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-14_2716_35_143_2012}

CAIE Further Paper 2 2024 June Q8

6 marks

8 The planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$ do not intersect and are both perpendicular to $\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$. The line $l$ intersects $\Pi _ { 1 }$ at the point $( 1,6,0 )$ and intersects $\Pi _ { 2 }$ at the point $( 3 , - 6,0 )$.

Find Cartesian equations of $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
Express the vector equation of $l$ in the form $\left( \begin{array} { l } x
y
z \end{array} \right) = \mathbf { a } + \lambda \mathbf { b }$, where $\mathbf { a }$ and $\mathbf { b }$ are vectors to be determined, and hence show that for points on $l , \frac { 1 } { 2 } x + \frac { 1 } { 12 } y = 1$ and $z = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-16_2715_40_144_2008}
Show that the characteristic equation of $\mathbf { A }$ is $- \lambda ^ { 3 } + 3 \lambda ^ { 2 } + \frac { 7 } { 4 } \lambda = 0$ and hence find the eigenvalues of $\mathbf { A }$. The matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 3
1 & 2 & 3
\frac { 1 } { 2 } & \frac { 1 } { 12 } & 0 \end{array} \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-17_194_1711_484_212}
Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }$, where $n$ is a positive integer. [6]
\includegraphics[max width=\textwidth, alt={}]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_65_1581_335_322} ........................................................................................................................................
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_72_1579_511_324}
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-18_2718_35_144_2012} 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 November Q1

By differentiating $\mathrm { e } ^ { - x ^ { 2 } }$, find the Maclaurin's series for $\mathrm { e } ^ { - x ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.
Deduce an approximation to $\int _ { 0 } ^ { \frac { 1 } { 5 } } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$, giving your answer as a rational fraction in its lowest terms.

CAIE Further Paper 2 2020 November Q2

2 The variables $x$ and $y$ are related by the differential equation $$9 \frac { d ^ { 2 } y } { d x ^ { 2 } } + 6 \frac { d y } { d x } + y = 3 x ^ { 2 } + 30 x$$

Find the general solution for $y$ in terms of $x$.
State an approximate solution for large positive values of $x$.

CAIE Further Paper 2 2020 November Q3

Show that the system of equations $$\begin{array} { r } x - 2 y - 4 z = 1
x - 2 y + k z = 1
- x + 2 y + 2 z = 1 \end{array}$$ where $k$ is a constant, does not have a unique solution.
Given that $k = - 4$, show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
Given instead that $k = - 2$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
For the case where $k \neq - 2$ and $k \neq - 4$, show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
\includegraphics[max width=\textwidth, alt={}, center]{7da7fa35-1b97-4708-a1a2-cba9e35c8bf0-06_894_841_260_612} The diagram shows the curve with equation $\mathrm { y } = 1 - \mathrm { x } ^ { 3 }$ for $0 \leqslant x \leqslant 1$, together with a set of $n$ rectangles of width $\frac { 1 } { n }$.

CAIE Further Paper 2 2020 November Q5

5 It is given that $$x = \sinh ^ { - 1 } t , \quad y = \cos ^ { - 1 } t$$ where $- 1 < t < 1$.

By differentiating $\cos y$ with respect to $t$, show that $\frac { d y } { d t } = - \frac { 1 } { \sqrt { 1 - t ^ { 2 } } }$.
Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $t$, simplifying your answer.

CAIE Further Paper 2 2020 November Q6

Use de Moivre's theorem to show that $\sin ^ { 4 } \theta = \frac { 1 } { 8 } ( \cos 4 \theta - 4 \cos 2 \theta + 3 )$.
Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } + y \cot \theta = \sin ^ { 3 } \theta$$ for which $y = 0$ when $\theta = \frac { 1 } { 2 } \pi$.

CAIE Further Paper 2 2020 November Q7

7 The matrix $\mathbf { P }$ is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 4 & 2
0 & - 1 & 1
0 & 0 & 2 \end{array} \right) .$$

State the eigenvalues of $\mathbf { P }$.
Use the characteristic equation of $\mathbf { P }$ to find $\mathbf { P } ^ { - 1 }$.
The $3 \times 3$ matrix $\mathbf { A }$ has distinct eigenvalues $b , - 1,1$ with corresponding eigenvectors $$\left( \begin{array} { l } 1
0
0 \end{array} \right) , \quad \left( \begin{array} { r } 4
- 1
0 \end{array} \right) , \quad \left( \begin{array} { l } 2
1
2 \end{array} \right)$$ respectively.
Find $\mathbf { A }$ in terms of b.

CAIE Further Paper 2 2020 November Q8

Sketch the graph of $\mathrm { y } = \operatorname { coth } \mathrm { x }$ for $x > 0$ and state the equations of the asymptotes.
Starting from the definitions of coth and cosech in terms of exponentials, prove that $$\operatorname { coth } ^ { 2 } x - \operatorname { cosech } ^ { 2 } x = 1$$ The curve $C$ has equation $\mathrm { y } = \ln \operatorname { coth } \left( \frac { 1 } { 2 } \mathrm { x } \right)$ for $x > 0$.
Show that $\frac { \mathrm { dy } } { \mathrm { dx } } = - \operatorname { cosechx }$.
It is given that the arc length of $C$ from $\mathrm { x } = \mathrm { a }$ to $\mathrm { x } = 2 \mathrm { a }$ is $\ln 4$, where $a$ is a positive constant. Show that $\cosh a = 2$ and find, in logarithmic form, the exact value of $a$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

Show that, at the point $( 1,1 )$ on $C , \frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 2 } { 3 }$.
Find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 1,1 )$.

CAIE Further Paper 2 2020 November Q6

6 Find the particular solution of the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 8 \frac { d x } { d t } + 15 x = 102 \cos 3 t$$ given that, when $t = 0 , x = 1$ and $\frac { \mathrm { dx } } { \mathrm { dt } } = 0$.

CAIE Further Paper 2 2020 November Q7

Show that $\sum _ { r = 1 } ^ { n } z ^ { 2 r } = \frac { z ^ { 2 n + 1 } - z } { z - z ^ { - 1 } }$, for $z \neq 0,1 , - 1$.
By letting $z = \cos \theta + i \sin \theta$, show that, if $\sin \theta \neq 0$, $$1 + 2 \sum _ { r = 1 } ^ { n } \cos ( 2 r \theta ) = \frac { \sin ( 2 n + 1 ) \theta } { \sin \theta }$$

Questions — CAIE Further Paper 2 (186 questions)

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