Questions - CAIE - A-Level Maths

CAIE FP1 2017 November Q9

Standard +0.8

9 The curve $C$ has equation $$y = \frac { 3 x - 9 } { ( x - 2 ) ( x + 1 ) }$$

Find the equations of the asymptotes of $C$.
\includegraphics[max width=\textwidth, alt={}, center]{68e31138-756a-433a-bf42-0fdfadad091e-14_61_1566_513_328}
Show that there is no point on $C$ for which $\frac { 1 } { 3 } < y < 3$.
Find the coordinates of the turning points of $C$.
Sketch $C$.

CAIE FP1 2018 November Q1

Moderate -0.5

1 The vectors $\mathbf { a } , \mathbf { b } , \mathbf { c }$ and $\mathbf { d }$ in $\mathbb { R } ^ { 3 }$ are given by $$\mathbf { a } = \left( \begin{array} { l } 1

CAIE FP1 2018 November Q2

Moderate -0.5

2
1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2
9
0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l }

CAIE FP1 2018 November Q4

Standard +0.8

4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0
- 8
3 \end{array} \right) .$$

Show that $\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}$ is a basis for $\mathbb { R } ^ { 3 }$.
Express $\mathbf { d }$ in terms of $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$.\\ 2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are $\alpha , 2 \alpha , 4 \alpha$, where $p , q , r$ and $\alpha$ are non-zero real constants.
Show that $$2 p \alpha + q = 0$$
Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$ 3 The sequence of positive numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is such that $u _ { 1 } < 3$ and, for $n \geqslant 1$, $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
By considering $3 - u _ { n + 1 }$, or otherwise, prove by mathematical induction that $u _ { n } < 3$ for all positive integers $n$.
Show that $u _ { n + 1 } > u _ { n }$ for $n \geqslant 1$.\\ 4 A curve is defined parametrically by $$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$ The arc of the curve joining the point where $t = 0$ to the point where $t = \pi$ is rotated through one complete revolution about the $x$-axis. The area of the surface generated is denoted by $S$.
Show that $$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$ where the constant $a$ is to be found.
Using the result $\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t$, find the exact value of $S$.

CAIE FP1 2018 November Q5

Standard +0.3

5 It is given that $\lambda$ is an eigenvalue of the matrix $\mathbf { A }$ with $\mathbf { e }$ as a corresponding eigenvector, and $\mu$ is an eigenvalue of the matrix $\mathbf { B }$ for which $\mathbf { e }$ is also a corresponding eigenvector.

Show that $\lambda + \mu$ is an eigenvalue of the matrix $\mathbf { A } + \mathbf { B }$ with $\mathbf { e }$ as a corresponding eigenvector.
The matrix $\mathbf { A }$, given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ has $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ as eigenvectors.
Find the corresponding eigenvalues.
The matrix $\mathbf { B }$ has eigenvalues 4, 5 and 1 with corresponding eigenvectors $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ respectively.
Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }$.

CAIE FP1 2018 November Q6

Standard +0.8

6 The curve $C$ has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where $a$ is constant and $a > 1$.

Find the equations of the asymptotes of $C$.
Show that $C$ intersects the $x$-axis twice.
Justifying your answer, find the number of stationary points on $C$.
Sketch $C$, stating the coordinates of its point of intersection with the $y$-axis.

CAIE FP1 2018 November Q7

Challenging +1.8

7

Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
Use the equation $\frac { \sin 8 \theta } { \sin 2 \theta } = 0$ to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form $\sin k \pi$, where $k$ is rational.

CAIE FP1 2018 November Q9

5 marks Moderate -0.5

9
0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 3
3
4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0
- 8
3 \end{array} \right) .$$

Show that $\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}$ is a basis for $\mathbb { R } ^ { 3 }$.
Express $\mathbf { d }$ in terms of $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$.\\ 2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are $\alpha , 2 \alpha , 4 \alpha$, where $p , q , r$ and $\alpha$ are non-zero real constants.
Show that $$2 p \alpha + q = 0$$
Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$ 3 The sequence of positive numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is such that $u _ { 1 } < 3$ and, for $n \geqslant 1$, $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
By considering $3 - u _ { n + 1 }$, or otherwise, prove by mathematical induction that $u _ { n } < 3$ for all positive integers $n$.
Show that $u _ { n + 1 } > u _ { n }$ for $n \geqslant 1$.\\ 4 A curve is defined parametrically by $$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$ The arc of the curve joining the point where $t = 0$ to the point where $t = \pi$ is rotated through one complete revolution about the $x$-axis. The area of the surface generated is denoted by $S$.
Show that $$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$ where the constant $a$ is to be found.
Using the result $\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t$, find the exact value of $S$.\\ 5 It is given that $\lambda$ is an eigenvalue of the matrix $\mathbf { A }$ with $\mathbf { e }$ as a corresponding eigenvector, and $\mu$ is an eigenvalue of the matrix $\mathbf { B }$ for which $\mathbf { e }$ is also a corresponding eigenvector.
Show that $\lambda + \mu$ is an eigenvalue of the matrix $\mathbf { A } + \mathbf { B }$ with $\mathbf { e }$ as a corresponding eigenvector.\\ The matrix $\mathbf { A }$, given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1
- 1 & 2 & 3
1 & 0 & 2 \end{array} \right)$$ has $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ as eigenvectors.
Find the corresponding eigenvalues.\\ The matrix $\mathbf { B }$ has eigenvalues 4, 5 and 1 with corresponding eigenvectors $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)$ and $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ respectively.
Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }$.\\ 6 The curve $C$ has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where $a$ is constant and $a > 1$.
Find the equations of the asymptotes of $C$.
Show that $C$ intersects the $x$-axis twice.
Justifying your answer, find the number of stationary points on $C$.
Sketch $C$, stating the coordinates of its point of intersection with the $y$-axis. 7
Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
Use the equation $\frac { \sin 8 \theta } { \sin 2 \theta } = 0$ to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form $\sin k \pi$, where $k$ is rational.\\ 8 The plane $\Pi _ { 1 }$ has equation $$\mathbf { r } = \left( \begin{array} { l } 5
1
0 \end{array} \right) + s \left( \begin{array} { r } - 4
1
3 \end{array} \right) + t \left( \begin{array} { l } 0
1
2 \end{array} \right)$$
Find a cartesian equation of $\Pi _ { 1 }$.\\ The plane $\Pi _ { 2 }$ has equation $3 x + y - z = 3$.
Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in degrees.
Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$. [5]\\ 9 The curve $C$ has polar equation $$r = 5 \sqrt { } ( \cot \theta ) ,$$ where $0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
Find the area of the finite region bounded by $C$ and the line $\theta = 0.01$, showing full working. Give your answer correct to 1 decimal place.
Let $P$ be the point on $C$ where $\theta = 0.01$.
Find the distance of $P$ from the initial line, giving your answer correct to 1 decimal place.
Find the maximum distance of $C$ from the initial line.
Sketch $C$.

CAIE FP1 2018 November Q10

Challenging +1.8

10

Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 37 \sin 3 t$$ given that $x = 3$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$ when $t = 0$.
Show that, for large positive values of $t$ and for any initial conditions, $$x \approx \sqrt { } ( 37 ) \sin ( 3 t - \phi ) ,$$ where the constant $\phi$ is such that $\tan \phi = 6$.

CAIE FP1 2018 November Q11 EITHER

Standard +0.8

By considering $( 2 r + 1 ) ^ { 2 } - ( 2 r - 1 ) ^ { 2 }$, use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
By considering $( 2 r + 1 ) ^ { 4 } - ( 2 r - 1 ) ^ { 4 }$, use the method of differences and the result given in part (i) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ The sums $S$ and $T$ are defined as follows: $$\begin{aligned} & S = 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + 4 ^ { 3 } + \ldots + ( 2 N ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } , \\ & T = 1 ^ { 3 } + 3 ^ { 3 } + 5 ^ { 3 } + 7 ^ { 3 } + \ldots + ( 2 N - 1 ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } . \end{aligned}$$
Use the result given in part (ii) to show that $S = ( 2 N + 1 ) ^ { 2 } ( N + 1 ) ^ { 2 }$.
Hence, or otherwise, find an expression in terms of $N$ for $T$, factorising your answer as far as possible.
Deduce the value of $\frac { S } { T }$ as $N \rightarrow \infty$.

CAIE FP1 2018 November Q11 OR

Challenging +1.8

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

Show that $A ( - 1,1 )$ is the only point on $C$ with $x$-coordinate equal to - 1 .
For $n \geqslant 1$, let $A _ { n }$ denote the value of $\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }$ at the point $A ( - 1,1 )$.
Show that $A _ { 1 } = 0$.
Show that $A _ { 2 } = \frac { 2 } { 5 }$.
Let $I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x$.
Show that for $n \geqslant 2$, $$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
Deduce the value of $I _ { 3 }$ in terms of $I _ { 1 }$.
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 FP1 2018 November Q1

Standard +0.3

1 The roots of the cubic equation $$x ^ { 3 } - 5 x ^ { 2 } + 13 x - 4 = 0$$ are $\alpha , \beta , \gamma$.

Find the value of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$.
Find the value of $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }$.

CAIE FP1 2018 November Q2

Standard +0.8

2 It is given that $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 3 & 1 \\ 0 & - 2 & 1 \\ 0 & 0 & 1 \end{array} \right)$$

Find the eigenvalue of $\mathbf { A }$ corresponding to the eigenvector $\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right)$.
Write down the negative eigenvalue of $\mathbf { A }$ and find a corresponding eigenvector.
Find an eigenvalue and a corresponding eigenvector of the matrix $\mathbf { A } + \mathbf { A } ^ { 6 }$.

CAIE FP1 2018 November Q3

Standard +0.8

3 The curve $C$ has polar equation $r = a \cos 3 \theta$, for $- \frac { 1 } { 6 } \pi \leqslant \theta \leqslant \frac { 1 } { 6 } \pi$, where $a$ is a positive constant.

Sketch $C$.
Find the area of the region enclosed by $C$, showing full working.
Using the identity $\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$, find a cartesian equation of $C$.

CAIE FP1 2018 November Q4

Standard +0.8

4

Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 4 \sin t$$
State an approximate solution for large positive values of $t$.

CAIE FP1 2018 November Q5

Challenging +1.2

5 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { r r r r } 3 & 2 & 0 & 1 \\ 6 & 5 & - 1 & 3 \\ 9 & 8 & - 2 & 5 \\ - 3 & - 2 & 0 & - 1 \end{array} \right)$$

Find the rank of $\mathbf { M }$.
Let $K$ be the null space of T .
Find a basis for $K$.
Find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } 2 \\ 5 \\ 8 \\ - 2 \end{array} \right) .$$

CAIE FP1 2018 November Q6

Challenging +1.2

6 It is given that $y = \mathrm { e } ^ { x } u$, where $u$ is a function of $x$. The $r$ th derivatives $\frac { \mathrm { d } ^ { r } y } { \mathrm {~d} x ^ { r } }$ and $\frac { \mathrm { d } ^ { r } u } { \mathrm {~d} x ^ { r } }$ are denoted by $y ^ { ( r ) }$ and $u ^ { ( r ) }$ respectively. Prove by mathematical induction that, for all positive integers $n$, $$y ^ { ( n ) } = \mathrm { e } ^ { x } \left( \binom { n } { 0 } u + \binom { n } { 1 } u ^ { ( 1 ) } + \binom { n } { 2 } u ^ { ( 2 ) } + \ldots + \binom { n } { r } u ^ { ( r ) } + \ldots + \binom { n } { n } u ^ { ( n ) } \right)$$ [You may use without proof the result $\binom { k } { r } + \binom { k } { r - 1 } = \binom { k + 1 } { r }$.]

CAIE FP1 2018 November Q8

Standard +0.8

8
- 2 \end{array} \right) .$$ 6 It is given that $y = \mathrm { e } ^ { x } u$, where $u$ is a function of $x$. The $r$ th derivatives $\frac { \mathrm { d } ^ { r } y } { \mathrm {~d} x ^ { r } }$ and $\frac { \mathrm { d } ^ { r } u } { \mathrm {~d} x ^ { r } }$ are denoted by $y ^ { ( r ) }$ and $u ^ { ( r ) }$ respectively. Prove by mathematical induction that, for all positive integers $n$, $$y ^ { ( n ) } = \mathrm { e } ^ { x } \left( \binom { n } { 0 } u + \binom { n } { 1 } u ^ { ( 1 ) } + \binom { n } { 2 } u ^ { ( 2 ) } + \ldots + \binom { n } { r } u ^ { ( r ) } + \ldots + \binom { n } { n } u ^ { ( n ) } \right)$$ [You may use without proof the result $\binom { k } { r } + \binom { k } { r - 1 } = \binom { k + 1 } { r }$.]\\ 7 Let $$S _ { N } = \sum _ { r = 1 } ^ { N } ( 3 r + 1 ) ( 3 r + 4 ) \quad \text { and } \quad T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r + 4 ) } .$$

Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = N \left( 3 N ^ { 2 } + 12 N + 13 \right)$$
Use the method of differences to show that $$T _ { N } = \frac { 1 } { 12 } - \frac { 1 } { 3 ( 3 N + 4 ) } .$$
Deduce that $\frac { S _ { N } } { T _ { N } }$ is an integer.
Find $\lim _ { N \rightarrow \infty } \frac { S _ { N } } { N ^ { 3 } T _ { N } }$.\\ 8
By considering the binomial expansion of $\left( z + \frac { 1 } { z } \right) ^ { 6 }$, where $z = \cos \theta + i \sin \theta$, express $\cos ^ { 6 } \theta$ in the form $$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$ where $p , q , r$ and $s$ are integers to be determined.
Hence find the exact value of $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x$$

CAIE FP1 2018 November Q9

Challenging +1.2

9 The curve $C$ has equation $$y = \frac { 5 x ^ { 2 } + 5 x + 1 } { x ^ { 2 } + x + 1 } .$$

Find the equation of the asymptote of $C$.
Show that, for all real values of $x , - \frac { 1 } { 3 } \leqslant y < 5$.
Find the coordinates of any stationary points of $C$.
Sketch $C$, stating the coordinates of any intersections with the $y$-axis.

CAIE FP1 2018 November Q10

Standard +0.8

10 The position vectors of the points $A , B , C , D$ are $$\mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k } , \quad - \mathbf { i } + 3 \mathbf { k } , \quad m \mathbf { j } + 4 \mathbf { k } ,$$ respectively, where $m$ is a constant.

Show that the lines $A B$ and $C D$ are parallel when $m = \frac { 3 } { 2 }$.
Given that $m \neq \frac { 3 } { 2 }$, find the shortest distance between the lines $A B$ and $C D$.
When $m = 2$, find the acute angle between the planes $A B C$ and $A B D$, giving your answer in degrees.

CAIE FP1 2018 November Q11 EITHER

Challenging +1.2

The curve $C$ is defined parametrically by $$x = 18 t - t ^ { 2 } \quad \text { and } \quad y = 8 t ^ { \frac { 3 } { 2 } }$$ where $0 < t \leqslant 4$.

Show that at all points of $C$, $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 3 ( 9 + t ) } { 2 t ^ { \frac { 1 } { 2 } } ( 9 - t ) ^ { 3 } }$$
Show that the mean value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ with respect to $x$ over the interval $0 < x \leqslant 56$ is $\frac { 3 } { 70 }$.
Find the area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis, showing full working.

CAIE FP1 2018 November Q11 OR

Challenging +1.8

Let $I _ { n } = \int _ { 1 } ^ { \sqrt { } 2 } \left( x ^ { 2 } - 1 \right) ^ { n } \mathrm {~d} x$.

Show that, for $n \geqslant 1$, $$( 2 n + 1 ) I _ { n } = \sqrt { } 2 - 2 n I _ { n - 1 } .$$
Using the substitution $x = \sec \theta$, show that $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { 2 n + 1 } \theta \sec \theta \mathrm {~d} \theta$$
Deduce the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 7 } \theta } { \cos ^ { 8 } \theta } \mathrm {~d} \theta$$ 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 FP1 2019 November Q1

Standard +0.8

1 The curve $C$ has equation $y = x ^ { a }$ for $0 \leqslant x \leqslant 1$, where $a$ is a positive constant. Find, in terms of $a$, the coordinates of the centroid of the region enclosed by $C$, the line $x = 1$ and the $x$-axis.

CAIE FP1 2019 November Q2

Challenging +1.2

2 It is given that $y = \ln ( a x + 1 )$, where $a$ is a positive constant. Prove by mathematical induction that, for every positive integer $n$, $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { ( n - 1 ) ! a ^ { n } } { ( a x + 1 ) ^ { n } }$$

CAIE FP1 2019 November Q3

Challenging +1.8

3 The integral $I _ { n }$, where $n$ is a positive integer, is defined by $$I _ { n } = \int _ { \frac { 1 } { 2 } } ^ { 1 } x ^ { - n } \sin \pi x \mathrm {~d} x$$

Show that $$n ( n + 1 ) I _ { n + 2 } = 2 ^ { n + 1 } n + \pi - \pi ^ { 2 } I _ { n }$$
Find $I _ { 5 }$ in terms of $\pi$ and $I _ { 1 }$.

Questions — CAIE (7279 questions)

Browse by module

Browse by board

CAIE FP1 2017 November Q9

CAIE FP1 2018 November Q1

CAIE FP1 2018 November Q2

CAIE FP1 2018 November Q4

CAIE FP1 2018 November Q5

CAIE FP1 2018 November Q6

CAIE FP1 2018 November Q7

CAIE FP1 2018 November Q9

CAIE FP1 2018 November Q10

CAIE FP1 2018 November Q11 EITHER

CAIE FP1 2018 November Q11 OR

CAIE FP1 2018 November Q1

CAIE FP1 2018 November Q2

CAIE FP1 2018 November Q3

CAIE FP1 2018 November Q4

CAIE FP1 2018 November Q5

CAIE FP1 2018 November Q6

CAIE FP1 2018 November Q8

CAIE FP1 2018 November Q9

CAIE FP1 2018 November Q10

CAIE FP1 2018 November Q11 EITHER

CAIE FP1 2018 November Q11 OR

CAIE FP1 2019 November Q1

CAIE FP1 2019 November Q2

CAIE FP1 2019 November Q3