Questions - CAIE - A-Level Maths

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 5 } { 6 }$ at the point $A ( 1,0 )$ on $C$.
Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$.

CAIE FP1 2004 November Q8

9 marks Challenging +1.8

8 The sequence of real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is such that $a _ { 1 } = 1$ and $$a _ { n + 1 } = \left( a _ { n } + \frac { 1 } { a _ { n } } \right) ^ { \lambda }$$ where $\lambda$ is a constant greater than 1 . Prove by mathematical induction that, for $n \geqslant 2$, $$a _ { n } \geqslant 2 ^ { \mathrm { g } ( n ) }$$ where $g ( n ) = \lambda ^ { n - 1 }$. Prove also that, for $n \geqslant 2 , \frac { a _ { n + 1 } } { a _ { n } } > 2 ^ { ( \lambda - 1 ) \mathrm { g } ( n ) }$.

CAIE FP1 2004 November Q9

10 marks Challenging +1.8

9 It is given that $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 3 } \right) ^ { - n } \mathrm {~d} x$$ where $n > 0$.

Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x \left( 1 + x ^ { 3 } \right) ^ { - n } \right] = - ( 3 n - 1 ) \left( 1 + x ^ { 3 } \right) ^ { - n } + 3 n \left( 1 + x ^ { 3 } \right) ^ { - n - 1 }$$ and hence, or otherwise, show that $$I _ { n + 1 } = \frac { 2 ^ { - n } } { 3 n } + \left( 1 - \frac { 1 } { 3 n } \right) I _ { n }$$
By considering the graph of $y = \frac { 1 } { 1 + x ^ { 3 } }$, show that $I _ { 1 } < 1$.
Deduce that $I _ { 3 } < \frac { 53 } { 72 }$.

CAIE FP1 2004 November Q10

11 marks Standard +0.8

10 The curve $C$ has equation $$y = \frac { x ^ { 2 } + 2 x - 3 } { ( \lambda x + 1 ) ( x + 4 ) }$$ where $\lambda$ is a constant.

Find the equations of the asymptotes of $C$ for the case where $\lambda = 0$.
Find the equations of the asymptotes of $C$ for the case where $\lambda$ is not equal to any of $- 1,0 , \frac { 1 } { 4 } , \frac { 1 } { 3 }$.
Sketch $C$ for the case where $\lambda = - 1$. Show, on your diagram, the equations of the asymptotes and the coordinates of the points of intersection of $C$ with the coordinate axes.

CAIE FP1 2004 November Q11

12 marks Challenging +1.8

11 The line $l _ { 1 }$ passes through the point $A$, whose position vector is $3 \mathbf { i } - 5 \mathbf { j } - 4 \mathbf { k }$, and is parallel to the vector $3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$. The line $l _ { 2 }$ passes through the point $B$, whose position vector is $2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k }$, and is parallel to the vector $\mathbf { i } - \mathbf { j } - 4 \mathbf { k }$. The point $P$ on $l _ { 1 }$ and the point $Q$ on $l _ { 2 }$ are such that $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$. The plane $\Pi _ { 1 }$ contains $P Q$ and $l _ { 1 }$, and the plane $\Pi _ { 2 }$ contains $P Q$ and $l _ { 2 }$.

Find the length of $P Q$.
Find a vector perpendicular to $\Pi _ { 1 }$.
Find the perpendicular distance from $B$ to $\Pi _ { 1 }$.
Find the angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.

CAIE FP1 2004 November Q12 EITHER

Challenging +1.8

The variable $y$ depends on $x$, and the variables $x$ and $t$ are related by $x = \mathrm { e } ^ { t }$. Show that $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t } \quad \text { and } \quad x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } .$$

Given that $y$ satisfies the differential equation $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 50 ( \ln x ) - 1$$ find a differential equation involving only $t$ and $y$.
Show that the complementary function of the differential equation in $t$ and $y$ may be written in the form $$R \mathrm { e } ^ { - \frac { 3 } { 2 } t } \sin ( 2 t + \phi )$$ where $R$ and $\phi$ are arbitrary constants.
Find a particular integral of the differential equation in $t$ and $y$.
Hence find the general solution of the differential equation in $x$ and $y$.

CAIE FP1 2004 November Q12 OR

Challenging +1.2

The matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that if $\mathbf { A }$ is non-singular then

$\lambda \neq 0$,
the matrix $\mathbf { A } ^ { - 1 }$ has $\lambda ^ { - 1 }$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 2 \\ 0 & - 2 & 4 \\ 0 & 0 & - 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 4 \mathbf { I } ) ^ { - 1 }$$ Find a non-singular matrix $\mathbf { P }$, and a diagonal matrix $\mathbf { D }$, such that $\mathbf { B } = \mathbf { P D P } ^ { - 1 }$.

CAIE FP1 2005 November Q1

4 marks Standard +0.8

1 Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z ^ { 5 } = - 16 + ( 16 \sqrt { } 3 ) i$$ giving each root in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$.

CAIE FP1 2005 November Q2

6 marks Standard +0.8

2 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is such that $u _ { 1 } = 1$ and $$u _ { n + 1 } = - 1 + \sqrt { } \left( u _ { n } + 7 \right)$$

Prove by induction that $u _ { n } < 2$ for all $n \geqslant 1$.
Show that if $u _ { n } = 2 - \varepsilon$, where $\varepsilon$ is small, then $$u _ { n + 1 } \approx 2 - \frac { 1 } { 6 } \varepsilon$$

CAIE FP1 2005 November Q3

7 marks Standard +0.3

3 The curve $C$ has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where $\lambda$ is a non-zero constant. Obtain the equations of the asymptotes of $C$. In separate diagrams, sketch $C$ for the cases where

$\lambda > 0$,
$\lambda < 0$.

CAIE FP1 2005 November Q4

7 marks Standard +0.3

4 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 24 \mathrm { e } ^ { 2 x }$$ given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 9$ when $x = 0$.

CAIE FP1 2005 November Q5

7 marks Challenging +1.8

5 In the equation $$x ^ { 3 } + a x ^ { 2 } + b x + c = 0$$ the coefficients $a , b$ and $c$ are real. It is given that all the roots are real and greater than 1 .

Prove that $a < - 3$.
By considering the sum of the squares of the roots, prove that $a ^ { 2 } > 2 b + 3$.
By considering the sum of the cubes of the roots, prove that $a ^ { 3 } < - 9 b - 3 c - 3$.

CAIE FP1 2005 November Q6

8 marks Challenging +1.8

6 Let $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { - n } \mathrm {~d} x$$ where $n \geqslant 1$. By considering $\frac { \mathrm { d } } { \mathrm { d } x } \left( x \left( 1 + x ^ { 2 } \right) ^ { - n } \right)$, or otherwise, prove that $$2 n I _ { n + 1 } = ( 2 n - 1 ) I _ { n } + 2 ^ { - n }$$ Deduce that $I _ { 3 } = \frac { 3 } { 32 } \pi + \frac { 1 } { 4 }$.
$7 \quad$ Write down an expression in terms of $z$ and $N$ for the sum of the series $$\sum _ { n = 1 } ^ { N } 2 ^ { - n } z ^ { n }$$ Use de Moivre's theorem to deduce that $$\sum _ { n = 1 } ^ { 10 } 2 ^ { - n } \sin \left( \frac { 1 } { 10 } n \pi \right) = \frac { 1025 \sin \left( \frac { 1 } { 10 } \pi \right) } { 2560 - 2048 \cos \left( \frac { 1 } { 10 } \pi \right) }$$

CAIE FP1 2005 November Q8

9 marks Standard +0.8

8 Find the coordinates of the centroid of the finite region bounded by the $x$-axis and the curve whose equation is $$y = x ^ { 2 } ( 1 - x )$$ Deduce the coordinates of the centroid of the finite region bounded by the $x$-axis and the curve whose equation is $$y = x ( 1 - x ) ^ { 2 }$$

CAIE FP1 2005 November Q9

10 marks Challenging +1.2

9 The planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$ have vector equations $$\mathbf { r } = \lambda _ { 1 } ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) + \mu _ { 1 } ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \lambda _ { 2 } ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) + \mu _ { 2 } ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )$$ respectively. The line $l$ passes through the point with position vector $4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }$ and is parallel to both $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find a vector equation for $l$. Find also the shortest distance between $l$ and the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$.

CAIE FP1 2005 November Q10

11 marks Challenging +1.2

10 It is given that the eigenvalues of the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { r r r } 4 & 1 & - 1 \\ - 4 & - 1 & 4 \\ 0 & - 1 & 5 \end{array} \right)$$ are $1,3,4$. Find a set of corresponding eigenvectors. Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $$\mathbf { M } ^ { n } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$ where $n$ is a positive integer. Find $\mathbf { P } ^ { - 1 }$ and deduce that $$\lim _ { n \rightarrow \infty } 4 ^ { - n } \mathbf { M } ^ { n } = \left( \begin{array} { r r r } - \frac { 1 } { 3 } & 0 & - \frac { 1 } { 3 } \\ \frac { 4 } { 3 } & 0 & \frac { 4 } { 3 } \\ \frac { 4 } { 3 } & 0 & \frac { 4 } { 3 } \end{array} \right)$$

CAIE FP1 2005 November Q11

11 marks Challenging +1.2

11 Find the rank of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 1 & 2 & 3 \\ 4 & 3 & 5 & 16 \\ 6 & 6 & 13 & 13 \\ 14 & 12 & 23 & 45 \end{array} \right)$$ Find vectors $\mathbf { x } _ { 0 }$ and $\mathbf { e }$ such that any solution of the equation $$\mathbf { A x } = \left( \begin{array} { r } 0 \\ 2 \\ - 1 \\ 3 \end{array} \right)$$ can be expressed in the form $\mathbf { x } _ { 0 } + \lambda \mathbf { e }$, where $\lambda \in \mathbb { R }$. Hence show that there is no vector which satisfies (*) and has all its elements positive.

CAIE FP1 2005 November Q12 EITHER

Challenging +1.8

Show that $\left( n + \frac { 1 } { 2 } \right) ^ { 3 } - \left( n - \frac { 1 } { 2 } \right) ^ { 3 } \equiv 3 n ^ { 2 } + \frac { 1 } { 4 }$. Use this result to prove that $\sum _ { n = 1 } ^ { N } n ^ { 2 } = \frac { 1 } { 6 } N ( N + 1 ) ( 2 N + 1 )$. The sums $S , T$ and $U$ are defined as follows: $$\begin{aligned} & S = 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } , \\ & T = 1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 2 N - 1 ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } , \\ & U = 1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } . \end{aligned}$$ Find and simplify expressions in terms of $N$ for each of $S , T$ and $U$. Hence

describe the behaviour of $\frac { S } { T }$ as $N \rightarrow \infty$,
prove that if $\frac { S } { U }$ is an integer then $\frac { T } { U }$ is an integer.

CAIE FP1 2005 November Q12 OR

Challenging +1.3

The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations $$r = 4 \cos \theta \quad \text { and } \quad r = 1 + \cos \theta$$ respectively, where $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.

Show that $C _ { 1 }$ and $C _ { 2 }$ meet at the points $A \left( \frac { 4 } { 3 } , \alpha \right)$ and $B \left( \frac { 4 } { 3 } , - \alpha \right)$, where $\alpha$ is the acute angle such that $\cos \alpha = \frac { 1 } { 3 }$.
In a single diagram, draw sketch graphs of $C _ { 1 }$ and $C _ { 2 }$.
Show that the area of the region bounded by the arcs $O A$ and $O B$ of $C _ { 1 }$, and the $\operatorname { arc } A B$ of $C _ { 2 }$, is $$4 \pi - \frac { 1 } { 3 } \sqrt { } 2 - \frac { 13 } { 2 } \alpha .$$