Questions - CAIE FP1 - A-Level Maths

Obtain the coordinates of the points of intersection of $C$ with the axes.
Obtain the equation of each of the asymptotes of $C$.
Draw a sketch of $C$.

CAIE FP1 2009 November Q4

4 It is given that $$x = t + \sin t , \quad y = t ^ { 2 } + 2 \cos t$$ where $- \pi < t < \pi$. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 t \sin t } { ( 1 + \cos t ) ^ { 3 } }$$ Show that $\frac { \mathrm { d } y } { \mathrm {~d} x }$ increases with $x$ over the given interval of $t$.

CAIE FP1 2009 November Q5

5 The equation $$x ^ { 3 } + 5 x + 3 = 0$$ has roots $\alpha , \beta , \gamma$. Use the substitution $x = - \frac { 3 } { y }$ to find a cubic equation in $y$ and show that the roots of this equation are $\beta \gamma , \gamma \alpha , \alpha \beta$. Find the exact values of $\beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 }$ and $\beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } + \alpha ^ { 3 } \beta ^ { 3 }$.

CAIE FP1 2009 November Q6

6 Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x ^ { n - 1 } \sqrt { } \left( 4 - x ^ { 2 } \right) \right] = \frac { 4 ( n - 1 ) x ^ { n - 2 } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } - \frac { n x ^ { n } } { \sqrt { } \left( 4 - x ^ { 2 } \right) }$$ Let $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { x ^ { n } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } \mathrm { d } x$$ where $n \geqslant 0$. Prove that $$n I _ { n } = 4 ( n - 1 ) I _ { n - 2 } - \sqrt { } 3$$ for $n \geq 2$. Given that $I _ { 0 } = \frac { 1 } { 6 } \pi$, find $I _ { 4 }$, leaving your answer in an exact form.

CAIE FP1 2009 November Q7

7 Use de Moivre's theorem to express $\sin ^ { 6 } \theta$ in the form $$a + b \cos 2 \theta + c \cos 4 \theta + d \cos 6 \theta$$ where $a , b , c , d$ are constants to be found. Hence evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } 2 x d x$$ leaving your answer in terms of $\pi$.

CAIE FP1 2009 November Q8

The curve $C _ { 1 }$ has equation $y = - \ln ( \cos x )$. Show that the length of the arc of $C _ { 1 }$ from the point where $x = 0$ to the point where $x = \frac { 1 } { 3 } \pi$ is $\ln ( 2 + \sqrt { 3 } )$.
The curve $C _ { 2 }$ has equation $y = 2 \sqrt { } ( x + 3 )$. The arc of $C _ { 2 }$ joining the point where $x = 0$ to the point where $x = 1$ is rotated through one complete revolution about the $x$-axis. Show that the area of the surface generated is $$\frac { 8 } { 3 } \pi ( 5 \sqrt { } 5 - 8 )$$

CAIE FP1 2009 November Q9

9 Show that if $y$ depends on $x$ and $x = \mathrm { e } ^ { u }$ then $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} u } .$$ Given that $y$ satisfies the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 30 x ^ { 2 }$$ use the substitution $x = \mathrm { e } ^ { u }$ to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 3 y = 30 \mathrm { e } ^ { 2 u }$$ Hence find the general solution for $y$ in terms of $x$.

CAIE FP1 2009 November Q10

10 The curve $C$ has polar equation $$r = a \sin 3 \theta$$ where $0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$.

Show that the area of the region enclosed by $C$ is $\frac { 1 } { 12 } \pi a ^ { 2 }$.
Show that, at the point of $C$ at maximum distance from the initial line, $$\tan 3 \theta + 3 \tan \theta = 0 .$$
Use the formula $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ to find this maximum distance.
Draw a sketch of $C$.

CAIE FP1 2009 November Q11 EITHER

Prove by induction that $$\sum _ { n = 1 } ^ { N } n ^ { 3 } = \frac { 1 } { 4 } N ^ { 2 } ( N + 1 ) ^ { 2 }$$ Use this result, together with the formula for $\sum _ { n = 1 } ^ { N } n ^ { 2 }$, to show that $$\sum _ { n = 1 } ^ { N } \left( 20 n ^ { 3 } + 36 n ^ { 2 } \right) = N ( N + 1 ) ( N + 3 ) ( 5 N + 2 ) .$$ Let $$S _ { N } = \sum _ { n = 1 } ^ { N } \left( 20 n ^ { 3 } + 36 n ^ { 2 } + \mu n \right)$$ Find the value of the constant $\mu$ such that $S _ { N }$ is of the form $N ^ { 2 } ( N + 1 ) ( a N + b )$, where the constants $a$ and $b$ are to be determined. Show that, for this value of $\mu$, $$5 + \frac { 22 } { N } < N ^ { - 4 } S _ { N } < 5 + \frac { 23 } { N }$$ for all $N \geqslant 18$.

CAIE FP1 2009 November Q11 OR

One of the eigenvalues of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 4 & 6
2 & - 4 & 2
- 3 & 4 & a \end{array} \right)$$ is - 2 . Find the value of $a$. Another eigenvalue of $\mathbf { A }$ is - 5 . Find eigenvectors $\mathbf { e } _ { 1 }$ and $\mathbf { e } _ { 2 }$ corresponding to the eigenvalues - 2 and - 5 respectively. The linear space spanned by $\mathbf { e } _ { 1 }$ and $\mathbf { e } _ { 2 }$ is denoted by $V$.

Prove that, for any vector $\mathbf { x }$ belonging to $V$, the vector $\mathbf { A x }$ also belongs to $V$.
Find a non-zero vector which is perpendicular to every vector in $V$, and determine whether it is an eigenvector of $\mathbf { A }$.

CAIE FP1 2010 November Q1

1 The curve $C$ has equation $y = \frac { 1 } { 4 } \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)$. Show that the length of the $\operatorname { arc }$ of $C$ from the point where $x = 0$ to the point where $x = \frac { 1 } { 2 }$ is $\frac { \mathrm { e } ^ { 2 } - 1 } { 4 \mathrm { e } }$.

CAIE FP1 2010 November Q2

2 Use the method of differences to find $S _ { N }$, where $$S _ { N } = \sum _ { n = 1 } ^ { N } \frac { 1 } { n ( n + 2 ) }$$ Deduce the value of $\lim _ { N \rightarrow \infty } S _ { N }$.

CAIE FP1 2010 November Q3

3 A finite region $R$ in the $x - y$ plane is bounded by the curve with equation $y = \sqrt { } x - \frac { 1 } { \sqrt { } x }$, the $x$-axis between $x = 1$ and $x = 4$, and the line $x = 4$. Find the exact value of the $y$-coordinate of the centroid of $R$.

CAIE FP1 2010 November Q4

4 Prove by mathematical induction that, for all non-negative integers $n , 7 ^ { 2 n + 1 } + 5 ^ { n + 3 }$ is divisible by 44 .

CAIE FP1 2010 November Q5

5 Let $I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \sin x \mathrm {~d} x$ for $n \geqslant 0$. Show that $$I _ { n + 2 } = 1 - ( n + 1 ) ( n + 2 ) I _ { n }$$ Hence find the value of $I _ { 6 }$, correct to 4 decimal places.

CAIE FP1 2010 November Q6

6 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 2 & - 1 & \alpha
2 & 3 & - 1 & 0
2 & 1 & 2 & - 2
0 & 1 & - 3 & - 2 \end{array} \right)$$ Given that the dimension of the range space of T is 4 , show that $\alpha \neq 1$. It is now given that $\alpha = 1$. Show that the vectors $$\left( \begin{array} { l } 1
2
2
0 \end{array} \right) , \quad \left( \begin{array} { l } 2
3
1
1 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } - 1
- 1
2
- 3 \end{array} \right)$$ form a basis for the range space of T . Given also that the vector $\left( \begin{array} { c } p
1
1
q \end{array} \right)$ is in the range space of T , find a condition satisfied by $p$ and $q$.

CAIE FP1 2010 November Q7

7 The roots of the equation $x ^ { 3 } + 4 x - 1 = 0$ are $\alpha , \beta$ and $\gamma$. Use the substitution $y = \frac { 1 } { 1 + x }$ to show that the equation $6 y ^ { 3 } - 7 y ^ { 2 } + 3 y - 1 = 0$ has roots $\frac { 1 } { \alpha + 1 } , \frac { 1 } { \beta + 1 }$ and $\frac { 1 } { \gamma + 1 }$. For the cases $n = 1$ and $n = 2$, find the value of $$\frac { 1 } { ( \alpha + 1 ) ^ { n } } + \frac { 1 } { ( \beta + 1 ) ^ { n } } + \frac { 1 } { ( \gamma + 1 ) ^ { n } }$$ Deduce the value of $\frac { 1 } { ( \alpha + 1 ) ^ { 3 } } + \frac { 1 } { ( \beta + 1 ) ^ { 3 } } + \frac { 1 } { ( \gamma + 1 ) ^ { 3 } }$. Hence show that $\frac { ( \beta + 1 ) ( \gamma + 1 ) } { ( \alpha + 1 ) ^ { 2 } } + \frac { ( \gamma + 1 ) ( \alpha + 1 ) } { ( \beta + 1 ) ^ { 2 } } + \frac { ( \alpha + 1 ) ( \beta + 1 ) } { ( \gamma + 1 ) ^ { 2 } } = \frac { 73 } { 36 }$.

CAIE FP1 2010 November Q8

8 The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations given by $$\begin{array} { l l r } C _ { 1 } : & r = 3 \sin \theta , & 0 \leqslant \theta < \pi ,
C _ { 2 } : & r = 1 + \sin \theta , & - \pi < \theta \leqslant \pi . \end{array}$$

Find the polar coordinates of the points, other than the pole, where $C _ { 1 }$ and $C _ { 2 }$ meet.
In a single diagram, draw sketch graphs of $C _ { 1 }$ and $C _ { 2 }$.
Show that the area of the region which is inside $C _ { 1 }$ but outside $C _ { 2 }$ is $\pi$.

CAIE FP1 2010 November Q9

9 Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0
- 1 & 2 & - 1
0 & - 1 & 3 \end{array} \right)$$ Find a non-singular matrix $\mathbf { M }$ and a diagonal matrix $\mathbf { D }$ such that $( \mathbf { A } - 2 \mathbf { I } ) ^ { 3 } = \mathbf { M D M } ^ { - 1 }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix.

CAIE FP1 2010 November Q10

10 By using de Moivre's theorem to express $\sin 5 \theta$ and $\cos 5 \theta$ in terms of $\sin \theta$ and $\cos \theta$, show that $$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$ where $t = \tan \theta$. Show that the roots of the equation $x ^ { 4 } - 10 x ^ { 2 } + 5 = 0$ are $\tan \left( \frac { 1 } { 5 } n \pi \right)$ for $n = 1,2,3,4$. By considering the product of the roots of this equation, find the exact value of $\tan \left( \frac { 1 } { 5 } \pi \right) \tan \left( \frac { 2 } { 5 } \pi \right)$.

CAIE FP1 2010 November Q11

11 It is given that $x \neq 0$ and $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 x y = 8 x ^ { 2 } + 16$$ Show that if $z = x y$ then $$\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 4 z = 8 x ^ { 2 } + 16$$ Find $y$ in terms of $x$, given that $y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2$ when $x = \frac { 1 } { 2 } \pi$.

CAIE FP1 2010 November Q12 EITHER

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

Show that $C$ has at most two stationary points.
Show that if $C$ has exactly two stationary points then $\lambda > - \frac { 5 } { 4 }$.
Find the set of values of $\lambda$ such that $C$ has two vertical asymptotes.
Find the $x$-coordinates of the points of intersection of $C$ with
(a) the $x$-axis,
(b) the horizontal asymptote.
Sketch $C$ in each of the cases
(a) $\lambda < - 2$,
(b) $\lambda > 2$.

CAIE FP1 2010 November Q12 OR

The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) + \mu ( - \mathbf { i } + \mathbf { k } )$. Obtain a cartesian equation of $\Pi _ { 1 }$ in the form $p x + q y + r z = d$. The plane $\Pi _ { 2 }$ has equation $\mathbf { r } . ( \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) = 12$. Find a vector equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$. The line $l$ passes through the point $A$ with position vector $a \mathbf { i } + ( 2 a + 1 ) \mathbf { j } - 3 \mathbf { k }$ and is parallel to $3 c \mathbf { i } - 3 \mathbf { j } + c \mathbf { k }$, where $a$ and $c$ are positive constants. Given that the perpendicular distance from $A$ to $\Pi _ { 1 }$ is $\frac { 15 } { \sqrt { } 6 }$ and that the acute angle between $l$ and $\Pi _ { 1 }$ is $\sin ^ { - 1 } \left( \frac { 2 } { \sqrt { } 6 } \right)$, find the values of $a$ and $c$. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

CAIE FP1 2011 November Q1

1 Verify that $\frac { 1 } { n ^ { 2 } } - \frac { 1 } { ( n + 1 ) ^ { 2 } } = \frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } }$. Let $S _ { N } = \sum _ { r = 1 } ^ { N } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$. Express $S _ { N }$ in terms of $N$. Let $S = \lim _ { N \rightarrow \infty } S _ { N }$. Find the least value of $N$ such that $S - S _ { N } < 10 ^ { - 16 }$.

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