Questions - CAIE - A-Level Maths

CAIE FP1 2014 June Q10

10 It is given that $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 2 n } x } { \cos x } \mathrm {~d} x$, where $n \geqslant 0$. Show that $$I _ { n } - I _ { n + 1 } = \frac { 2 ^ { - \left( n + \frac { 1 } { 2 } \right) } } { 2 n + 1 }$$ Hence show that $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 6 } x } { \cos x } \mathrm {~d} x = \ln ( 1 + \sqrt { } 2 ) - \frac { 73 } { 120 } \sqrt { } 2$.

CAIE FP1 2014 June Q11

11 The line $l _ { 1 }$ passes through the points $A ( 2,3 , - 5 )$ and $B ( 8,7 , - 13 )$. The line $l _ { 2 }$ passes through the points $C ( - 2,1,8 )$ and $D ( 3 , - 1,4 )$. Find the shortest distance between the lines $l _ { 1 }$ and $l _ { 2 }$. The plane $\Pi _ { 1 }$ passes through the points $A , B$ and $D$. The plane $\Pi _ { 2 }$ passes though the points $A , C$ and $D$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in degrees.

CAIE FP1 2014 June Q12 EITHER

The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = ( 2 - t ) ^ { \frac { 1 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 2 .$$ Find

$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ in terms of $t$,
the mean value of $y$ with respect to $x$ over the interval $0 \leqslant x \leqslant 4$,
the $y$-coordinate of the centroid of the region enclosed by $C$, the $x$-axis and the $y$-axis.

CAIE FP1 2014 June Q12 OR

The curve $C$ has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + d }$$ where $a , b , c$ and $d$ are constants. The curve cuts the $y$-axis at $( 0 , - 2 )$ and has asymptotes $x = 2$ and $y = x + 1$.

Write down the value of $d$.
Determine the values of $a , b$ and $c$.
Show that, at all points on $C$, either $y \leqslant 3 - 2 \sqrt { 6 }$ or $y \geqslant 3 + 2 \sqrt { 6 }$.

CAIE FP1 2015 June Q1

1 Use the List of Formulae (MF10) to show that $\sum _ { r = 1 } ^ { 13 } \left( 3 r ^ { 2 } - 5 r + 1 \right)$ and $\sum _ { r = 0 } ^ { 9 } \left( r ^ { 3 } - 1 \right)$ have the same numerical value.

CAIE FP1 2015 June Q2

2 Find the value of the constant $k$ for which the system of equations $$\begin{aligned} 2 x - 3 y + 4 z & = 1 \\ 3 x - y & = 2 \\ x + 2 y + k z & = 1 \end{aligned}$$ does not have a unique solution. For this value of $k$, solve the system of equations.

CAIE FP1 2015 June Q3

3 The sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is such that $a _ { 1 } > 5$ and $a _ { n + 1 } = \frac { 4 a _ { n } } { 5 } + \frac { 5 } { a _ { n } }$ for every positive integer $n$.
Prove by mathematical induction that $a _ { n } > 5$ for every positive integer $n$. Prove also that $a _ { n } > a _ { n + 1 }$ for every positive integer $n$.

CAIE FP1 2015 June Q4

4 The roots of the cubic equation $x ^ { 3 } - 7 x ^ { 2 } + 2 x - 3 = 0$ are $\alpha , \beta$ and $\gamma$. Find the values of

$\frac { 1 } { ( \alpha \beta ) ( \beta \gamma ) ( \gamma \alpha ) }$,
$\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }$,
$\frac { 1 } { \alpha ^ { 2 } \beta \gamma } + \frac { 1 } { \alpha \beta ^ { 2 } \gamma } + \frac { 1 } { \alpha \beta \gamma ^ { 2 } }$. Deduce a cubic equation, with integer coefficients, having roots $\frac { 1 } { \alpha \beta } , \frac { 1 } { \beta \gamma }$ and $\frac { 1 } { \gamma \alpha }$.

CAIE FP1 2015 June Q5

5 The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations $$\begin{array} { l l } C _ { 1 } : & r = \frac { 1 } { \sqrt { 2 } } , \quad \text { for } 0 \leqslant \theta < 2 \pi \\ C _ { 2 } : & r = \sqrt { } \left( \sin \frac { 1 } { 2 } \theta \right) , \quad \text { for } 0 \leqslant \theta \leqslant \pi \end{array}$$ Find the polar coordinates of the point of intersection of $C _ { 1 }$ and $C _ { 2 }$. Sketch $C _ { 1 }$ and $C _ { 2 }$ on the same diagram. Find the exact value of the area of the region enclosed by $C _ { 1 } , C _ { 2 }$ and the half-line $\theta = 0$.

CAIE FP1 2015 June Q6

6 A curve has equation $x ^ { 2 } - 6 x y + 25 y ^ { 2 } = 16$. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ at the point $( 3,1 )$. By finding the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $( 3,1 )$, determine the nature of this turning point.

CAIE FP1 2015 June Q7

7 Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x$, where $n$ is a non-negative integer. Show that $$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2$$ Find the exact value of $I _ { 4 }$.

CAIE FP1 2015 June Q8

8 By considering $\sum _ { r = 1 } ^ { n } z ^ { 2 r - 1 }$, where $z = \cos \theta + \mathrm { i } \sin \theta$, show that, if $\sin \theta \neq 0$, $$\sum _ { r = 1 } ^ { n } \sin ( 2 r - 1 ) \theta = \frac { \sin ^ { 2 } n \theta } { \sin \theta }$$ Deduce that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) \cos \left[ \frac { ( 2 r - 1 ) \pi } { 2 n } \right] = - \operatorname { cosec } \left( \frac { \pi } { 2 n } \right) \cot \left( \frac { \pi } { 2 n } \right)$$

CAIE FP1 2015 June Q9

9 The curve $C$ has parametric equations $$x = 4 t + 2 t ^ { \frac { 3 } { 2 } } , \quad y = 4 t - 2 t ^ { \frac { 3 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 4$$ Find the arc length of $C$, giving your answer correct to 3 significant figures. Find the mean value of $y$ with respect to $x$ over the interval $0 \leqslant x \leqslant 32$.

CAIE FP1 2015 June Q10

10 The matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 2 & - 3 \\ 2 & 2 & 3 \\ - 3 & 3 & 3 \end{array} \right)$$ The matrix $\mathbf { A }$ has an eigenvector $\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)$. Find the corresponding eigenvalue. The matrix $\mathbf { A }$ also has eigenvalues 4 and 6. Find corresponding eigenvectors. Hence find a matrix $\mathbf { P }$ such that $\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$, where $\mathbf { D }$ is a diagonal matrix which is to be determined. The matrix $\mathbf { B }$ is such that $\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }$, where $$\mathbf { Q } = \left( \begin{array} { r r r } 4 & 11 & 5 \\ 1 & 4 & 2 \\ 1 & 2 & 1 \end{array} \right)$$ By using the expression $\mathbf { P D P } ^ { - 1 }$ for $\mathbf { A }$, find the set of eigenvalues and a corresponding set of eigenvectors for $\mathbf { B }$.
[0pt] [Question 11 is printed on the next page.]

CAIE FP1 2015 June Q11 EITHER

Show that the substitution $v = \frac { 1 } { y }$ reduces the differential equation $$\frac { 2 } { y ^ { 3 } } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - \frac { 1 } { y ^ { 2 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { 2 } { y ^ { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \frac { 5 } { y } = 17 + 6 x - 5 x ^ { 2 }$$ to the differential equation $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 5 v = 17 + 6 x - 5 x ^ { 2 }$$ Hence find $y$ in terms of $x$, given that when $x = 0 , y = \frac { 1 } { 2 }$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 1$.

CAIE FP1 2015 June Q11 OR

The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $\mathbf { r } = 8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } )$ and $\mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 14 \mathbf { k } + \mu ( 2 \mathbf { j } - 3 \mathbf { k } )$ respectively. 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 $l _ { 2 }$. Find the position vector of the point $P$ and the position vector of the point $Q$. The points with position vectors $8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ and $5 \mathbf { i } + 3 \mathbf { j } - 14 \mathbf { k }$ are denoted by $A$ and $B$ respectively. Find

$\overrightarrow { A P } \times \overrightarrow { A Q }$ and hence the area of the triangle $A P Q$,
the volume of the tetrahedron $A P Q B$. (You are given that the volume of a tetrahedron is $\frac { 1 } { 3 } \times$ area of base × perpendicular height.) \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.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series.
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 2017 June Q1

1 It is given that $\sum _ { r = 1 } ^ { n } u _ { r } = n ^ { 2 } ( 2 n + 3 )$, where $n$ is a positive integer.

Find $\sum _ { r = n + 1 } ^ { 2 n } u _ { r }$.
Find $u _ { r }$.

CAIE FP1 2017 June Q2

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

CAIE FP1 2017 June Q3

3 A curve $C$ has equation $\tan y = x$, for $x > 0$.

Use implicit differentiation to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - 2 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$
Hence find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $\left( 1 , \frac { 1 } { 4 } \pi \right)$ on $C$.

CAIE FP1 2017 June Q4

4

Find the value of $k$ for which the set of linear equations $$\begin{aligned} x + 3 y + k z & = 4 \\ 4 x - 2 y - 10 z & = - 5 \\ x + y + 2 z & = 1 \end{aligned}$$ has no unique solution.
For this value of $k$, find the set of possible solutions, giving your answer in the form $$\left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \mathbf { a } + t \mathbf { b } ,$$ where $\mathbf { a }$ and $\mathbf { b }$ are vectors and $t$ is a scalar.

CAIE FP1 2017 June Q5

5 The matrix $\mathbf { A }$, given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & - 2 \\ 6 & 4 & - 6 \\ 6 & 5 & - 7 \end{array} \right)$$ has eigenvalues $1 , - 1$ and - 2 .

Find a set of corresponding eigenvectors.
The matrix $\mathbf { B }$ is given by $\mathbf { B } = \mathbf { A } - 2 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix. Write down the eigenvalues of $\mathbf { B }$, and state a set of corresponding eigenvectors.

CAIE FP1 2017 June Q6

6 Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x$.

Prove that, for $n \geqslant 2$, $$I _ { n } + n ( n - 1 ) I _ { n - 2 } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } .$$
Calculate the exact value of $I _ { 1 }$ and deduce the exact value of $I _ { 3 }$.

CAIE FP1 2017 June Q7

7 By finding a cubic equation whose roots are $\alpha , \beta$ and $\gamma$, solve the set of simultaneous equations $$\begin{aligned} \alpha + \beta + \gamma & = - 1 , \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 29 , \\ \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - 1 . \end{aligned}$$

CAIE FP1 2017 June Q8

8

Let $z = \cos \theta + \mathrm { i } \sin \theta$. Show that $z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta$ and hence express $16 \sin ^ { 5 } \theta$ in the form $\sin 5 \theta + p \sin 3 \theta + q \sin \theta$, where $p$ and $q$ are integers to be determined.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } 16 \sin ^ { 5 } \theta \mathrm {~d} \theta$.

CAIE FP1 2017 June Q9

9 The curve $C$ has equation $y = \frac { x ^ { 2 } - 3 x + 6 } { 1 - x }$.

Find the equations of the asymptotes of $C$.
Find the coordinates of the turning points of $C$.
Find the coordinates of any intersections with the coordinate axes.
Sketch $C$.

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