Questions - CAIE FP1 - A-Level Maths

$\lambda \neq 0$,
the matrix $\mathbf { A } ^ { - 1 }$ has $\lambda ^ { - 1 }$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. The $3 \times 3$ matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 2 & - 4
0 & - 1 & 5
0 & 0 & 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 3 \mathbf { I } ) ^ { - 1 }$$ where $\mathbf { I }$ is the $3 \times 3$ identity matrix. Find a non-singular matrix $\mathbf { P }$, and a diagonal matrix $\mathbf { D }$, such that $\mathbf { B } = \mathbf { P D P } ^ { - 1 }$. \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. 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 2014 November Q11 OR

The square 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 $3 \times 3$ matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 2 & - 4
0 & - 1 & 5
0 & 0 & 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 3 \mathbf { I } ) ^ { - 1 }$$ where $\mathbf { I }$ is the $3 \times 3$ identity matrix. Find a non-singular matrix $\mathbf { P }$, and a diagonal matrix $\mathbf { D }$, such that $\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$. \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. 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 2016 November Q1

1 Use the method of differences to find $\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }$. Deduce the value of $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }$.

CAIE FP1 2016 November Q2

2 Find the cubic equation with roots $\alpha , \beta$ and $\gamma$ such that $$\begin{aligned} \alpha + \beta + \gamma & = 3
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 1
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = - 30 \end{aligned}$$ giving your answer in the form $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers to be found.

CAIE FP1 2016 November Q3

3 Find a matrix $\mathbf { A }$ whose eigenvalues are $- 1,1,2$ and for which corresponding eigenvectors are $$\left( \begin{array} { l } 1
0
0 \end{array} \right) , \quad \left( \begin{array} { l } 1
1
0 \end{array} \right) , \quad \left( \begin{array} { l } 0
1
1 \end{array} \right) ,$$ respectively.

CAIE FP1 2016 November Q4

4 Using factorials, show that $\binom { n } { r - 1 } + \binom { n } { r } = \binom { n + 1 } { r }$. Hence prove by mathematical induction that $$( a + x ) ^ { n } = \binom { n } { 0 } a ^ { n } + \binom { n } { 1 } a ^ { n - 1 } x + \ldots + \binom { n } { r } a ^ { n - r } x ^ { r } + \ldots + \binom { n } { n } x ^ { n }$$ for every positive integer $n$.

CAIE FP1 2016 November Q5

5 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 3 & 5 & 7
2 & 8 & 7 & 9
3 & 13 & 9 & 11
6 & 24 & 21 & 27 \end{array} \right)$$ Find

the rank of $\mathbf { A }$,
a basis for the range space of T ,
a basis for the null space of T .

CAIE FP1 2016 November Q6

6 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 7 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 116 \sin 2 t$$ State an approximate solution for large positive values of $t$.

CAIE FP1 2016 November Q7

7 The curve $C$ has equation $y = \mathrm { e } ^ { - 2 x }$. Find, giving your answers correct to 3 significant figures,

the mean value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ over the interval $0 \leqslant x \leqslant 2$,
the coordinates of the centroid of the region bounded by $C$, $x = 0$, $x = 2$ and $y = 0$.

CAIE FP1 2016 November Q8

8 A curve $C$ has equation $x ^ { 2 } + 4 x y - y ^ { 2 } + 20 = 0$. Show that, at stationary points on $C , x = - 2 y$. Find the coordinates of the stationary points on $C$, and determine their nature by considering the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the stationary points.

CAIE FP1 2016 November Q9

9 Evaluate $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x$. Given that $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x$, prove that, for $n > 1$, $$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$ By first using the substitution $x = \cos ^ { - 1 } u$, find the value of $$\int _ { 0 } ^ { 1 } \left( \cos ^ { - 1 } u \right) ^ { 3 } \mathrm {~d} u$$ giving your answer in an exact form.

CAIE FP1 2016 November Q10

10 Let $z = \cos \theta + \mathrm { i } \sin \theta$. Show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ By considering $\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }$, show that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta - 2 \cos 4 \theta - \cos 2 \theta + 2 ) .$$ Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 4 } \theta \cos ^ { 2 } \theta d \theta$.
[0pt] [Question 11 is printed on the next page.]

CAIE FP1 2016 November Q11 EITHER

The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $$\mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + s ( 3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } - \mathbf { k } )$$ 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 $l _ { 2 }$. Show that the position vector of $P$ is $3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ and find the position vector of $Q$. Find, in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }$, an equation of the plane $\Pi$ which passes through $P$ and is perpendicular to $l _ { 1 }$. The plane $\Pi$ meets the plane $\mathbf { r } = p \mathbf { i } + q \mathbf { j }$ in the line $l _ { 3 }$. Find a vector equation of $l _ { 3 }$.

CAIE FP1 2016 November Q11 OR

A curve $C$ has parametric equations $$x = 1 - 3 t ^ { 2 } , \quad y = t \left( 1 - 3 t ^ { 2 } \right) , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }$$ Show that $\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + 9 t ^ { 2 } \right) ^ { 2 }$. Hence find

the arc length of $C$,
the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis. Use the fact that $t = \frac { y } { x }$ to find a cartesian equation of $C$. Hence show that the polar equation of $C$ is $r = \sec \theta \left( 1 - 3 \tan ^ { 2 } \theta \right)$, and state the domain of $\theta$. Find the area of the region enclosed between $C$ and the initial line. \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 November Q1

1 Find $\sum _ { r = 1 } ^ { n } ( 4 r - 3 ) ( 4 r + 1 )$, giving your answer in its simplest form.

CAIE FP1 2017 November Q2

2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 4 - 5 t ^ { 2 }$$

CAIE FP1 2017 November Q3

Show that $\frac { \mathrm { d } ^ { n + 1 } } { \mathrm {~d} x ^ { n + 1 } } \left( x ^ { n + 1 } \ln x \right) = \frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x ^ { n } + ( n + 1 ) x ^ { n } \ln x \right)$.
Prove by mathematical induction that, for all positive integers $n$, $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x ^ { n } \ln x \right) = n ! \left( \ln x + 1 + \frac { 1 } { 2 } + \ldots + \frac { 1 } { n } \right)$$

CAIE FP1 2017 November Q4

4 The cubic equation $2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 10 = 0$ has roots $\alpha , \beta$ and $\gamma$.

Find the value of $( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )$.
Find the value of $( \beta + \gamma ) ( \gamma + \alpha ) ( \alpha + \beta )$.

CAIE FP1 2017 November Q5

5 The curve $C$ has equation $2 x ^ { 3 } + 3 x ^ { 2 } y - 3 y ^ { 3 } - 16 = 0$.

Find the coordinates of the point $A$ on $C$ at which $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ and $x \neq 0$.
Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$.

CAIE FP1 2017 November Q6

6 The points $A , B$ and $C$ have position vectors $2 \mathbf { i } - \mathbf { j } + \mathbf { k } , 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$ and $- \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$ respectively.

Find the area of the triangle $A B C$.
....................................................................................................................................
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\includegraphics[max width=\textwidth, alt={}, center]{9221f480-4af6-44be-a535-d2ceb0f8b5d2-08_71_1563_868_331}
Find the perpendicular distance of the point $A$ from the line $B C$.
Find the cartesian equation of the plane through $A , B$ and $C$.

CAIE FP1 2017 November Q7

7 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & - 1 & - 2 & 3
5 & - 3 & - 4 & 25
6 & - 4 & - 6 & 28
7 & - 5 & - 8 & 31 \end{array} \right)$$

Find the rank of $\mathbf { A }$ and a basis for the null space of T .
Find the matrix product $\mathbf { A } \left( \begin{array} { r } - 1
1
- 1
1 \end{array} \right)$ and hence find the general solution of the equation $\mathbf { A } \mathbf { x } = \left( \begin{array} { r } 3
21
24
27 \end{array} \right)$.
$8 \quad$ Let $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x$ for $n > 0$.

CAIE FP1 2017 November Q9

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]{9221f480-4af6-44be-a535-d2ceb0f8b5d2-14_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 2017 November Q10

Use de Moivre's theorem to show that $$\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta$$
Hence explain why the roots of the equation $16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0$ are $x = \pm \sin \frac { 1 } { 5 } \pi$ and $x = \pm \sin \frac { 2 } { 5 } \pi$.
Without using a calculator, find the exact values of $$\sin \frac { 1 } { 5 } \pi \sin \frac { 2 } { 5 } \pi \sin \frac { 3 } { 5 } \pi \sin \frac { 4 } { 5 } \pi \quad \text { and } \quad \sin ^ { 2 } \left( \frac { 1 } { 5 } \pi \right) + \sin ^ { 2 } \left( \frac { 2 } { 5 } \pi \right) .$$

CAIE FP1 2017 November Q11 EITHER

The vector $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A }$, with corresponding eigenvalue $\lambda$, and is also an eigenvector of the matrix $\mathbf { B }$, with corresponding eigenvalue $\mu$. Show that $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A B }$ with corresponding eigenvalue $\lambda \mu$.
Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 0 & 1 & 3
3 & 2 & - 3
1 & 1 & 2 \end{array} \right) .$$
The matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { r r r } 3 & 6 & 1
1 & - 2 & - 1
6 & 6 & - 2 \end{array} \right) ,$$ has eigenvectors $\left( \begin{array} { r } 1
- 1
0 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right)$ and $\left( \begin{array} { l } 1
0
1 \end{array} \right)$. Find the eigenvalues of the matrix $\mathbf { A B }$, and state corresponding eigenvectors.

CAIE FP1 2017 November Q11 OR

The polar equation of a curve $C$ is $r = a ( 1 + \cos \theta )$ for $0 \leqslant \theta < 2 \pi$, where $a$ is a positive constant.

Sketch $C$.
Show that the cartesian equation of $C$ is $$x ^ { 2 } + y ^ { 2 } = a \left( x + \sqrt { } \left( x ^ { 2 } + y ^ { 2 } \right) \right)$$
Find the area of the sector of $C$ between $\theta = 0$ and $\theta = \frac { 1 } { 3 } \pi$.
Find the arc length of $C$ between the point where $\theta = 0$ and the point where $\theta = \frac { 1 } { 3 } \pi$.

Questions — CAIE FP1 (549 questions)

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