Questions - CAIE FP1 - A-Level Maths

CAIE FP1 2011 November Q2

2 Prove by mathematical induction that, for all positive integers $n$, $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \frac { 1 } { 2 x + 3 } \right) = ( - 1 ) ^ { n } \frac { n ! 2 ^ { n } } { ( 2 x + 3 ) ^ { n + 1 } }$$

CAIE FP1 2011 November Q3

3 The equation $$x ^ { 3 } + 5 x ^ { 2 } - 3 x - 15 = 0$$ has roots $\alpha , \beta , \gamma$. Find the value of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$. Hence show that the matrix $\left( \begin{array} { c c c } 1 & \alpha & \beta
\alpha & 1 & \gamma
\beta & \gamma & 1 \end{array} \right)$ is singular.

CAIE FP1 2011 November Q4

4 A curve has parametric equations $$x = 2 \sin 2 t , \quad y = 3 \cos 2 t$$ for $0 < t < \frac { 1 } { 2 } \pi$. For the point on the curve where $t = \frac { 1 } { 3 } \pi$, find the value of

$\frac { \mathrm { d } y } { \mathrm {~d} x }$,
$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.

CAIE FP1 2011 November Q5

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

CAIE FP1 2011 November Q6

6 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = \sin 2 t$$ Describe the behaviour of $x$ as $t \rightarrow \infty$, justifying your answer.

CAIE FP1 2011 November Q7

7 Show that $\frac { \mathrm { d } } { \mathrm { d } t } \left( t \left( 1 + t ^ { 3 } \right) ^ { n } \right) = ( 3 n + 1 ) \left( 1 + t ^ { 3 } \right) ^ { n } - 3 n \left( 1 + t ^ { 3 } \right) ^ { n - 1 }$. Let $I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + t ^ { 3 } \right) ^ { n } \mathrm {~d} t$. Using the above result, or otherwise, show that $$( 3 n + 1 ) I _ { n } = 2 ^ { n } + 3 n I _ { n - 1 }$$ Hence evaluate $I _ { 3 }$.

CAIE FP1 2011 November Q8

8 The curve $C$ has polar equation $r = 1 + \sin \theta$ for $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. Draw a sketch of $C$. The area of the region enclosed by the initial line, the half-line $\theta = \frac { 1 } { 2 } \pi$, and the part of $C$ for which $\theta$ is positive, is denoted by $A _ { 1 }$. The area of the region enclosed by the initial line, and the part of $C$ for which $\theta$ is negative, is denoted by $A _ { 2 }$. Find the ratio $A _ { 1 } : A _ { 2 }$, giving your answer correct to 1 decimal place.

CAIE FP1 2011 November Q9

9 Find a cartesian equation of the plane $\Pi$ containing the lines $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - 7 \mathbf { j } + 10 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )$$ The line $l$ passes through the point $P$ with position vector $6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }$ and is parallel to the vector $2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }$. Find

the position vector of the point where $l$ meets $\Pi$,
the perpendicular distance from $P$ to $\Pi$,
the acute angle between $l$ and $\Pi$.

CAIE FP1 2011 November Q10

10 A curve $C$ has equation $$y = \frac { 5 \left( x ^ { 2 } - x - 2 \right) } { x ^ { 2 } + 5 x + 10 }$$ Find the coordinates of the points of intersection of $C$ with the axes. Show that, for all real values of $x , - 1 \leqslant y \leqslant 15$. Sketch $C$, stating the coordinates of any turning points and the equation of the horizontal asymptote.
[0pt] [Question 11 is printed on the next page.]

CAIE FP1 2011 November Q11 EITHER

The curve $C$ has equation $y = \frac { 1 } { 3 } x ^ { \frac { 1 } { 2 } } ( 3 - x )$, for $0 \leqslant x \leqslant 3$. Find the mean value of $y$ with respect to $x$ over the interval $0 \leqslant x \leqslant 3$. Show that $$\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( x ^ { - \frac { 1 } { 2 } } + x ^ { \frac { 1 } { 2 } } \right)$$ where $s$ denotes arc length, and find the arc length of $C$. Find the area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis.

CAIE FP1 2011 November Q11 OR

Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 1 & 2
0 & 2 & 2
- 1 & 1 & 3 \end{array} \right)$$ The linear transformation $\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }$ is defined by $\mathbf { x } \mapsto \mathbf { A x }$. Let $\mathbf { e } , \mathbf { f }$ be two linearly independent eigenvectors of $\mathbf { A }$, with corresponding eigenvalues $\lambda$ and $\mu$ respectively, and let $\Pi$ be the plane, through the origin, containing $\mathbf { e }$ and $\mathbf { f }$. By considering the parametric equation of $\Pi$, show that all points of $\Pi$ are mapped by T onto points of $\Pi$. Find cartesian equations of three planes, each with the property that all points of the plane are mapped by T onto points of the same plane.

CAIE FP1 2012 November Q1

1 Find the cartesian equation corresponding to the polar equation $r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)$. Sketch the the graph of $r = ( \sqrt { } 2 ) \sec \left( \theta - \frac { 1 } { 4 } \pi \right)$, for $- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi$, indicating clearly the polar coordinates of the intersection with the initial line.

CAIE FP1 2012 November Q2

2 The curve $C$ has equation $y = 2 x ^ { \frac { 1 } { 2 } }$ for $0 \leqslant x \leqslant 4$. Find

the mean value of $y$ with respect to $x$ for $0 \leqslant x \leqslant 4$,
the $y$-coordinate of the centroid of the region enclosed by $C$, the line $x = 4$ and the $x$-axis.

CAIE FP1 2012 November Q3

3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 26 t ^ { 2 } + 3 t + 13$$

CAIE FP1 2012 November Q4

4 Let $\mathrm { f } ( r ) = r ( r + 1 ) ( r + 2 )$. Show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = 3 r ( r + 1 )$$ Hence show that $\sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )$. Using the standard result for $\sum _ { r = 1 } ^ { n } r$, deduce that $\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$. Find the sum of the series $$1 ^ { 2 } + 2 \times 2 ^ { 2 } + 3 ^ { 2 } + 2 \times 4 ^ { 2 } + 5 ^ { 2 } + 2 \times 6 ^ { 2 } + \ldots + 2 ( n - 1 ) ^ { 2 } + n ^ { 2 }$$ where $n$ is odd.

CAIE FP1 2012 November Q5

5 Let $I _ { n }$ denote $\int _ { 0 } ^ { \infty } x ^ { n } \mathrm { e } ^ { - 2 x } \mathrm {~d} x$. Show that $I _ { n } = \frac { 1 } { 2 } n I _ { n - 1 }$, for $n \geqslant 1$. Prove by mathematical induction that, for all positive integers $n , I _ { n } = \frac { n ! } { 2 ^ { n + 1 } }$.

CAIE FP1 2012 November Q6

6 Use de Moivre's theorem to show that $$\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1$$ Without using a calculator, verify that $\cos 4 \theta = - \cos 3 \theta$ for each of the values $\theta = \frac { 1 } { 7 } \pi , \frac { 3 } { 7 } \pi , \frac { 5 } { 7 } \pi , \pi$. Using the result $\cos 3 \theta = 4 \cos ^ { 3 } \theta - 3 \cos \theta$, show that the roots of the equation $$8 c ^ { 4 } + 4 c ^ { 3 } - 8 c ^ { 2 } - 3 c + 1 = 0$$ are $\cos \frac { 1 } { 7 } \pi , \cos \frac { 3 } { 7 } \pi , \cos \frac { 5 } { 7 } \pi , - 1$. Deduce that $\cos \frac { 1 } { 7 } \pi + \cos \frac { 3 } { 7 } \pi + \cos \frac { 5 } { 7 } \pi = \frac { 1 } { 2 }$.

CAIE FP1 2012 November Q7

7 The curve $C$ has equation $$y = \lambda x + \frac { x } { x - 2 }$$ where $\lambda$ is a non-zero constant. Find the equations of the asymptotes of $C$. Show that $C$ has no turning points if $\lambda < 0$. Sketch $C$ in the case $\lambda = - 1$, stating the coordinates of the intersections with the axes.

CAIE FP1 2012 November Q8

8 The curve $C$ has parametric equations $$x = \frac { 1 } { 3 } t ^ { 3 } - \ln t , \quad y = \frac { 4 } { 3 } t ^ { \frac { 3 } { 2 } }$$ for $1 \leqslant t \leqslant 3$. Find the arc length of $C$. Find also the area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis.

CAIE FP1 2012 November Q9

9 The plane $\Pi$ has equation $$\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ The line $l$, which does not lie in $\Pi$, has equation $$\mathbf { r } = 3 \mathbf { i } + 6 \mathbf { j } + 12 \mathbf { k } + t ( 8 \mathbf { i } + 5 \mathbf { j } - 8 \mathbf { k } )$$ Show that $l$ is parallel to $\Pi$. Find the position vector of the point at which the line with equation $\mathbf { r } = 5 \mathbf { i } - 4 \mathbf { j } + 7 \mathbf { k } + s ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )$ meets $\Pi$. Find the perpendicular distance from the point with position vector $9 \mathbf { i } + 11 \mathbf { j } + 2 \mathbf { k }$ to $l$.

CAIE FP1 2012 November Q10

10 Write down the eigenvalues of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16
0 & 2 & 3
0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let $n$ be a positive integer. Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $$\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$ Find $\mathbf { P } ^ { - 1 }$ and $\mathbf { A } ^ { n }$. Hence find $\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)$.

CAIE FP1 2012 November Q11 EITHER

The roots of the equation $x ^ { 4 } - 3 x ^ { 2 } + 5 x - 2 = 0$ are $\alpha , \beta , \gamma , \delta$, and $\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }$ is denoted by $S _ { n }$. Show that $$S _ { n + 4 } - 3 S _ { n + 2 } + 5 S _ { n + 1 } - 2 S _ { n } = 0$$ Find the values of

$S _ { 2 }$ and $S _ { 4 }$,
$S _ { 3 }$ and $S _ { 5 }$. Hence find the value of $$\alpha ^ { 2 } \left( \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } \right) + \beta ^ { 2 } \left( \gamma ^ { 3 } + \delta ^ { 3 } + \alpha ^ { 3 } \right) + \gamma ^ { 2 } \left( \delta ^ { 3 } + \alpha ^ { 3 } + \beta ^ { 3 } \right) + \delta ^ { 2 } \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) .$$

CAIE FP1 2012 November Q11 OR

The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }$ is represented by the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4
3 & 4 & 6 & 1
- 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is $R$. In any order,

show that the dimension of $R$ is 2 ,
find a basis for $R$ and obtain a cartesian equation for $R$,
find a basis for the null space of T . The vector $\left( \begin{array} { l } 8
7
k \end{array} \right)$ belongs to $R$. Find the value of $k$ and, with this value of $k$, find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { l } 8
7
k \end{array} \right) .$$

CAIE FP1 2013 November Q6

6 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 3 & - 1 & 2
4 & - 10 & 0 & 2
1 & - 1 & 3 & - 4
5 & - 12 & 1 & 1 \end{array} \right)$$ Find, in either order, the rank of $\mathbf { M }$ and a basis for the null space $K$ of T. Evaluate $$\mathbf { M } \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right)$$ and hence show that every solution of $$\mathbf { M x } = \left( \begin{array} { r } 2
16
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }$$ where $\lambda$ and $\mu$ are real numbers and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$.

CAIE FP1 2013 November Q10

10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }$$ where $\lambda$ and $\mu$ are real numbers and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$. 7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue. Find the eigenvalues of the matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix. 8 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2
3
- 1 \end{array} \right) + s \left( \begin{array} { l } 1
0
1 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
- 2 \end{array} \right)$. Find a cartesian equation of $\Pi _ { 1 }$. The plane $\Pi _ { 2 }$ has equation $2 x - y + z = 10$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$. 9 The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis. Find the coordinates of the centroid of the region bounded by $C$, the $x$-axis and the line $x = 1$. 10 The curve $C$ has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 }$$ where $p$ is a positive constant and $p \neq 3$.

Obtain the equations of the asymptotes of $C$.
Find the value of $p$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case.
For the case $p = 1$, show that $C$ has no turning points, and sketch $C$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis.

Questions — CAIE FP1 (549 questions)

Browse by module

Browse by board