Questions - CAIE FP1 - A-Level Maths

CAIE FP1 2013 November Q16 EITHER

State the fifth roots of unity in the form $\cos \theta + \mathrm { i } \sin \theta$, where $- \pi < \theta \leqslant \pi$. Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form $\cos \theta \pm \mathrm { i } \sin \theta$. Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$

CAIE FP1 2013 November Q16 OR

Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that $v = y ^ { 3 }$, show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for $y$ in terms of $x$, given that when $x = 0 , y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$. \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. 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 2013 November Q1

1 Express $\frac { 1 } { r ( r + 1 ) ( r - 1 ) }$ in partial fractions. Find $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$ State the value of $$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$

CAIE FP1 2013 November Q2

2 Show that the matrix $\left( \begin{array} { r r r } 1 & 4 & 2
3 & 0 & - 2
3 & - 3 & - 4 \end{array} \right)$ has no inverse. Solve the system of equations $$\begin{array} { r } x + 4 y + 2 z = 0
3 x - 2 z = 4
3 x - 3 y - 4 z = 5 \end{array}$$

CAIE FP1 2013 November Q3

3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 x ^ { 2 } + 8$$

CAIE FP1 2013 November Q4

4 A curve has parametric equations $$x = 2 \theta - \sin 2 \theta , \quad y = 1 - \cos 2 \theta , \quad \text { for } - 3 \pi \leqslant \theta \leqslant 3 \pi$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$$ except for certain values of $\theta$, which should be stated. Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $\theta = \frac { 1 } { 4 } \pi$.

CAIE FP1 2013 November Q5

5 The equation $$8 x ^ { 3 } + 36 x ^ { 2 } + k x - 21 = 0$$ where $k$ is a constant, has roots $a - d , a , a + d$. Find the numerical values of the roots and determine the value of $k$.

CAIE FP1 2013 November Q6

6 [In this question you may use, without proof, the formula $\int \sec x \mathrm {~d} x = \ln ( \sec x + \tan x ) + \operatorname { const }$.]

Let $y = \sec x$. Find the mean value of $y$ with respect to $x$ over the interval $\frac { 1 } { 6 } \pi \leqslant x \leqslant \frac { 1 } { 3 } \pi$.
The curve $C$ has equation $y = - \ln ( \cos x )$, for $0 \leqslant x \leqslant \frac { 1 } { 3 } \pi$. Find the arc length of $C$.

CAIE FP1 2013 November Q7

7 The curve $C$ has equation $$y = \frac { 2 x ^ { 2 } + 5 x - 1 } { x + 2 }$$ Find the equations of the asymptotes of $C$. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } > 2$ at all points on $C$. Sketch C.

CAIE FP1 2013 November Q8

8 The points $A , B , C$ have position vectors $$4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k } , \quad 5 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin $O$. Find a cartesian equation of the plane $A B C$. The point $D$ has position vector $6 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }$. Find the coordinates of $E$, the point of intersection of the line $O D$ with the plane $A B C$. Find the acute angle between the line $E D$ and the plane $A B C$.

CAIE FP1 2013 November Q9

9 Prove by mathematical induction that, for every positive integer $n$, $$( \cos \theta + i \sin \theta ) ^ { n } = \cos n \theta + i \sin n \theta$$ Express $\sin ^ { 5 } \theta$ in the form $p \sin 5 \theta + q \sin 3 \theta + r \sin \theta$, where $p , q$ and $r$ are rational numbers to be determined.

CAIE FP1 2013 November Q10

10 The curve $C$ has polar equation $r = 2 \sin \theta ( 1 - \cos \theta )$, for $0 \leqslant \theta \leqslant \pi$. Find $\frac { \mathrm { d } r } { \mathrm {~d} \theta }$ and hence find the polar coordinates of the point of $C$ that is furthest from the pole. Sketch $C$. Find the exact area of the sector from $\theta = 0$ to $\theta = \frac { 1 } { 4 } \pi$.

CAIE FP1 2013 November Q11 EITHER

Let $I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { n } \mathrm {~d} x$. Show that, for all integers $n$, $$( 2 n + 1 ) I _ { n } = 2 n I _ { n - 1 } + 2 ^ { n }$$ Evaluate $I _ { 0 }$ and hence find $I _ { 3 }$. Given that $I _ { - 1 } = \frac { 1 } { 4 } \pi$, find $I _ { - 3 }$.

CAIE FP1 2013 November Q11 OR

The vector $\mathbf { e }$ is an eigenvector of each of the $3 \times 3$ matrices $\mathbf { A }$ and $\mathbf { B }$, with corresponding eigenvalues $\lambda$ and $\mu$ respectively. Justifying your answer, state an eigenvalue of $\mathbf { A } + \mathbf { B }$. The matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 1 & - 6
1 & 0 & - 2
3 & - 1 & - 3 \end{array} \right)$$ has eigenvectors $\left( \begin{array} { l } 1
1
1 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { l } 2
0
1 \end{array} \right)$. Find the corresponding eigenvalues. The matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { r r r } 8 & - 2 & - 8
2 & 0 & - 4
4 & - 2 & - 4 \end{array} \right) ,$$ also has eigenvectors $\left( \begin{array} { l } 1
1
1 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { l } 2
0
1 \end{array} \right)$, for which $- 2,2,4$, respectively, are corresponding eigenvalues. The matrix $\mathbf { M }$ is given by $\mathbf { M } = \mathbf { A } + \mathbf { B } - 5 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix. State the eigenvalues of $\mathbf { M }$. Find matrices $\mathbf { R }$ and $\mathbf { S }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { M } ^ { 5 } = \mathbf { R D S }$.
[0pt] [You should show clearly all the elements of the matrices $\mathbf { R } , \mathbf { S }$ and $\mathbf { D }$.] \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 2014 November Q1

1 Given that $$u _ { k } = \frac { 1 } { \sqrt { } ( 2 k - 1 ) } - \frac { 1 } { \sqrt { } ( 2 k + 1 ) }$$ express $\sum _ { k = 13 } ^ { n } u _ { k }$ in terms of $n$. Deduce the value of $\sum _ { k = 13 } ^ { \infty } u _ { k }$.

CAIE FP1 2014 November Q2

2 A curve $C$ has parametric equations $$x = \mathrm { e } ^ { t } \cos t , \quad y = \mathrm { e } ^ { t } \sin t , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ Find the arc length of $C$.

CAIE FP1 2014 November Q3

3 It is given that $u _ { r } = r \times r !$ for $r = 1,2,3 , \ldots$. Let $S _ { n } = u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots + u _ { n }$. Write down the values of $$2 ! - S _ { 1 } , \quad 3 ! - S _ { 2 } , \quad 4 ! - S _ { 3 } , \quad 5 ! - S _ { 4 }$$ Conjecture a formula for $S _ { n }$. Prove, by mathematical induction, a formula for $S _ { n }$, for all positive integers $n$.

CAIE FP1 2014 November Q4

4 A curve $C$ has equation $y = \frac { 2 x ^ { 2 } + x - 1 } { x - 1 }$. Find the equations of the asymptotes of $C$. Show that there is no point on $C$ for which $1 < y < 9$.

CAIE FP1 2014 November Q5

5 Find the value of $a$ for which the system of equations $$\begin{aligned} & x - y + 2 z = 4
& x + a y - 3 z = b
& x - y + 7 z = 13 \end{aligned}$$ where $a$ and $b$ are constants, has no unique solution. Taking $a$ as the value just found,

find the general solution in the case $b = - 5$,
interpret the situation geometrically in the case $b \neq - 5$.

CAIE FP1 2014 November Q6

6 Use de Moivre's theorem to show that $$\cos 5 \theta \equiv \cos \theta \left( 16 \sin ^ { 4 } \theta - 12 \sin ^ { 2 } \theta + 1 \right)$$ By considering the equation $\cos 5 \theta = 0$, show that the exact value of $\sin ^ { 2 } \left( \frac { 1 } { 10 } \pi \right)$ is $\frac { 3 - \sqrt { 5 } } { 8 }$.

CAIE FP1 2014 November Q7

7 Let $I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x$. Show that, for all positive integers $n$, $$I _ { n } = n I _ { n - 1 } - 1$$ Find the exact value of $I _ { 4 }$. By considering the area of the region enclosed by the $x$-axis, the $y$-axis and the curve with equation $y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }$ in the interval $0 \leqslant x \leqslant 1$, show that $$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$

CAIE FP1 2014 November Q8

8 A circle has polar equation $r = a$, for $0 \leqslant \theta < 2 \pi$, and a cardioid has polar equation $r = a ( 1 - \cos \theta )$, for $0 \leqslant \theta < 2 \pi$, where $a$ is a positive constant. Draw sketches of the circle and the cardioid on the same diagram. Write down the polar coordinates of the points of intersection of the circle and the cardioid. Show that the area of the region that is both inside the circle and inside the cardioid is $$\left( \frac { 5 } { 4 } \pi - 2 \right) a ^ { 2 }$$

CAIE FP1 2014 November Q9

9 Given that $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + ( 2 - 3 x ) y = 10 \mathrm { e } ^ { 2 x }$$ and that $v = x y$, show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } - 3 v = 10 \mathrm { e } ^ { 2 x }$$ Find the general solution for $y$ in terms of $x$.

CAIE FP1 2014 November Q10

10 The line $l _ { 1 }$ is parallel to the vector $\mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }$ and passes through the point $A$, whose position vector is $3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$. The line $l _ { 2 }$ is parallel to the vector $- 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }$ and passes through the point $B$, whose position vector is $- 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$. 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 $l _ { 2 }$. Find

the length $P Q$,
the cartesian equation of the plane $\Pi$ containing $P Q$ and $l _ { 2 }$,
the perpendicular distance of $A$ from $\Pi$.

CAIE FP1 2014 November Q11 EITHER

The roots of the quartic equation $x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0$ are $\alpha , \beta , \gamma$ and $\delta$. Find the values of

$\alpha + \beta + \gamma + \delta$,
$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }$,
$\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } + \frac { 1 } { \delta }$,
$\frac { \alpha } { \beta \gamma \delta } + \frac { \beta } { \alpha \gamma \delta } + \frac { \gamma } { \alpha \beta \delta } + \frac { \delta } { \alpha \beta \gamma }$. Using the substitution $y = x + 1$, find a quartic equation in $y$. Solve this quartic equation and hence find the roots of the equation $x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0$.

Questions — CAIE FP1 (549 questions)

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