Questions - CAIE FP1 - A-Level Maths

(a) By differentiating $\ln y$ with respect to $t$, show that $\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a$.
(b) Write down $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }$.
Determine the set of values of $a$ for which the infinite series $$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$ is convergent.
A curve has parametric equations $$x = t ^ { a } , \quad y = a ^ { t }$$
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ in terms of $a$ and $t$, and show that, when $t = 2$, $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE FP1 2002 November Q1

1 Given that $$u _ { n } = \mathrm { e } ^ { n x } - \mathrm { e } ^ { ( n + 1 ) x }$$ find $\sum _ { n = 1 } ^ { N } \| _ { n }$ in terms of $N$ and $x$. Hence determine the set of values of $x$ for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity for cases where this exists.

CAIE FP1 2002 November Q2

2 The equation $$x ^ { 4 } + x ^ { 3 } + A x ^ { 2 } + 4 x - 2 = 0$$ where $A$ is a constant, has roots $\alpha , \beta , \gamma , \delta$. Find a polynomial equation whose roots are $$\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }$$ Given that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$$ find the value of $A$.

CAIE FP1 2002 November Q3

3 It is given that, for $n = 0,1,2,3 , \ldots$, $$a _ { n } = 17 ^ { 2 n } + 3 ( 9 ) ^ { n } + 20$$ Simplify $a _ { n + 1 } - a _ { n }$, and hence prove by induction that $a _ { n }$ is divisible by 24 for all $n \geqslant 0$.

CAIE FP1 2002 November Q4

4 It is given that, for $n \geqslant 0$, $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } e ^ { - x ^ { 2 } } d x$$

Find $I _ { 1 }$ in terms of c .
Show that $$I _ { n + 2 } = \frac { n + 1 } { 2 } I _ { n } - \frac { 1 } { 2 \mathrm { e } }$$
Find $I _ { 5 }$ in terms of $e$.

CAIE FP1 2002 November Q5

5 The curve $C$ has polar equation $r \theta = 1$, for $0 < \theta \leqslant 2 \pi$.

Use the fact that $\frac { \sin \theta } { \theta }$ tends to 1 as $\theta$ tends to 0 to show that the line with carresian equation $y = 1$ is an asymptote to $C$.
Sketch $C$. The points $P$ and $Q$ on $C$ correspond to $\theta = \frac { 1 } { 6 } \pi$ and $\theta = \frac { 1 } { 3 } \pi$ respectively.
Find the area of the sector $O P Q$, where $O$ is the origin.
Show that the length of the are $P Q$ is $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \sqrt { } \left( 1 + \theta ^ { 2 } \right) } { \theta ^ { 2 } } \mathrm {~d} \theta$$

CAIE FP1 2002 November Q6

6 A curve has equation $x ^ { 3 } + x y ^ { 2 } - y ^ { 3 } = 3$.

Show that there is no point of the curve at which $\frac { d y } { d x } = 0$.
Find the values of $\frac { d y } { d x }$ and $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 1 , - 1 )$.

CAIE FP1 2002 November Q7

7 Given that $z = \cos \theta + \mathrm { i } \sin \theta$, show that

$z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta$.
$z ^ { n } + z ^ { - n } = 2 \cos n \theta$. Hence show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta )$$ Find a similar expression for $\cos ^ { 6 } \theta$, and hence express $\cos ^ { 6 } \theta - \sin ^ { 6 } \theta$ in the fom $a \cos 2 \theta + b \cos 6 \theta$.

CAIE FP1 2002 November Q8

8 The value of the assets of a large commercial organisation at time $t$, measured in years, is $
) \left( 10 ^ { 8 } y + 10 ^ { 9 } \right)\(. The variables $y$ and $t$ are related by the differential equation $$\frac { d ^ { 2 } y } { d t ^ { 2 } } + 5 \frac { d y } { d t } + 6 y = 15 \cos 3 t - 3 \sin 3 t$$ Find $y$ in terms of $t$, given that $y = 3$ and $\frac { \mathrm { d } y } { \mathrm {~d} t } = - 2$ when $t = 0$. Show that, for large values of $t$, the value of the assets is less than \(
) 9.5 \times 10 ^ { 8 }$ for about a third of the time.

CAIE FP1 2002 November Q9

9 The planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$, which meet in the line $/$, have vector equations $$\begin{aligned} & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 1 } ( 2 \mathbf { i } + 3 \mathbf { k } ) + \phi _ { 1 } ( - 4 \mathbf { j } + 5 \mathbf { k } ) ,
& \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 2 } ( 3 \mathbf { j } + \mathbf { k } ) + \phi _ { 2 } ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) , \end{aligned}$$ respectively. Find a vector equation of the line $l$ in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$. Find a vector equation of the plane $\Pi _ { 3 }$ which contains $l$ and which passes through the point with position vector $4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$. Find also the equation of $\Pi _ { 3 }$ in the form $a x + b y + c z = d$. Deduce, or prove otherwise, that the system of equations $$\begin{aligned} & 6 x - 5 y - 4 z = - 32
& 5 x - y + 3 z = 24
& 9 x - 2 y + 5 z = 40 \end{aligned}$$ has an infinite number of solutions.

CAIE FP1 2002 November Q10

10 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { H }$, where $$\mathbf { H } = \left( \begin{array} { r r r r } 1 & 2 & - 3 & - 5
- 1 & 4 & 5 & 1
2 & 3 & 0 & - 3
- 3 & 5 & 7 & 2 \end{array} \right)$$

Find the dimension of the range space of T .
Find a basis for the null space of $T$.
It is given that $\mathbf { x }$ satisfies the equation $$\mathbf { H } \mathbf { x } = \left( \begin{array} { r } 2
- 10
- 1
- 15 \end{array} \right)$$ Using the fact that $$\mathbf { H } \left( \begin{array} { r } 1
- 3
1
- 2 \end{array} \right) = \left( \begin{array} { r } 2
- 10
- 1
- 15 \end{array} \right) ,$$ find the least possible value of $| \mathbf { x } |$.
[0pt] [For the vector $\mathbf { x } = \left( \begin{array} { c } x _ { 1 }
x _ { 2 }
x _ { 3 }
x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)$.]

CAIE FP1 2002 November Q11 EITHER

The vector $\mathbf { e }$ is an eigenvector of the square matrix $\mathbf { G }$. Show that

$\mathbf { e }$ is an eigenvector of $\mathbf { G } + k \mathbf { I }$, where $k$ is a scalar and $\mathbf { I }$ is an identity matrix,
$\mathbf { e }$ is an cigenvector of $\mathbf { G } ^ { 2 }$. Find the eigenvalues, and corresponding eigenvectors, of the matrices $\mathbf { A }$ and $\mathbf { B } ^ { 2 }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 3 & 0
1 & 0 & 1
- 1 & 3 & 2 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r } - 5 & - 3 & 0
1 & - 8 & 1
- 1 & 3 & - 6 \end{array} \right)$$

CAIE FP1 2002 November Q11 OR

The curve $C$ has equation $$y = \frac { ( x - a ) ( x - b ) } { x - c }$$ where $a , b , c$ are constants, and it is given that $0 < a < b < c$.

Express $y$ in the form $$x + P + \frac { Q } { x - c }$$ giving the constants $P$ and $Q$ in terms of $a , b$ and $c$.
Find the equations of the asymptotes of $C$.
Show that $C$ has two stationary points.
Given also that $a + b > c$, sketch $C$, showing the asymptotes and the coordinates of the points of intersection of $C$ with the axes.

CAIE FP1 2003 November Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{653d57aa-7775-4063-a8c9-11c8bc964fae-2_566_606_264_772} The curve $C$ has polar equation $$r = \theta ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { \theta ^ { 2 } / \pi }$$ where $0 \leqslant \theta \leqslant \pi$. The area of the finite region bounded by $C$ and the line $\theta = \beta$ is $\pi$ (see diagram). Show that $$\beta = ( \pi \ln 3 ) ^ { \frac { 1 } { 2 } }$$

CAIE FP1 2003 November Q2

2 Given that $$u _ { n } = \frac { 1 } { n ^ { 2 } - n + 1 } - \frac { 1 } { n ^ { 2 } + n + 1 } ,$$ find $S _ { N } = \sum _ { n = N + 1 } ^ { 2 N } u _ { n }$ in terms of $N$. Find a number $M$ such that $S _ { N } < 10 ^ { - 20 }$ for all $N > M$.

CAIE FP1 2003 November Q3

3 Three $n \times 1$ column vectors are denoted by $\mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 }$, and $\mathbf { M }$ is an $n \times n$ matrix. Show that if $\mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 }$ are linearly dependent then the vectors $\mathbf { M x } _ { 1 } , \mathbf { M x } _ { 2 } , \mathbf { M x } _ { 3 }$ are also linearly dependent. The vectors $\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }$ and the matrix $\mathbf { P }$ are defined as follows: $$\begin{gathered} \mathbf { y } _ { 1 } = \left( \begin{array} { l } 1
5
7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3

CAIE FP1 2003 November Q5

5
7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) , \quad \mathbf { y } _ { 3 } = \left( \begin{array} { r } 5
51
55 \end{array} \right)
\mathbf { P } = \left( \begin{array} { r r r } 1 & - 4 & 3
0 & 2 & 5
0 & 0 & - 7 \end{array} \right) . \end{gathered}$$

Show that $\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }$ are linearly dependent.
Find a basis for the linear space spanned by the vectors $\mathbf { P y } _ { 1 } , \mathbf { P y } _ { 2 } , \mathbf { P y } _ { 3 }$. 4 Given that $y = x \sin x$, find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$, simplifying your results as far as possible, and show that $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = - x \sin x + 6 \cos x$$ Use induction to establish an expression for $\frac { \mathrm { d } ^ { 2 n } y } { \mathrm {~d} x ^ { 2 n } }$, where $n$ is a positive integer. 5 The integral $I _ { n }$ is defined by $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x$$ By considering $\frac { \mathrm { d } } { \mathrm { d } x } \left( \tan x \sec ^ { n } x \right)$, or otherwise, show that $$( n + 1 ) I _ { n + 2 } = 2 ^ { \frac { 1 } { 2 } n } + n I _ { n }$$ Find the value of $I _ { 6 }$.

CAIE FP1 2003 November Q7

7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) , \quad \mathbf { y } _ { 3 } = \left( \begin{array} { r } 5
51
55 \end{array} \right)
\mathbf { P } = \left( \begin{array} { r r r } 1 & - 4 & 3
0 & 2 & 5
0 & 0 & - 7 \end{array} \right) . \end{gathered}$$

Show that $\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }$ are linearly dependent.
Find a basis for the linear space spanned by the vectors $\mathbf { P y } _ { 1 } , \mathbf { P y } _ { 2 } , \mathbf { P y } _ { 3 }$. 4 Given that $y = x \sin x$, find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$, simplifying your results as far as possible, and show that $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = - x \sin x + 6 \cos x$$ Use induction to establish an expression for $\frac { \mathrm { d } ^ { 2 n } y } { \mathrm {~d} x ^ { 2 n } }$, where $n$ is a positive integer. 5 The integral $I _ { n }$ is defined by $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x$$ By considering $\frac { \mathrm { d } } { \mathrm { d } x } \left( \tan x \sec ^ { n } x \right)$, or otherwise, show that $$( n + 1 ) I _ { n + 2 } = 2 ^ { \frac { 1 } { 2 } n } + n I _ { n }$$ Find the value of $I _ { 6 }$. 6 Find the sum of the squares of the roots of the equation $$x ^ { 3 } + x + 12 = 0$$ and deduce that only one of the roots is real. The real root of the equation is denoted by $\alpha$. Prove that $- 3 < \alpha < - 2$, and hence prove that the modulus of each of the other roots lies between 2 and $\sqrt { } 6$. 7 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 4 y = \mathrm { e } ^ { - \alpha t }$$ where $\alpha$ is a constant and $\alpha \neq 2$. Show that if $\alpha < 2$ then, whatever the initial conditions, $y \mathrm { e } ^ { \alpha t } \rightarrow \frac { 1 } { ( 2 - \alpha ) ^ { 2 } }$ as $t \rightarrow \infty$.

CAIE FP1 2003 November Q8

8 Given that $z = \mathrm { e } ^ { \mathrm { i } \theta }$ and $n$ is a positive integer, show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ Hence express $\sin ^ { 6 } \theta$ in the form $$p \cos 6 \theta + q \cos 4 \theta + r \cos 2 \theta + s$$ where the constants $p , q , r , s$ are to be determined. Hence find the mean value of $\sin ^ { 6 } \theta$ with respect to $\theta$ over the interval $0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$.

CAIE FP1 2003 November Q9

9 The line $l _ { 1 }$ passes through the point $A$ with position vector $\mathbf { i } - \mathbf { j } - 2 \mathbf { k }$ and is parallel to the vector $3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }$. The variable line $l _ { 2 }$ passes through the point $( 1 + 5 \cos t ) \mathbf { i } - ( 1 + 5 \sin t ) \mathbf { j } - 14 \mathbf { k }$, where $0 \leqslant t < 2 \pi$, and is parallel to the vector $15 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }$. The points $P$ and $Q$ are on $l _ { 1 }$ and $l _ { 2 }$ respectively, and $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$.

Find the length of $P Q$ in terms of $t$.
Hence show that the lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect, and find the maximum length of $P Q$ as $t$ varies.
The plane $\Pi _ { 1 }$ contains $l _ { 1 }$ and $P Q$; the plane $\Pi _ { 2 }$ contains $l _ { 2 }$ and $P Q$. Find the angle between the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$, correct to the nearest tenth of a degree.

CAIE FP1 2003 November Q10

10 Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 4 & 1
- 6 & - 1 & 3
8 & 8 & 4 \end{array} \right)$$ Hence find a non-singular matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } + \mathbf { A } ^ { 2 } + \mathbf { A } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.

CAIE FP1 2003 November Q11 EITHER

The curve $C$ has equation $y = \frac { 5 ( x - 1 ) ( x + 2 ) } { ( x - 2 ) ( x + 3 ) }$.

Express $y$ in the form $P + \frac { Q } { x - 2 } + \frac { R } { x + 3 }$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ for exactly one value of $x$ and find the corresponding value of $y$.
Write down the equations of all the asymptotes of $C$.
Find the set of values of $k$ for which the line $y = k$ does not intersect $C$.

CAIE FP1 2003 November Q11 OR

A curve has equation $y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } }$, for $x \geqslant 0$. The arc of the curve joining the origin to the point where $x = 3$ is denoted by $R$.

Find the length of $R$.
Find the $y$-coordinate of the centroid of the region bounded by the $x$-axis, the line $x = 3$ and $R$.
Show that the area of the surface generated when $R$ is rotated through one revolution about the $y$-axis is $\frac { 232 } { 15 } \pi$.

CAIE FP1 2004 November Q1

1 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 5 & 2 & 6
2 & 0 & - 1 & 7
3 & - 1 & - 2 & 10
4 & 10 & 13 & 29 \end{array} \right)$$ Find the dimension of the null space of T .

CAIE FP1 2004 November Q2

2 The curve $C$ is defined parametrically by $$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t , \quad 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ where $a$ is a positive constant. Find the area of the surface generated when $C$ is rotated through one complete revolution about the $x$-axis.

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