Questions - CAIE FP1 - A-Level Maths

$- 1 < x < 1$,
$x = 1$.

CAIE FP1 2008 June Q3

3 Show that if $\lambda$ is an eigenvalue of the square matrix $\mathbf { A }$ with $\mathbf { e }$ as a corresponding eigenvector, and $\mu$ is an eigenvalue of the square matrix $\mathbf { B }$ for which $\mathbf { e }$ is also a corresponding eigenvector, then $\lambda + \mu$ is an eigenvalue of the matrix $\mathbf { A } + \mathbf { B }$ with $\mathbf { e }$ as a corresponding eigenvector. The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0
- 4 & - 6 & - 6
5 & 11 & 10 \end{array} \right)$$ has $\left( \begin{array} { r } 1
- 1
1 \end{array} \right)$ as an eigenvector. Find the corresponding eigenvalue. The other two eigenvalues of $\mathbf { A }$ are 1 and 2, with corresponding eigenvectors $\left( \begin{array} { r } 1
2
- 3 \end{array} \right)$ and $\left( \begin{array} { r } 1
1
- 2 \end{array} \right)$ respectively. The matrix $\mathbf { B }$ has eigenvalues $2,3,1$ with corresponding eigenvectors $\left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { r } 1
2
- 3 \end{array} \right)$, $\left( \begin{array} { r } 1
1
- 2 \end{array} \right)$ respectively. Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $( \mathbf { A } + \mathbf { B } ) ^ { 4 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.
[0pt] [You are not required to evaluate $\mathbf { P } ^ { - 1 }$.]

CAIE FP1 2008 June Q4

4 The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations $$r = \theta + 2 \quad \text { and } \quad r = \theta ^ { 2 }$$ respectively, where $0 \leqslant \theta \leqslant \pi$.

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 area bounded by $C _ { 1 } , C _ { 2 }$ and the line $\theta = 0$.

CAIE FP1 2008 June Q5

5 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots $\alpha , \beta , \gamma$. Show that the equation with roots $\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }$ is $$y ^ { 3 } - 3 y ^ { 2 } + 4 y - 1 = 0$$ Hence find the value of $\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 }$.

CAIE FP1 2008 June Q6

6 The curve $C$ is defined parametrically by $$x = 4 t - t ^ { 2 } \quad \text { and } \quad y = 1 - \mathrm { e } ^ { - t }$$ where $0 \leqslant t < 2$. Show that at all points of $C$, $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { ( t - 1 ) \mathrm { e } ^ { - t } } { 4 ( 2 - t ) ^ { 3 } }$$ Show that the mean value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ with respect to $x$ over the interval $0 \leqslant x \leqslant \frac { 7 } { 4 }$ is $$\frac { 4 e ^ { - \frac { 1 } { 2 } } - 3 } { 21 }$$

CAIE FP1 2008 June Q7

7 Prove by induction that $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 5 } + r ^ { 3 } \right) = \frac { 1 } { 2 } n ^ { 3 } ( n + 1 ) ^ { 3 }$$ for all $n \geqslant 1$. Use this result together with the List of Formulae (MF10) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 5 } = \frac { 1 } { 12 } n ^ { 2 } ( n + 1 ) ^ { 2 } \mathrm { Q } ( n )$$ where $\mathrm { Q } ( n )$ is a quadratic function of $n$ which is to be determined.

CAIE FP1 2008 June Q8

Given that $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } t ^ { n } \sin t \mathrm {~d} t$$ show that, for $n \geqslant 2$, $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } .$$
A curve $C$ in the $x - y$ plane is defined parametrically in terms of $t$. It is given that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t ^ { 4 } ( 1 - \cos 2 t ) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = t ^ { 4 } \sin 2 t .$$ Find the length of the arc of $C$ from the point where $t = 0$ to the point where $t = \frac { 1 } { 2 } \pi$.

CAIE FP1 2008 June Q9

9 The curve $C$ has equation $$y = \frac { x ^ { 2 } - 2 x + \lambda } { x + 1 }$$ where $\lambda$ is a constant. Show that the equations of the asymptotes of $C$ are independent of $\lambda$. Find the value of $\lambda$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case. Sketch $C$ in the case $\lambda = - 4$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis.

CAIE FP1 2008 June Q10

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

CAIE FP1 2008 June Q11

11 Show that, with a suitable value of the constant $\alpha$, the substitution $y = x ^ { \alpha } w$ reduces the differential equation $$2 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( 3 x ^ { 2 } + 8 x \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + \left( x ^ { 2 } + 6 x + 4 \right) y = \mathrm { f } ( x )$$ to $$2 \frac { \mathrm {~d} ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} w } { \mathrm {~d} x } + w = \mathrm { f } ( x )$$ Find the general solution for $y$ in the case where $\mathrm { f } ( x ) = 6 \sin 2 x + 7 \cos 2 x$.

CAIE FP1 2008 June Q12 EITHER

The position vectors of the points $A , B , C , D$ are
$7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$,
$3 \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }$,
$2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }$,
$2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }$
respectively. It is given that the shortest distance between the line $A B$ and the line $C D$ is 3 .

Show that $\lambda ^ { 2 } - 5 \lambda + 4 = 0$.
Find the acute angle between the planes through $A , B , D$ corresponding to the values of $\lambda$ satisfying the equation in part (i).

CAIE FP1 2008 June Q12 OR

The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 2 & - 1 & - 1
1 & 3 & - 1 & 0
1 & 0 & 3 & 1
0 & 3 & - 4 & - 1 \end{array} \right) .$$ The range space of T is denoted by $V$.

Determine the dimension of $V$.
Show that the vectors $\left( \begin{array} { l } 1
1
1
0 \end{array} \right) , \left( \begin{array} { l } 2
3
0
3 \end{array} \right) , \left( \begin{array} { r } - 1
- 1
3
- 4 \end{array} \right)$ are linearly independent.
Write down a basis of $V$. The set of elements of $\mathbb { R } ^ { 4 }$ which do not belong to $V$ is denoted by $W$.
State, with a reason, whether $W$ is a vector space.
Show that if the vector $\left( \begin{array} { l } x
y
z
t \end{array} \right)$ belongs to $W$ then $y - z - t \neq 0$.

CAIE FP1 2009 June Q1

1 The equation $$x ^ { 4 } - x ^ { 3 } - 1 = 0$$ has roots $\alpha , \beta , \gamma , \delta$. By using the substitution $y = x ^ { 3 }$, or by any other method, find the exact value of $\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } + \delta ^ { 6 }$.

CAIE FP1 2009 June Q2

2 Verify that, for all positive values of $n$, $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) } .$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find

the sum to $N$ terms,
the sum to infinity.

CAIE FP1 2009 June Q3

3 The equation of a curve is $y = \lambda x ^ { 2 }$, where $\lambda > 0$. The region bounded by the curve, the $x$-axis and the line $x = a$, where $a > 0$, is denoted by $R$. The $y$-coordinate of the centroid of $R$ is $a$. Show that $\lambda = \frac { 10 } { 3 a }$.

CAIE FP1 2009 June Q4

4 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where $x = 0$ to the point where $x = 1$ is denoted by $s$. Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { } \left( 1 + x ^ { 4 } \right) \mathrm { d } x$$ The surface area generated when this arc is rotated through one complete revolution about the $x$-axis is denoted by $S$. Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { } 2 - 1 )$$ [Do not attempt to evaluate $s$ or $S$.]

CAIE FP1 2009 June Q5

5 Draw a sketch of the curve $C$ whose polar equation is $r = \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. On the same diagram draw the line $\theta = \alpha$, where $0 < \alpha < \frac { 1 } { 2 } \pi$. The region bounded by $C$ and the line $\theta = \frac { 1 } { 2 } \pi$ is denoted by $R$. Find the exact value of $\alpha$ for which the line $\theta = \alpha$ divides $R$ into two regions of equal area.

CAIE FP1 2009 June Q6

6 A curve has equation $$( x + y ) \left( x ^ { 2 } + y ^ { 2 } \right) = 1$$ Find the values of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $( 0,1 )$.

CAIE FP1 2009 June Q7

7 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where $n \geqslant 0$. Show that, for all $n \geqslant 1$, $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 }$$ Hence prove by induction that, for all positive integers $n$, $$I _ { n } < n ! .$$

CAIE FP1 2009 June Q8

8 Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73$$ Show that, whatever the initial conditions, $\frac { y } { x ^ { 2 } } \rightarrow 1$ as $x \rightarrow \infty$.

CAIE FP1 2009 June Q9

9 The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 4
1 & 5 & - 1
2 & 1 & 5 \end{array} \right)$$ has eigenvalues $1,5,7$. Find a set of corresponding eigenvectors. Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }$.
[0pt] [The evaluation of $\mathbf { P } ^ { - 1 }$ is not required.]
Determine the set of values of the real constant $k$ such that $k ^ { n } \mathbf { A } ^ { n }$ tends to the zero matrix as $n \rightarrow \infty$.

CAIE FP1 2009 June Q10

10 The curve $C$ has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where $\lambda$ is a non-zero constant. Obtain the equation of each of the asymptotes of $C$. In separate diagrams, sketch $C$ for the cases $\lambda > 0$ and $\lambda < 0$. In both cases the coordinates of the turning points must be indicated.

CAIE FP1 2009 June Q11

11 The line $l _ { 1 }$ is parallel to the vector $4 \mathbf { j } - \mathbf { k }$ and passes through the point $A$ whose position vector is $2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }$. The variable line $l _ { 2 }$ is parallel to the vector $\mathbf { i } - ( 2 \sin t ) \mathbf { j }$, where $0 \leqslant t < 2 \pi$, and passes through the point $B$ whose position vector is $\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$. 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 find the values of $t$ for which $l _ { 1 }$ and $l _ { 2 }$ intersect.
For the case $t = \frac { 1 } { 4 } \pi$, find the perpendicular distance from $A$ to the plane $B P Q$, giving your answer correct to 3 decimal places.

CAIE FP1 2009 June Q12 EITHER

By considering $\sum _ { k = 0 } ^ { n - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { k }$, show that $$\sum _ { k = 0 } ^ { n - 1 } \cos k \theta \sec ^ { k } \theta = \cot \theta \sin n \theta \sec ^ { n } \theta$$ provided $\theta$ is not an integer multiple of $\frac { 1 } { 2 } \pi$. Hence or otherwise show that $$\sum _ { k = 0 } ^ { n - 1 } 2 ^ { k } \cos \left( \frac { 1 } { 3 } k \pi \right) = \frac { 2 ^ { n } } { \sqrt { 3 } } \sin \left( \frac { 1 } { 3 } n \pi \right)$$ Given that $0 < x < 1$, show that $$\sum _ { k = 0 } ^ { n - 1 } \frac { \cos \left( k \cos ^ { - 1 } x \right) } { x ^ { k } } = \frac { \sin \left( n \cos ^ { - 1 } x \right) } { x ^ { n - 1 } \sqrt { } \left( 1 - x ^ { 2 } \right) }$$

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