Questions - CAIE FP1 - A-Level Maths

Show that the dimension of $V$ is 3 .
Express $\mathbf { v } _ { 4 }$ as a linear combination of $\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }$ and $\mathbf { v } _ { 3 }$.
Write down a basis for $V$.
Let $\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 0 & 0
2 & - 5 & - 3 & - 2
0 & 5 & 15 & 10
2 & 6 & 18 & 8 \end{array} \right)$.
Find the general solution of $\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }$.
The set of elements of $\mathbb { R } ^ { 4 }$ which are not solutions of $\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }$ is denoted by $W$.
State, with a reason, whether $W$ is a vector space.
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 2019 June Q1

1 A curve $C$ has equation $\cos y = x$, for $- \pi < x < \pi$.

Use implicit differentiation to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \cot y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$
Hence find the exact value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } \pi \right)$ on $C$.

CAIE FP1 2019 June Q2

2 Let $u _ { n } = \frac { 4 \sin \left( n - \frac { 1 } { 2 } \right) \sin \frac { 1 } { 2 } } { \cos ( 2 n - 1 ) + \cos 1 }$.

Using the formulae for $\cos P \pm \cos Q$ given in the List of Formulae MF10, show that $$u _ { n } = \frac { 1 } { \cos n } - \frac { 1 } { \cos ( n - 1 ) }$$
Use the method of differences to find $\sum _ { n = 1 } ^ { N } u _ { n }$.
Explain why the infinite series $u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$ does not converge.

CAIE FP1 2019 June Q3

3 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $\mathbf { r } = 6 \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } )$ and $\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + \mu ( - 6 \mathbf { j } + \mathbf { k } )$ 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 $l _ { 2 }$. Find the position vectors of $P$ and $Q$.

CAIE FP1 2019 June Q4

4 It is given that, for $n \geqslant 0$, $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { x ^ { 3 } } \mathrm {~d} x$$

Show that $I _ { 2 } = \frac { 1 } { 3 } ( \mathrm { e } - 1 )$.
Show that, for $n \geqslant 3$, $$3 I _ { n } = \mathrm { e } - ( n - 2 ) I _ { n - 3 }$$
Hence find the exact value of $I _ { 8 }$.

CAIE FP1 2019 June Q5

5 A curve $C$ is defined parametrically by $$x = \frac { 2 } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } } \quad \text { and } \quad y = \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } }$$ for $0 \leqslant t \leqslant 1$. The area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis is denoted by $S$.

Show that $S = 4 \pi \int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \left( \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } \right) ^ { 2 } } \mathrm {~d} t$.
Using the substitution $u = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }$, or otherwise, find $S$ in terms of $\pi$ and e .

CAIE FP1 2019 June Q6

6 The equation $$x ^ { 3 } - x + 1 = 0$$ has roots $\alpha , \beta , \gamma$.

Use the relation $x = y ^ { \frac { 1 } { 3 } }$ to show that the equation $$y ^ { 3 } + 3 y ^ { 2 } + 2 y + 1 = 0$$ has roots $\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }$. Hence write down the value of $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }$.
Let $S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }$.
Find the value of $S _ { - 3 }$.
Show that $S _ { 6 } = 5$ and find the value of $S _ { 9 }$.

CAIE FP1 2019 June Q7

7 Find the particular solution of the differential equation $$10 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 3 \frac { \mathrm {~d} x } { \mathrm {~d} t } - x = t + 2$$ given that when $t = 0 , x = 0$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$.

CAIE FP1 2019 June Q8

Prove by mathematical induction that, for $z \neq 1$ and all positive integers $n$, $$1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = \frac { z ^ { n } - 1 } { z - 1 }$$
By letting $z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )$, use de Moivre's theorem to deduce that $$\sum _ { m = 1 } ^ { \infty } \left( \frac { 1 } { 2 } \right) ^ { m } \sin m \theta = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$

CAIE FP1 2019 June Q9

9 It is given that $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A }$, with corresponding eigenvalue $\lambda$.

Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$, with corresponding eigenvalue $\lambda ^ { 2 }$.
The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $$\mathbf { A } = \left( \begin{array} { c c c } n & 1 & 3
0 & 2 n & 0
0 & 0 & 3 n \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + n \mathbf { I } ) ^ { 2 }$$ where $\mathbf { I }$ is the $3 \times 3$ identity matrix and $n$ is a non-zero integer.
Find, in terms of $n$, a non-singular matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.

CAIE FP1 2019 June Q10

10 The curves $C _ { 1 }$ and $C _ { 2 }$ have equations $$y = \frac { a x } { x + 5 } \quad \text { and } \quad y = \frac { x ^ { 2 } + ( a + 10 ) x + 5 a + 26 } { x + 5 }$$ respectively, where $a$ is a constant and $a > 2$.

Find the equations of the asymptotes of $C _ { 1 }$.
Find the equation of the oblique asymptote of $C _ { 2 }$.
Show that $C _ { 1 }$ and $C _ { 2 }$ do not intersect.
Find the coordinates of the stationary points of $C _ { 2 }$.
Sketch $C _ { 1 }$ and $C _ { 2 }$ on a single diagram. [You do not need to calculate the coordinates of any points where $C _ { 2 }$ crosses the axes.]

CAIE FP1 2019 June Q11 EITHER

The curve $C _ { 1 }$ has polar equation $r ^ { 2 } = 2 \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.

The point on $C _ { 1 }$ furthest from the line $\theta = \frac { 1 } { 2 } \pi$ is denoted by $P$. Show that, at $P$, $$2 \theta \tan \theta = 1$$ and verify that this equation has a root between 0.6 and 0.7 .
The curve $C _ { 2 }$ has polar equation $r ^ { 2 } = \theta \sec ^ { 2 } \theta$, for $0 \leqslant \theta < \frac { 1 } { 2 } \pi$. The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the pole, denoted by $O$, and at another point $Q$.
Find the exact value of $\theta$ at $Q$.
The diagram below shows the curve $C _ { 2 }$. Sketch $C _ { 1 }$ on this diagram.
Find, in exact form, the area of the region $O P Q$ enclosed by $C _ { 1 }$ and $C _ { 2 }$.

CAIE FP1 2019 June Q11 OR

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

For $a \neq - 4$, the range space of T is denoted by $V$.
(a) Find the dimension of $V$ and show that $$\left( \begin{array} { r } - 1
1
1
1 \end{array} \right) , \quad \left( \begin{array} { r } 2
0
- 2
2 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 4
- 1
a
2 \end{array} \right)$$ form a basis for $V$.
(b) Show that if $\left( \begin{array} { l } x
y
z
t \end{array} \right)$ belongs to $V$ then $x + 2 y = t$.
For $a = - 4$, find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } - 1
1
1
1 \end{array} \right)$$ 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 2019 June Q1

1 Prove by mathematical induction that $3 ^ { 3 n } - 1$ is divisible by 13 for every positive integer $n$.

CAIE FP1 2019 June Q2

2 The curve $C$ has polar equation $r ^ { 2 } = \ln ( 1 + \theta )$, for $0 \leqslant \theta \leqslant 2 \pi$.

Sketch $C$.
Using the substitution $u = 1 + \theta$, or otherwise, find the area of the region bounded by $C$ and the initial line, leaving your answer in an exact form.

CAIE FP1 2019 June Q3

Write down the fifth roots of unity.
Find all the roots of the equation $$z ^ { 10 } + z ^ { 5 } + 1 = 0$$ giving each root in the form $\mathrm { e } ^ { \mathrm { i } \theta }$.

CAIE FP1 2019 June Q4

Use the method of differences to show that $\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r - 2 ) } = \frac { 1 } { 3 } - \frac { 1 } { 3 ( 3 N + 1 ) }$.
Find the limit, as $N \rightarrow \infty$, of $\sum _ { r = N + 1 } ^ { N ^ { 2 } } \frac { N } { ( 3 r + 1 ) ( 3 r - 2 ) }$.

CAIE FP1 2019 June Q5

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

Find the value of $\alpha$ and state a basis for the range space of T .
Obtain a basis for the null space of T .

CAIE FP1 2019 June Q6

6 The curve $C$ has equation $$y = \frac { x ^ { 2 } } { k x - 1 }$$ where $k$ is a positive constant.

Obtain the equations of the asymptotes of $C$.
Find the coordinates of the stationary points of $C$.
Sketch $C$.

CAIE FP1 2019 June Q7

7 The line $l _ { 1 }$ passes through the points $A ( - 3,1,4 )$ and $B ( - 1,5,9 )$. The line $l _ { 2 }$ passes through the points $C ( - 2,6,5 )$ and $D ( - 1,7,5 )$.

Find the shortest distance between the lines $l _ { 1 }$ and $l _ { 2 }$.
Find the acute angle between the line $l _ { 2 }$ and the plane containing $A , B$ and $D$.

CAIE FP1 2019 June Q8

8 Find the particular solution of the differential equation $$9 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 50 \sin t$$ given that when $t = 0 , x = 0$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$.

CAIE FP1 2019 June Q9

9 A cubic equation $x ^ { 3 } + b x ^ { 2 } + c x + d = 0$ has real roots $\alpha , \beta$ and $\gamma$ such that $$\begin{aligned} \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - \frac { 5 } { 12 }
\alpha \beta \gamma & = - 12
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 90 \end{aligned}$$

Find the values of $c$ and $d$.
Express $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ in terms of $b$.
Show that $b ^ { 3 } - 15 b + 126 = 0$.
Given that $3 + \mathrm { i } \sqrt { } ( 12 )$ is a root of $y ^ { 3 } - 15 y + 126 = 0$, deduce the value of $b$.

CAIE FP1 2019 June Q10

10 Let $I _ { n } = \int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { n } x \mathrm {~d} x$, where $n \geqslant 0$.

By considering $\frac { \mathrm { d } } { \mathrm { d } x } \left( \cot ^ { n + 1 } x \right)$, or otherwise, show that $$I _ { n + 2 } = \frac { 1 } { n + 1 } - I _ { n }$$ The curve $C$ has equation $y = \cot x$, for $\frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi$.
Find, in an exact form, the $y$-coordinate of the centroid of the region enclosed by $C$, the line $x = \frac { 1 } { 4 } \pi$ and the $x$-axis.

CAIE FP1 2019 June Q11 EITHER

A $3 \times 3$ matrix $\mathbf { A }$ has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors $$\left( \begin{array} { l } 1
1
0 \end{array} \right) , \quad \left( \begin{array} { r } - 1
0
b \end{array} \right) , \quad \left( \begin{array} { r } 0
1
- 1 \end{array} \right)$$ respectively, where $b$ is a positive constant.

Find $\mathbf { A }$ in terms of $b$.
Find $\mathbf { A } ^ { - 1 } \left( \begin{array} { r } 0
2
- 2 \end{array} \right)$.
It is given that $$\mathbf { A } ^ { n } \left( \begin{array} { l } 1
1
0 \end{array} \right) = \left( \begin{array} { l } 4
4
0 \end{array} \right) \quad \text { and } \quad \mathbf { A } ^ { n } \left( \begin{array} { r } - 1
0
b \end{array} \right) = \left( \begin{array} { c } - 1
0
b ^ { - 1 } \end{array} \right) .$$ Find the values of $n$ and $b$.

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