Questions - CAIE FP1 - A-Level Maths

Find a basis for $R _ { 1 }$, the range space of $\mathrm { T } _ { 1 }$.
Find a basis for $K _ { 2 }$, the null space of $\mathrm { T } _ { 2 }$, and hence show that $K _ { 2 }$ is a subspace of $R _ { 1 }$. The set of vectors which belong to $R _ { 1 }$ but do not belong to $K _ { 2 }$ is denoted by $W$.
State whether $W$ is a vector space, justifying your answer. The linear transformation $\mathrm { T } _ { 3 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is the result of applying $\mathrm { T } _ { 1 }$ and then $\mathrm { T } _ { 2 }$, in that order.
Find the dimension of the null space of $\mathrm { T } _ { 3 }$.

CAIE FP1 2010 June Q1

1 The variables $x$ and $y$ are such that $y = - 1$ when $x = 1$ and $$x ^ { 2 } + y ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = 29$$ Find the values of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 1$.

CAIE FP1 2010 June Q2

2 The curve $C$ has polar equation $$r = a \left( 1 - \mathrm { e } ^ { - \theta } \right)$$ where $a$ is a positive constant and $0 \leqslant \theta < 2 \pi$.

Draw a sketch of $C$.
Show that the area of the region bounded by $C$ and the lines $\theta = \ln 2$ and $\theta = \ln 4$ is $$\frac { 1 } { 2 } a ^ { 2 } \left( \ln 2 - \frac { 13 } { 32 } \right)$$

CAIE FP1 2010 June Q3

3 At any point $( x , y )$ on the curve $C$, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t \sqrt { } \left( t ^ { 2 } + 4 \right) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = - t \sqrt { } \left( 4 - t ^ { 2 } \right)$$ where the parameter $t$ is such that $0 \leqslant t \leqslant 2$. Show that the length of $C$ is $4 \sqrt { } 2$. Given that $y = 0$ when $t = 2$, determine the area of the surface generated when $C$ is rotated through one complete revolution about the $x$-axis, leaving your answer in an exact form.

CAIE FP1 2010 June Q4

4 The sum $S _ { N }$ is defined by $S _ { N } = \sum _ { n = 1 } ^ { N } n ^ { 5 }$. Using the identity $$\left( n + \frac { 1 } { 2 } \right) ^ { 6 } - \left( n - \frac { 1 } { 2 } \right) ^ { 6 } \equiv 6 n ^ { 5 } + 5 n ^ { 3 } + \frac { 3 } { 8 } n$$ find $S _ { N }$ in terms of $N$. [You need not simplify your result.] Hence find $\lim _ { N \rightarrow \infty } N ^ { - \lambda } S _ { N }$, for each of the two cases

$\lambda = 6$,
$\lambda > 6$.

CAIE FP1 2010 June Q5

5 Let $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { n } \mathrm {~d} x$$ where $n \geqslant 1$. Show that $$I _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } ( n + 1 ) I _ { n }$$ Hence prove by induction that, for all positive integers $n , I _ { n }$ is of the form $A _ { n } \mathrm { e } ^ { 2 } + B _ { n }$, where $A _ { n }$ and $B _ { n }$ are rational numbers.

CAIE FP1 2010 June Q6

6 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots $\alpha , \beta , \gamma$. Use the relation $x = \sqrt { } y$ to show that the equation $$y ^ { 3 } + 2 y ^ { 2 } + y - 1 = 0$$ has roots $\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }$. Let $S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }$.

Write down the value of $S _ { 2 }$ and show that $S _ { 4 } = 2$.
Find the values of $S _ { 6 }$ and $S _ { 8 }$.

CAIE FP1 2010 June Q7

7 The lines $l _ { 1 }$ and $l _ { 2 }$ have vector equations $$\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$$ respectively.

Show that $l _ { 1 }$ and $l _ { 2 }$ intersect.
Find the perpendicular distance from the point $P$ whose position vector is $3 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }$ to the plane containing $l _ { 1 }$ and $l _ { 2 }$.
Find the perpendicular distance from $P$ to $l _ { 1 }$.

CAIE FP1 2010 June Q8

8 The matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 1 & - 1
- 4 & - 1 & 4
0 & - 1 & 5 \end{array} \right)$$ Given that one eigenvector of $\mathbf { A }$ is $\left( \begin{array} { r } 1
- 2
- 1 \end{array} \right)$, find the corresponding eigenvalue. Given also that another eigenvalue of $\mathbf { A }$ is 4, find a corresponding eigenvector. Given further that $\left( \begin{array} { r } 1
- 4
- 1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$, with corresponding eigenvalue 1 , find matrices $\mathbf { P }$ and $\mathbf { Q }$, together with a diagonal matrix $\mathbf { D }$, such that $\mathbf { A } ^ { 5 } = \mathbf { P D Q }$.

CAIE FP1 2010 June Q9

Write down the five fifth roots of unity.
Hence find all the roots of the equation $$z ^ { 5 } + 16 + ( 16 \sqrt { } 3 ) i = 0$$ giving answers in the form $r \mathrm { e } ^ { \mathrm { i } q \pi }$, where $r > 0$ and $q$ is a rational number. Show these roots on an Argand diagram. Let $w$ be a root of the equation in part (ii).
Show that $$\sum _ { k = 0 } ^ { 4 } \left( \frac { w } { 2 } \right) ^ { k } = \frac { 3 + \mathrm { i } \sqrt { } 3 } { 2 - w }$$
Identify the root for which $| 2 - w |$ is least.

CAIE FP1 2010 June Q10

10 Find the set of values of $a$ for which the system of equations $$\begin{aligned} x + 4 y + 12 z & = 5
2 x + a y + 12 z & = a - 1
3 x + 12 y + 2 a z & = 10 \end{aligned}$$ has a unique solution. Show that the system does not have any solution in the case $a = 18$. Given that $a = 8$, show that the number of solutions is infinite and find the solution for which $x + y + z = 1$.

CAIE FP1 2010 June Q11 EITHER

The variables $z$ and $x$ are related by the differential equation $$3 z ^ { 2 } \frac { \mathrm {~d} ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 6 z ^ { 2 } \frac { \mathrm {~d} z } { \mathrm {~d} x } + 6 z \left( \frac { \mathrm {~d} z } { \mathrm {~d} x } \right) ^ { 2 } + 5 z ^ { 3 } = 5 x + 2$$ Use the substitution $y = z ^ { 3 }$ to show that $y$ and $x$ are related by the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 5 x + 2$$ Given that $z = 1$ and $\frac { \mathrm { d } z } { \mathrm {~d} x } = - \frac { 2 } { 3 }$ when $x = 0$, find $z$ in terms of $x$. Deduce that, for large positive values of $x , z \approx x ^ { \frac { 1 } { 3 } }$.

CAIE FP1 2010 June Q11 OR

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

Obtain the equations of the asymptotes of $C$.
Show that there is exactly one point of intersection of $C$ with the asymptotes and find its coordinates.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence
(a) find the coordinates of any stationary points of $C$,
(b) state the set of values of $x$ for which the gradient of $C$ is negative.
Draw a sketch of $C$.

CAIE FP1 2010 June Q1

1 Given that 5 is an eigenvalue of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 3 & 0
1 & 2 & 1
- 1 & 3 & 4 \end{array} \right)$$ find a corresponding eigenvector. Hence find an eigenvalue and a corresponding eigenvector of the matrix $\mathbf { A } + \mathbf { A } ^ { 2 }$.

CAIE FP1 2010 June Q2

2 By considering the identity $$\cos [ ( 2 n - 1 ) \alpha ] - \cos [ ( 2 n + 1 ) \alpha ] \equiv 2 \sin \alpha \sin 2 n \alpha$$ show that if $\alpha$ is not an integer multiple of $\pi$ then $$\sum _ { n = 1 } ^ { N } \sin ( 2 n \alpha ) = \frac { 1 } { 2 } \cot \alpha - \frac { 1 } { 2 } \operatorname { cosec } \alpha \cos [ ( 2 N + 1 ) \alpha ]$$ Deduce that the infinite series $$\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 2 } { 3 } n \pi \right)$$ does not converge.

CAIE FP1 2010 June Q3

3 The sequence $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is such that $x _ { 1 } = 3$ and $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + 4 x _ { n } - 2 } { 2 x _ { n } + 3 }$$ for $n = 1,2,3 , \ldots$. Prove by induction that $x _ { n } > 2$ for all $n$.

CAIE FP1 2010 June Q4

4 The parametric equations of a curve are $$x = \cos t + t \sin t , \quad y = \sin t - t \cos t$$ The arc of the curve joining the point where $t = 0$ to the point where $t = \frac { 1 } { 2 } \pi$ is rotated about the $x$-axis through one complete revolution. Find the area of the surface generated, leaving your result in terms of $\pi$.

CAIE FP1 2010 June Q5

5 Use de Moivre's theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ Hence find all the roots of the equation $$32 x ^ { 5 } - 40 x ^ { 3 } + 10 x + 1 = 0$$ in the form $\sin ( q \pi )$, where $q$ is a positive rational number.

CAIE FP1 2010 June Q6

6 The curve $C$ has equation $$y = \frac { x ^ { 2 } - 3 x - 7 } { x + 1 }$$

Obtain the equations of the asymptotes of $C$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } > 1$ at all points of $C$.
Draw a sketch of $C$.

CAIE FP1 2010 June Q7

7 It is given that $$x = t ^ { 2 } \mathrm { e } ^ { - t ^ { 2 } } \quad \text { and } \quad y = t \mathrm { e } ^ { - t ^ { 2 } }$$

Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 t ^ { 2 } } { 2 t - 2 t ^ { 3 } }$$
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ in terms of $t$.

CAIE FP1 2010 June Q8

8 Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 10 \sin 3 x - 20 \cos 3 x$$ Show that, for large positive $x$ and independently of the initial conditions, $$y \approx R \sin ( 3 x + \phi )$$ where the constants $R$ and $\phi$, such that $R > 0$ and $0 < \phi < 2 \pi$, are to be determined correct to 2 decimal places.

CAIE FP1 2010 June Q9

9 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ where $n$ is a non-negative integer. Show that $I _ { n + 2 } = \frac { n + 1 } { n + 2 } I _ { n }$. The region $R$ of the $x - y$ plane is bounded by the $x$-axis, the line $x = \frac { \pi } { 2 m }$ and the curve whose equation is $y = \sin ^ { 4 } m x$, where $m > 0$. Find the $y$-coordinate of the centroid of $R$.

CAIE FP1 2010 June Q10

10 The equation $$x ^ { 4 } + x ^ { 3 } + c x ^ { 2 } + 4 x - 2 = 0$$ where $c$ is a constant, has roots $\alpha , \beta , \gamma , \delta$.

Use the substitution $y = \frac { 1 } { x }$ to find an equation which has roots $\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }$.
Find, in terms of $c$, the values of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }$ and $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$.
Hence find $$\left( \alpha - \frac { 1 } { \alpha } \right) ^ { 2 } + \left( \beta - \frac { 1 } { \beta } \right) ^ { 2 } + \left( \gamma - \frac { 1 } { \gamma } \right) ^ { 2 } + \left( \delta - \frac { 1 } { \delta } \right) ^ { 2 }$$ in terms of $c$.
Deduce that when $c = - 3$ the roots of the given equation are not all real.

CAIE FP1 2010 June Q11

11 The curve $C$ has polar equation $$r = \frac { a } { 1 + \theta }$$ where $a$ is a positive constant and $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.

Show that $r$ decreases as $\theta$ increases.
The point $P$ of $C$ is further from the initial line than any other point of $C$. Show that, at $P$, $$\tan \theta = 1 + \theta$$ and verify that this equation has a root between 1.1 and 1.2.
Draw a sketch of $C$.
Find the area of the region bounded by the initial line, the line $\theta = \frac { 1 } { 2 } \pi$ and $C$, leaving your answer in terms of $\pi$ and $a$.

CAIE FP1 2010 June Q12 EITHER

The line $l _ { 1 }$ passes through the point $A$ whose position vector is $3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$ and is parallel to the vector $\mathbf { i } + \mathbf { j }$. The line $l _ { 2 }$ passes through the point $B$ whose position vector is $- \mathbf { i } - \mathbf { k }$ and is parallel to the vector $\mathbf { j } + 2 \mathbf { k }$. The point $P$ is on $l _ { 1 }$ and the point $Q$ is on $l _ { 2 }$ and $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$.

Find the length of $P Q$.
Find the position vector of $Q$.
Show that the perpendicular distance from $Q$ to the plane containing $A B$ and the line $l _ { 1 }$ is $\sqrt { } 3$.

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