Questions - CAIE FP2 - A-Level Maths

Number of letters	0	1	2	3	4	5	$\geqslant 6$
Number of days	57	60	53	25	4	1	0

Number of letters	0	1	2	3	4	5	$\geqslant 6$
Expected number of days	53.964	70.693	$p$	$q$	6.622	1.735	0.463

Show that $p = 46.304$, correct to 3 decimal places, and find $q$.
Carry out a goodness of fit test at the $10 \%$ significance level.

CAIE FP2 2012 June Q1

1 A circular flywheel of radius 0.3 m , with moment of inertia about its axis $18 \mathrm {~kg} \mathrm {~m} ^ { 2 }$, is rotating freely with angular speed $6 \mathrm { rad } \mathrm { s } ^ { - 1 }$. A tangential force of constant magnitude 48 N is applied to the rim of the flywheel, in order to slow the flywheel down. Find the time taken for the angular speed of the flywheel to be reduced to $2 \mathrm { rad } \mathrm { s } ^ { - 1 }$.

CAIE FP2 2012 June Q2

2 Two particles, of masses $3 m$ and $m$, are moving in the same straight line towards each other with speeds $2 u$ and $u$ respectively. When they collide, the impulse acting on each particle has magnitude $4 m u$. Show that the total loss in kinetic energy is $\frac { 4 } { 3 } m u ^ { 2 }$.

CAIE FP2 2012 June Q3

3 A particle $P$ of mass $m$ is projected horizontally with speed $\sqrt { } \left( \frac { 7 } { 2 } g a \right)$ from the lowest point of the inside of a fixed hollow smooth sphere of internal radius $a$ and centre $O$. The angle between $O P$ and the downward vertical at $O$ is denoted by $\theta$. Show that, as long as $P$ remains in contact with the inner surface of the sphere, the magnitude of the reaction between the sphere and the particle is $\frac { 3 } { 2 } m g ( 1 + 2 \cos \theta )$. Find the speed of $P$

when it loses contact with the sphere,
when, in the subsequent motion, it passes through the horizontal plane containing $O$. (You may assume that this happens before $P$ comes into contact with the sphere again.)
$4 \quad A B$ is a diameter of a uniform circular disc $D$ of mass $9 m$, radius $3 a$ and centre $O$. A lamina is formed by removing a circular disc, with centre $O$ and radius $a$, from $D$. Show that the moment of inertia of the lamina, about a fixed horizontal axis $l$ through $A$ and perpendicular to the plane of the lamina, is $112 m a ^ { 2 }$. A particle of mass $3 m$ is now attached to the lamina at $B$. The system is free to rotate about the axis $l$. The system is held with $B$ vertically above $A$ and is then slightly displaced and released from rest. The greatest speed of $B$ in the subsequent motion is $k \sqrt { } ( g a )$. Find the value of $k$, correct to 3 significant figures.

CAIE FP2 2012 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{39282b82-5229-484a-beb9-7a845dbb5727-2_478_867_1816_641} Two uniform rods $A B$ and $B C$ are smoothly jointed at $B$ and rest in equilibrium with $C$ on a rough horizontal floor and with $A$ against a rough vertical wall. The rod $A B$ is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod $A B$ has mass $3 m$ and length $3 a$, the rod $B C$ has mass $5 m$ and length $5 a$, and $C$ is at a distance $6 a$ from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod $B C$ at $C$ has magnitude $\frac { 13 } { 2 } m g$. The coefficient of friction at both $A$ and $C$ is $\mu$. Find the least possible value of $\mu$ for which the rods do not slip at either $A$ or $C$.

CAIE FP2 2012 June Q6

6 The probability that a particular type of light bulb is defective is 0.01 . A large number of these bulbs are tested, one by one. Assuming independence, find the probability that the tenth bulb tested is the first to be found defective. The first defective bulb is the $N$ th to be tested. Write down the value of $\mathrm { E } ( N )$. Find the least value of $n$ such that $\mathrm { P } ( N \leqslant n )$ is greater than 0.9 .

CAIE FP2 2012 June Q7

7 A random sample of 8 swimmers from a swimming club were timed over a distance of 100 metres, once in an outdoor pool and once in an indoor pool. Their times, in seconds, are given in the following table.

Swimmer	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Outdoor time	66.2	62.4	60.8	65.4	68.8	64.3	65.2	67.2
Indoor time	66.1	60.3	60.9	65.2	66.4	63.8	62.4	69.8

Assuming a normal distribution, test, at the $5 \%$ significance level, whether there is a non-zero difference between mean time in the outdoor pool and mean time in the indoor pool.

CAIE FP2 2012 June Q8

8 The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable $X$ metres is the distance between two successive flaws. Show that the distribution function of $X$ is given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.016 x } & x \geqslant 0 ,
0 & x < 0 , \end{cases}$$ and deduce that $X$ has a negative exponential distribution, stating its mean. Find

the median distance between two successive flaws,
the probability that there is a distance of at least 50 metres between two successive flaws.

CAIE FP2 2012 June Q9

9 A random sample of 8 observations of a normal random variable $X$ gave the following summarised data, where $\bar { x }$ denotes the sample mean. $$\Sigma x = 42.5 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 15.519$$ Test, at the $5 \%$ significance level, whether the population mean of $X$ is greater than 4.5. Calculate a 95\% confidence interval for the population mean of $X$.

CAIE FP2 2012 June Q10

10 Random samples of employees are taken from two companies, $A$ and $B$. Each employee is asked which of three types of coffee (Cappuccino, Latte, Ground) they prefer. The results are shown in the following table.

	Cappuccino	Latte	Ground
Company $A$	60	52	32
Company $B$	35	40	31

Test, at the 5\% significance level, whether coffee preferences of employees are independent of their company. Larger random samples, consisting of $N$ times as many employees from each company, are taken. In each company, the proportions of employees preferring the three types of coffee remain unchanged. Find the least possible value of $N$ that would lead to the conclusion, at the $1 \%$ significance level, that coffee preferences of employees are not independent of their company.

CAIE FP2 2012 June Q11 EITHER

A particle $P$ of mass $m$ is attached to one end of a light elastic string of modulus of elasticity $4 m g$ and natural length $l$. The other end of the string is attached to a fixed point $O$. The particle rests in equilibrium at the point $E$, vertically below $O$. The particle is pulled down a vertical distance $\frac { 1 } { 8 } l$ from $E$ and released from rest. Show that the motion of $P$ is simple harmonic with period $\pi \sqrt { } \left( \frac { l } { g } \right)$. At an instant when $P$ is moving vertically downwards through $E$, the string is cut. When $P$ has descended a further distance $\frac { 7 } { 16 } l$ under gravity, it strikes a fixed smooth plane which is inclined at $30 ^ { \circ }$ to the horizontal. The coefficient of restitution between $P$ and the plane is $\frac { 1 } { 3 }$. Show that the speed of $P$ immediately after the impact is $\frac { 1 } { 4 } \sqrt { } ( 5 g l )$.

CAIE FP2 2012 June Q11 OR

A new restaurant $S$ has recently opened in a particular town. In order to investigate any effect of $S$ on an existing restaurant $R$, the daily takings, $x$ and $y$ in thousands of dollars, at $R$ and $S$ respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

Day	1	2	3	4	5	6	7	8
$x$	1.2	1.4	0.9	1.1	0.8	1.0	0.6	1.5
$y$	0.3	0.4	0.6	0.6	0.25	0.75	0.6	0.35

Calculate the product moment correlation coefficient for this sample.
Stating your hypotheses, test, at the $2.5 \%$ significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. Another sample is taken over $N$ randomly chosen days and the product moment correlation coefficient is found to be - 0.431 . A test, at the $5 \%$ significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.
Find the range of possible values of $N$. \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 FP2 2012 June Q3

when it loses contact with the sphere,
when, in the subsequent motion, it passes through the horizontal plane containing $O$. (You may assume that this happens before $P$ comes into contact with the sphere again.)
$4 A B$ is a diameter of a uniform circular disc $D$ of mass $9 m$, radius $3 a$ and centre $O$. A lamina is formed by removing a circular disc, with centre $O$ and radius $a$, from $D$. Show that the moment of inertia of the lamina, about a fixed horizontal axis $l$ through $A$ and perpendicular to the plane of the lamina, is $112 m a ^ { 2 }$. A particle of mass $3 m$ is now attached to the lamina at $B$. The system is free to rotate about the axis $l$. The system is held with $B$ vertically above $A$ and is then slightly displaced and released from rest. The greatest speed of $B$ in the subsequent motion is $k \sqrt { } ( g a )$. Find the value of $k$, correct to 3 significant figures.

CAIE FP2 2012 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{621b50d6-44e8-435d-ac6e-bb2ee5bcdd38-2_478_867_1816_641} Two uniform rods $A B$ and $B C$ are smoothly jointed at $B$ and rest in equilibrium with $C$ on a rough horizontal floor and with $A$ against a rough vertical wall. The $\operatorname { rod } A B$ is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod $A B$ has mass $3 m$ and length $3 a$, the rod $B C$ has mass $5 m$ and length $5 a$, and $C$ is at a distance $6 a$ from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod $B C$ at $C$ has magnitude $\frac { 13 } { 2 } m g$. The coefficient of friction at both $A$ and $C$ is $\mu$. Find the least possible value of $\mu$ for which the rods do not slip at either $A$ or $C$.

CAIE FP2 2012 June Q8

8 The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable $X$ metres is the distance between two successive flaws. Show that the distribution function of $X$ is given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.016 x } & x \geqslant 0
0 & x < 0 \end{cases}$$ and deduce that $X$ has a negative exponential distribution, stating its mean. Find

the median distance between two successive flaws,
the probability that there is a distance of at least 50 metres between two successive flaws.

CAIE FP2 2012 June Q1

1 Two smooth spheres $A$ and $B$, of equal radii and of masses $3 m$ and $6 m$ respectively, are at rest on a smooth horizontal surface. Sphere $A$ is projected directly towards $B$ with speed $u$. The coefficient of restitution between $A$ and $B$ is $e$. Show that the kinetic energy lost in the collision between $A$ and $B$ is $m u ^ { 2 } \left( 1 - e ^ { 2 } \right)$.

CAIE FP2 2012 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-2_293_875_525_635} Two light elastic strings, each of natural length $a$ and modulus of elasticity $2 m g$, are attached to a particle $P$ of mass $m$. The strings join the particle to the points $A$ and $B$ which are fixed and at a distance $4 a$ apart on a smooth horizontal surface. The particle is at rest at the mid-point $O$ of $A B$. The particle is now displaced a small distance in a direction perpendicular to $A B$, on the surface, and released from rest. At time $t$, the displacement of $P$ from $O$ is $x$ (see diagram). Show that $$\ddot { x } = - \frac { 4 g x } { a } \left( 1 - \frac { 1 } { 2 } \left( 1 + \frac { x ^ { 2 } } { 4 a ^ { 2 } } \right) ^ { - \frac { 1 } { 2 } } \right) .$$ Given that $\frac { x } { a }$ is so small that $\left( \frac { x } { a } \right) ^ { 2 }$ and higher powers may be neglected, show that the motion of $P$ is approximately simple harmonic and state the period of the motion.

CAIE FP2 2012 June Q3

3 The point $O$ is on the fixed line $l$. Points $A$ and $B$ on $l$ are such that $O A = 6 \mathrm {~m}$ and $O B = 8 \mathrm {~m}$, with $O$ between $A$ and $B$. A particle $P$ oscillates on $l$ in simple harmonic motion with centre $O$. When $P$ is at $A$ its speed is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and when $P$ is at $B$ its speed is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Show that the amplitude of the motion is 10 m and find the period of the motion. Find the time taken by $P$ to travel directly from $A$ to $B$, through $O$.

CAIE FP2 2012 June Q4

4 A smooth sphere, with centre $O$ and radius $a$, has its lowest point fixed on a horizontal plane. A particle $P$ of mass $m$ is projected horizontally with speed $u$ from the highest point on the outer surface of the sphere. In the subsequent motion, $O P$ makes an angle $\theta$ with the upward vertical through $O$. Show that, while $P$ remains in contact with the sphere, the magnitude of the reaction of the sphere on
$P$ is $m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }$. The particle loses contact with the surface of the sphere when $\theta = \alpha$. Given that $u = \frac { 1 } { 2 } \sqrt { } ( g a )$, find

$\cos \alpha$,
the vertical component of the velocity of $P$ as it strikes the horizontal plane.

CAIE FP2 2012 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-3_319_794_255_678} A uniform rod $A B$, of mass $m$ and length $6 a$, is rigidly attached at $B$ to a point on the circumference of a uniform circular lamina of mass $m$, radius $2 a$ and centre $O$. The lamina and the rod are in the same vertical plane, and $A B O$ is a straight line (see diagram). Show that the moment of inertia of the system about an axis $l$ through $A$ perpendicular to the plane of the lamina is $78 m a ^ { 2 }$. A particle of mass $2 m$ is now attached at $B$ and the system is free to rotate in a vertical plane about the fixed axis $l$ which is horizontal. Initially $A B$ is horizontal, with $O$ moving downwards and the system having angular velocity $\frac { 3 } { 5 } \sqrt { } \left( \frac { g } { a } \right)$. At time $t , A B$ makes an angle $\theta$ with the downward vertical through $A$.

Find, in terms of $a , g$ and $\theta$, an expression for $\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } }$.
Find the angular velocity of the system when $B$ is vertically below $A$.

CAIE FP2 2012 June Q6

6 A random sample of 10 observations of a normal random variable $X$ has mean $\bar { x }$, where $$\bar { x } = 8.254 , \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.912 .$$ Using a $5 \%$ significance level, test whether the mean of $X$ is greater than 8.05.

CAIE FP2 2012 June Q7

7 The waiting time, $T$ minutes, before a customer is served in a restaurant has distribution function F given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - \lambda t } & t \geqslant 0
0 & t < 0 \end{cases}$$ where $\lambda$ is a positive constant. The standard deviation of $T$ is 8 . Find

the value of $\lambda$,
the probability that a customer has to wait between 5 and 10 minutes before being served,
the median value of $T$.

CAIE FP2 2012 June Q8

8 Residents of three towns $A , B$ and $C$ were asked to grade the reliability of their digital television signal as good, satisfactory or poor. A random sample of responses from each town is taken and the numbers in each category are given in the following table.

	Good	Satisfactory	Poor
Town $A$	24	34	14
Town $B$	58	60	26
Town $C$	20	34	30

Test, at the 2.5\% significance level, whether grade of reliability is independent of town. Identify which town makes the greatest contribution to the test statistic and relate your answer to the context of the question.

CAIE FP2 2012 June Q9

9 The continuous random variable $X$ has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 a } & - a \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where $a$ is a positive constant. Find the distribution function of $X$. The random variable $Y$ is defined by $Y = \mathrm { e } ^ { X }$. Find the distribution function of $Y$. Given that $a = 4$, find the value of $k$ for which $\mathrm { P } ( Y \geqslant k ) = 0.25$.

Number of letters	0	1	2	3	4	5	\(\geqslant 6\)
Number of days	57	60	53	25	4	1	0

Swimmer	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Outdoor time	66.2	62.4	60.8	65.4	68.8	64.3	65.2	67.2
Indoor time	66.1	60.3	60.9	65.2	66.4	63.8	62.4	69.8

Day	1	2	3	4	5	6	7	8
\(x\)	1.2	1.4	0.9	1.1	0.8	1.0	0.6	1.5
\(y\)	0.3	0.4	0.6	0.6	0.25	0.75	0.6	0.35

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