Questions - CAIE FP2 - A-Level Maths

the maximum speed of $P$,
the magnitude of the maximum acceleration of $P$.
$2 \quad$ A particle $P$ of mass $m$ is projected horizontally with speed $u$ from the lowest point on the inside of a fixed hollow sphere with centre $O$. The sphere has a smooth internal surface of radius $a$. Assuming that the particle does not lose contact with the sphere, show that when the speed of the particle has been reduced to $\frac { 1 } { 2 } u$ the angle $\theta$ between $O P$ and the downward vertical satisfies the equation $$8 g a ( 1 - \cos \theta ) = 3 u ^ { 2 }$$ Find, in terms of $m , u , a$ and $g$, an expression for the magnitude of the contact force acting on the particle in this position.

CAIE FP2 2010 November Q3

3 Two smooth spheres $A$ and $B$, of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere $A$ has mass $m$ and speed $u$ and sphere $B$ has mass $\alpha m$ and speed $\frac { 1 } { 4 } u$. The spheres collide and $A$ is brought to rest by the collision. Find the coefficient of restitution in terms of $\alpha$. Deduce that $\alpha \geqslant 2$.

CAIE FP2 2010 November Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{c7844913-5c2e-47b4-87b6-f822f4d4bf22-2_426_862_1553_644} A hemispherical bowl of radius $r$ is fixed with its rim horizontal. A thin uniform rod rests in equilibrium on the rim of the bowl with one end resting on the inner surface of the bowl at $A$, as shown in the diagram. The rod has length $2 a$ and weight $W$. The point of contact between the rod and the rim is $B$, and the rim has centre $C$. The rod is in a vertical plane containing $C$. The rod is inclined at $\theta$ to the horizontal and the line $A C$ is inclined at $2 \theta$ to the horizontal. The contacts at $A$ and $B$ are smooth. In any order, show that

the contact force acting on the rod at $A$ has magnitude $W \tan \theta$,
the contact force acting on the rod at $B$ has magnitude $\frac { W \cos 2 \theta } { \cos \theta }$,
$2 r \cos 2 \theta = a \cos \theta$.

CAIE FP2 2010 November Q5

5 A uniform circular disc has diameter $A B$, mass $2 m$ and radius $a$. A particle of mass $m$ is attached to the disc at $B$. The disc is able to rotate about a smooth fixed horizontal axis through $A$. The axis is tangential to the disc. Show that the moment of inertia of the system about the axis is $\frac { 13 } { 2 } m a ^ { 2 }$. The disc is held with $A B$ horizontal and released. Find the angular speed of the system when $B$ is directly below $A$. The disc is slightly displaced from the position of equilibrium in which $B$ is below $A$. At time $t$ the angle between $A B$ and the vertical is $\theta$. Write down the equation of motion, and find the approximate period of small oscillations about the equilibrium position.

CAIE FP2 2010 November Q6

6 The mean Intelligence Quotient (IQ) of a random sample of 15 pupils at $\operatorname { School } A$ is 109 . The mean IQ of a random sample of 20 pupils at School $B$ is 112 . You may assume that the IQs for the populations from which these samples are taken are normally distributed, and that both distributions have standard deviation 15. Find a $90 \%$ confidence interval for $\mu _ { B } - \mu _ { A }$, where $\mu _ { A }$ and $\mu _ { B }$ are the population mean IQs.

CAIE FP2 2010 November Q7

7 The discrete random variable $X$ has a geometric distribution with mean 4.
Find

$\mathrm { P } ( X = 5 )$,
$\mathrm { P } ( X \geqslant 5 )$,
the least integer $N$ such that $\mathrm { P } ( X \leqslant N ) > 0.9995$.

CAIE FP2 2010 November Q8

8 The owner of three driving schools, $A , B$ and $C$, wished to assess whether there was an association between passing the driving test and the school attended. He selected a random sample of learner drivers from each of his schools and recorded the numbers of passes and failures at each school. The results that he obtained are shown in the table below.

\multirow{2}{*}{}	Driving school attended
\cline { 2 - 4 }	$A$	$B$	$C$
Passes	23	15	17
Failures	27	25	43

Using a $\chi ^ { 2 }$-test and a $5 \%$ level of significance, test whether there is an association between passing or failing the driving test and the driving school attended.

CAIE FP2 2010 November Q9

9 A national athletics coach suspects that, on average, 200-metre runners' indoor times exceed their outdoor times by more than 0.1 seconds. In order to test this, the coach randomly selects eight 200 -metre runners and records their indoor and outdoor times. The results, in seconds, are shown in the table.

Runner	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Indoor time	21.5	21.8	20.9	21.2	21.4	21.4	21.2	21.0
Outdoor time	21.1	21.7	20.7	20.9	21.3	21.0	21.1	20.8

Stating suitable hypotheses and any necessary assumption that you make, test the coach's suspicion at the 2.5\% level of significance.

CAIE FP2 2010 November Q10

10 For each month of a certain year, a weather station recorded the average rainfall per day, $x \mathrm {~mm}$, and the average amount of sunshine per day, $y$ hours. The results are summarised below. $$n = 12 , \quad \Sigma x = 24.29 , \quad \Sigma x ^ { 2 } = 50.146 , \quad \Sigma y = 45.8 , \quad \Sigma y ^ { 2 } = 211.16 , \quad \Sigma x y = 88.415 .$$

Find the mean values, $\bar { x }$ and $\bar { y }$.
Calculate the gradient of the line of regression of $y$ on $x$.
Use the answers to parts (i) and (ii) to obtain the equation of the line of regression of $y$ on $x$.
Find the product moment correlation coefficient and comment, in context, on its value.
Stating your hypotheses, test at the $1 \%$ level of significance whether there is negative correlation between average rainfall per day and average amount of sunshine per day.

CAIE FP2 2010 November Q11 EITHER

A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points $A$ and $B$ on the table, which are 6 m apart. The particle is joined to $A$ by a light elastic string of natural length 2 m and modulus of elasticity 60 N , and to $B$ by a light elastic string of natural length 1 m and modulus of elasticity 20 N . The mid-point of $A B$ is $M$, and $O$ is the point between $M$ and $B$ at which the particle can rest in equilibrium. Show that $M O = 0.2 \mathrm {~m}$. The particle is held at $M$ and then released. Show that the equation of motion is $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } = - 500 y$$ where $y$ metres is the displacement from $O$ in the direction $O B$ at time $t$ seconds, and state the period of the motion. For the instant when the particle is 0.3 m from $M$ for the first time, find

the speed of the particle,
the time taken, after release, to reach this position.

CAIE FP2 2010 November Q11 OR

The continuous random variable $T$ has a negative exponential distribution with probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$ Show that for $t \geqslant 0$ the distribution function is given by $\mathrm { F } ( t ) = 1 - \mathrm { e } ^ { - \lambda t }$. The table below shows some values of $\mathrm { F } ( t )$ for the case when the mean is 20 . Find the missing value.

$t$	0	5	10	15	20	25	30	35	40
$\mathrm {~F} ( t )$	0	0.2212	0.3935		0.6321	0.7135	0.7769	0.8262	0.8647

It is thought that the lifetime of a species of insect under laboratory conditions has a negative exponential distribution with mean 20 hours. When observation starts there are 100 insects, which have been randomly selected. The lifetimes of the insects, in hours, are summarised in the table below.

Lifetime (hours)	$0 - 5$	$5 - 10$	$10 - 15$	$15 - 20$	$20 - 25$	$25 - 30$	$30 - 35$	$35 - 40$	$\geqslant 40$
Frequency	20	20	11	9	9	8	5	1	17

Calculate the expected values for each interval, assuming a negative exponential model with a mean of 20 hours, giving your values correct to 2 decimal places. Perform a $\chi ^ { 2 }$-test of goodness of fit, at the $5 \%$ level of significance, in order to test whether a negative exponential distribution, with a mean of 20 hours, is a suitable model for the lifetime of this species of insect under laboratory conditions.

CAIE FP2 2011 November Q1

1 A particle is moving in a circle of radius 2 m . At time $t \mathrm {~s}$ its velocity is $\left( t ^ { 2 } - 12 \right) \mathrm { m } \mathrm { s } ^ { - 1 }$. Find the magnitude of the resultant acceleration of the particle when $t = 4$.

CAIE FP2 2011 November Q2

2 A particle $P$ is moving in simple harmonic motion with centre $O$. When $P$ is 5 m from $O$ its speed is $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and when it is 9 m from $O$ its speed is $\frac { 3 } { 5 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Show that the amplitude of the motion is $\frac { 15 } { 2 } \sqrt { } 2 \mathrm {~m}$. Given that the greatest speed of $P$ is $3 \sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find $V$.

CAIE FP2 2011 November Q3

3 A fixed hollow sphere with centre $O$ has a smooth inner surface of radius $a$. A particle $P$ of mass $m$ is projected horizontally with speed $2 \sqrt { } ( a g )$ from the lowest point of the inner surface of the sphere. The particle loses contact with the inner surface of the sphere when $O P$ makes an angle $\theta$ with the upward vertical.

Show that $\cos \theta = \frac { 2 } { 3 }$.
Find the greatest height that $P$ reaches above the level of $O$.

CAIE FP2 2011 November Q4

4 Two smooth spheres $P$ and $Q$, of equal radius, have masses $m$ and $3 m$ respectively. They are moving in the same direction in the same straight line on a smooth horizontal table. Sphere $P$ has speed $u$ and collides directly with sphere $Q$ which has speed $k u$, where $0 < k < 1$. Sphere $P$ is brought to rest by the collision. Show that the coefficient of restitution between $P$ and $Q$ is $\frac { 3 k + 1 } { 3 ( 1 - k ) }$. One third of the total kinetic energy of the spheres is lost in the collision. Show that $$k = \frac { 1 } { 3 } ( 2 \sqrt { } 3 - 3 )$$

CAIE FP2 2011 November Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{96b6c92d-6d13-452f-84ec-37c45651b232-2_529_493_1667_826} A uniform solid sphere with centre $C$, radius $2 a$ and mass $3 M$, is pivoted about a smooth horizontal axis and hangs at rest. The point $O$ on the axis is vertically above $C$ and $O C = a$. A particle $P$ of mass $M$ is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through $O$ is $\frac { 84 } { 5 } M a ^ { 2 }$. The system is released from rest with $O P$ making a small angle $\alpha$ with the downward vertical. Find

the period of small oscillations,
the time from release until $O P$ makes an angle $\frac { 1 } { 2 } \alpha$ with the downward vertical for the first time.

CAIE FP2 2011 November Q6

6 The continuous random variable $X$ has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 1
\frac { 1 } { 2 } & 1 \leqslant x \leqslant 3
0 & x > 3 \end{cases}$$ Find the distribution function of $X$. The random variable $Y$ is defined by $Y = X ^ { 3 }$. Find

the probability density function of $Y$,
the expected value and variance of $Y$.

CAIE FP2 2011 November Q7

7 The lifetime, in hours, of a 'Trulite' light bulb is a random variable $T$. The probability density function f of $T$ is given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0
\lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \end{cases}$$ where $\lambda$ is a positive constant. Given that the mean lifetime of Trulite bulbs is 2000 hours, find the probability that a randomly chosen Trulite bulb has a lifetime of at least 1000 hours. A particular light fitting has 6 randomly chosen Trulite bulbs. Find the probability that no more than one of these bulbs has a lifetime less than 1000 hours. By using new technology, the proportion of Trulite bulbs with very short lifetimes is to be reduced. Find the least value of the new mean lifetime that will ensure that the probability that a randomly chosen Trulite bulb has a lifetime of no more than 4 hours is less than 0.001 .

CAIE FP2 2011 November Q8

8 A sample of 216 observations of the continuous random variable $X$ was obtained and the results are summarised in the following table.

Interval	$0 \leqslant x < 1$	$1 \leqslant x < 2$	$2 \leqslant x < 3$	$3 \leqslant x < 4$	$4 \leqslant x < 5$	$5 \leqslant x < 6$
Observed frequency	1	3	15	31	59	107

Interval	$0 \leqslant x < 1$	$1 \leqslant x < 2$	$2 \leqslant x < 3$	$3 \leqslant x < 4$	$4 \leqslant x < 5$	$5 \leqslant x < 6$
Expected frequency	1	7	$a$	$b$	$c$	91

Show that $a = 19$ and find the values of $b$ and $c$.
Carry out a goodness of fit test at the $10 \%$ significance level.

CAIE FP2 2011 November Q9

9 A random sample of five metal rods produced by a machine is taken. Each rod is tested for hardness. The results, in suitable units, are as follows. $$\begin{array} { l l l l l } 524 & 526 & 520 & 523 & 530 \end{array}$$ Assuming a normal distribution, calculate a $95 \%$ confidence interval for the population mean. Some adjustments are made to the machine. Assume that a normal distribution is still appropriate and that the population variance remains unchanged. A second random sample, this time of ten metal rods, is now taken. The results for hardness are as follows. $$\begin{array} { l l l l l l l l l l } 525 & 520 & 522 & 524 & 518 & 520 & 519 & 525 & 527 & 516 \end{array}$$ Stating suitable hypotheses, test at the $10 \%$ significance level whether there is any difference between the population means before and after the adjustments.

CAIE FP2 2011 November Q10 EITHER

\includegraphics[max width=\textwidth, alt={}]{96b6c92d-6d13-452f-84ec-37c45651b232-5_606_787_411_680}

A uniform rod $A B$, of weight $W$ and length $2 a$, rests with the end $A$ on a rough horizontal plane. A light inextensible string $B C$ is attached to the rod at $B$ and passes over a small smooth fixed peg $P$, which is at a distance $h$ vertically above $A$. A particle is attached at $C$ and hangs vertically. The points $A , B$ and $C$ are all in the same vertical plane. In equilibrium the rod is inclined at an angle $\theta$ to the horizontal (see diagram). The coefficient of friction between the rod and the plane is $\mu$. Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at $C$ has weight $k W$, angle $A B P = 90 ^ { \circ }$ and $h = 3 a$, find

the value of $k$,
the horizontal component of the force on $P$, in terms of $W$.

CAIE FP2 2011 November Q10 OR

The regression line of $y$ on $x$ obtained from a random sample of five pairs of values of $x$ and $y$ is $$y = 2.5 x - 1.5$$ The data is given in the following table.

$x$	1	2	4	2	6
$y$	2	3	6	$p$	$q$

Show that $p + q = 19$.
Find the values of $p$ and $q$.
Determine the value of the product moment correlation coefficient for this sample.
It is later discovered that the values of $x$ given in the table have each been divided by 10 (that is, the actual values are $10,20,40,20,60$ ). Without any further calculation, state
(a) the equation of the actual regression line of $y$ on $x$,
(b) the value of the actual product moment correlation coefficient.

CAIE FP2 2011 November Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{d7370e24-f2b2-451b-bc66-e6a6cae78cc6-2_529_493_1667_826} A uniform solid sphere with centre $C$, radius $2 a$ and mass $3 M$, is pivoted about a smooth horizontal axis and hangs at rest. The point $O$ on the axis is vertically above $C$ and $O C = a$. A particle $P$ of mass $M$ is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through $O$ is $\frac { 84 } { 5 } M a ^ { 2 }$. The system is released from rest with $O P$ making a small angle $\alpha$ with the downward vertical. Find

the period of small oscillations,
the time from release until $O P$ makes an angle $\frac { 1 } { 2 } \alpha$ with the downward vertical for the first time.

CAIE FP2 2011 November Q10 EITHER

\includegraphics[max width=\textwidth, alt={}]{d7370e24-f2b2-451b-bc66-e6a6cae78cc6-5_606_787_411_680}

the value of $k$,
the horizontal component of the force on $P$, in terms of $W$.

CAIE FP2 2011 November Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{0d4a352c-4eda-45b4-9284-60c6fc680f02-2_529_493_1667_826} A uniform solid sphere with centre $C$, radius $2 a$ and mass $3 M$, is pivoted about a smooth horizontal axis and hangs at rest. The point $O$ on the axis is vertically above $C$ and $O C = a$. A particle $P$ of mass $M$ is attached to the sphere at its lowest point (see diagram). Show that the moment of inertia of the system about the axis through $O$ is $\frac { 84 } { 5 } M a ^ { 2 }$. The system is released from rest with $O P$ making a small angle $\alpha$ with the downward vertical. Find

the period of small oscillations,
the time from release until $O P$ makes an angle $\frac { 1 } { 2 } \alpha$ with the downward vertical for the first time.

Lifetime (hours)	\(0 - 5\)	\(5 - 10\)	\(10 - 15\)	\(15 - 20\)	\(20 - 25\)	\(25 - 30\)	\(30 - 35\)	\(35 - 40\)	\(\geqslant 40\)
Frequency	20	20	11	9	9	8	5	1	17

Interval	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)	\(5 \leqslant x < 6\)
Expected frequency	1	7	\(a\)	\(b\)	\(c\)	91

\(x\)	1	2	4	2	6
\(y\)	2	3	6	\(p\)	\(q\)

\(t\)	0	5	10	15	20	25	30	35	40
\(\mathrm {~F} ( t )\)	0	0.2212	0.3935		0.6321	0.7135	0.7769	0.8262	0.8647

Questions — CAIE FP2 (474 questions)

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CAIE FP2 2010 November Q1

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CAIE FP2 2011 November Q1

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