Questions - CAIE FP2 - A-Level Maths

the angular speed of the smaller disc,
the magnitude of the acceleration of a point $P$ on the circumference of the larger disc, and the angle between the direction of this acceleration and $P A$.

CAIE FP2 2010 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-3_378_625_1272_758} A light elastic band, of total natural length $a$ and modulus of elasticity $\frac { 1 } { 2 } m g$, is stretched over two small smooth pins fixed at the same horizontal level and at a distance $a$ apart. A particle of mass $m$ is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle $\beta$ with the vertical (see diagram). Express the tension in the band in terms of $m , g$ and $\beta$, and hence show that $\beta = \frac { 1 } { 4 } \pi$. The particle is given a velocity of magnitude $\sqrt { } ( a g )$ vertically downwards. At time $t$ the displacement of the particle from its equilibrium position is $x$. Show that, neglecting air resistance, $$\ddot { x } = - \frac { 2 g } { a } x .$$ Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time.

CAIE FP2 2010 June Q6

6 The lifetime, $X$ days, of a particular insect is such that $\log _ { 10 } X$ has a normal distribution with mean 1.5 and standard deviation 0.2. Find the median lifetime. Find also $\mathrm { P } ( X \geq 50 )$.

CAIE FP2 2010 June Q7

7 The continuous random variable $X$ has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
1 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0 \end{cases}$$ For a random value of $X$, find the probability that 2 lies between $X$ and $4 X$. Find also the expected value of the width of the interval ( $X , 4 X$ ).

CAIE FP2 2010 June Q8

8 An examination involved writing an essay. In order to compare the time taken to write the essay by students in two large colleges, a sample of 12 students from college $A$ and a sample of 8 students from college $B$ were randomly selected. The times, $t _ { A }$ and $t _ { B }$, taken for these students to write the essay were measured, correct to the nearest minute, and are summarised by $$n _ { A } = 12 , \quad \Sigma t _ { A } = 257 , \quad \Sigma t _ { A } ^ { 2 } = 5629 , \quad n _ { B } = 8 , \quad \Sigma t _ { B } = 206 , \quad \Sigma t _ { B } ^ { 2 } = 5359$$ Stating any required assumptions, calculate a $95 \%$ confidence interval for the difference in the population means. State, giving a reason, whether your confidence interval supports the statement that the population means, for the two colleges, are equal.

CAIE FP2 2010 June Q9

9 A set of 20 pairs of bivariate data $( x , y )$ is summarised by $$\Sigma x = 200 , \quad \Sigma x ^ { 2 } = 2125 , \quad \Sigma y = 240 , \quad \Sigma y ^ { 2 } = 8245 .$$ The product moment correlation coefficient is - 0.992 .

What does the value of the product moment correlation coefficient indicate about a scatter diagram of the data points?
Find the equation of the regression line of $y$ on $x$.
The equation of the regression line of $x$ on $y$ is $x = a ^ { \prime } + b ^ { \prime } y$. Find the value of $b ^ { \prime }$.

CAIE FP2 2010 June Q10

10 Three new flu vaccines, $A , B$ and $C$, were tested on 500 volunteers. The vaccines were assigned randomly to the volunteers and 178 received $A , 149$ received $B$ and 173 received $C$. During the following year, 30 of the volunteers given $A$ caught flu, 29 of the volunteers given $B$ caught flu, and 16 of the volunteers given $C$ caught flu. Carry out a suitable test for independence at the 5\% significance level. Without using a statistical test, decide which of the vaccines appears to be most effective.

CAIE FP2 2010 June Q11 EITHER

\includegraphics[max width=\textwidth, alt={}]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-5_538_572_456_788}

A uniform disc, of mass $4 m$ and radius $a$, and a uniform ring, of mass $m$ and radius $2 a$, each have centre $O$. A wheel is made by fixing three uniform rods, $O A , O B$ and $O C$, each of mass $m$ and length $2 a$, to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through $A$, perpendicular to the plane of the wheel, is $42 m a ^ { 2 }$. The axis through $A$ is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with $O$ above the level of $A$ and $A O$ making an angle of $30 ^ { \circ }$ with the horizontal. Find the angular speed of the wheel when $A O$ is horizontal. When $A O$ is horizontal the disc becomes detached from the wheel. Find the angle that $A O$ makes with the horizontal when the wheel first comes to instantaneous rest.

CAIE FP2 2010 June Q11 OR

The continuous random variable $T$ has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 2
\frac { 2 } { ( t - 1 ) ^ { 3 } } & t \geqslant 2 \end{cases}$$

Find the distribution function of $T$, and find also $\mathrm { P } ( T > 5 )$.
Consecutive independent observations of $T$ are made until the first observation that exceeds 5 is obtained. The random variable $N$ is the total number of observations that have been made up to and including the observation exceeding 5. Find $\mathrm { P } ( N > \mathrm { E } ( N ) )$.
Find the probability density function of $Y$, where $Y = \frac { 1 } { T - 1 }$.

CAIE FP2 2010 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{d24c9c0b-b8f6-4407-8b93-81d90285b60d-2_159_707_1443_721} Two perfectly elastic small smooth spheres $A$ and $B$ have masses $3 m$ and $m$ respectively. They lie at rest on a smooth horizontal plane with $B$ at a distance $a$ from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and $B$ is between $A$ and the barrier (see diagram). Sphere $A$ is projected towards sphere $B$ with speed $u$ and, after the collision between the spheres, $B$ hits the barrier. The coefficient of restitution between $B$ and the barrier is $\frac { 1 } { 2 }$. Find the speeds of $A$ and $B$ immediately after they first collide, and the distance from the barrier of the point where they collide for the second time.

CAIE FP2 2010 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{d24c9c0b-b8f6-4407-8b93-81d90285b60d-3_506_969_255_587} Two coplanar discs, of radii 0.5 m and 0.3 m , rotate about their centres $A$ and $B$ respectively, where $A B = 0.8 \mathrm {~m}$. At time $t$ seconds the angular speed of the larger disc is $\frac { 1 } { 2 } t \mathrm { rad } \mathrm { s } ^ { - 1 }$ (see diagram). There is no slipping at the point of contact. For the instant when $t = 2$, find

the angular speed of the smaller disc,
the magnitude of the acceleration of a point $P$ on the circumference of the larger disc, and the angle between the direction of this acceleration and $P A$.

CAIE FP2 2010 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{d24c9c0b-b8f6-4407-8b93-81d90285b60d-3_378_625_1272_758} A light elastic band, of total natural length $a$ and modulus of elasticity $\frac { 1 } { 2 } m g$, is stretched over two small smooth pins fixed at the same horizontal level and at a distance $a$ apart. A particle of mass $m$ is attached to the lower part of the band and when the particle is in equilibrium the sloping parts of the band each make an angle $\beta$ with the vertical (see diagram). Express the tension in the band in terms of $m , g$ and $\beta$, and hence show that $\beta = \frac { 1 } { 4 } \pi$. The particle is given a velocity of magnitude $\sqrt { } ( a g )$ vertically downwards. At time $t$ the displacement of the particle from its equilibrium position is $x$. Show that, neglecting air resistance, $$\ddot { x } = - \frac { 2 g } { a } x .$$ Show that the particle passes through the level of the pins in the subsequent motion, and find the time taken to reach this level for the first time.

CAIE FP2 2010 June Q11 EITHER

\includegraphics[max width=\textwidth, alt={}]{d24c9c0b-b8f6-4407-8b93-81d90285b60d-5_538_572_456_788}

CAIE FP2 2010 June Q1

1 A uniform disc with centre $O$ has mass $m$ and radius $a$. It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through $O$. One end of a light inextensible string is attached to a point on the circumference and is wrapped several times round the circumference. A particle $P$, of mass $2 m$, is attached to the free end of the string and the disc is held at rest with $P$ hanging freely. The system is released from rest. Assuming that resistances may be neglected, find the acceleration of $P$.

CAIE FP2 2010 June Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-2_582_798_616_671} A particle of mass $m$ is attached to the end $B$ of a light inextensible string. The other end of the string is attached to a fixed point $A$ which is at a distance $a$ above the vertex $V$ of a circular cone of semi-vertical angle $60 ^ { \circ }$. The axis of the cone is vertical. The particle moves with constant speed $u$ in a horizontal circle on the smooth surface of the cone. The string makes a constant angle of $30 ^ { \circ }$ with the vertical (see diagram). The tension in the string and the magnitude of the normal force acting on the particle are denoted by $T$ and $R$ respectively. Show that $$T = \frac { m } { \sqrt { } 3 } \left( g + \frac { 2 u ^ { 2 } } { a } \right) ,$$ and find a similar expression for $R$. Deduce that $u ^ { 2 } \leqslant \frac { 1 } { 2 } g a$.

CAIE FP2 2010 June Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-3_351_314_255_918} A spring balance is modelled by a vertical light elastic spring $A B$, of natural length 0.25 m and modulus of elasticity $\lambda \mathrm { N }$. The bottom end $B$ of the spring is fixed, and the top end $A$ is attached to a small tray of mass 0.1 kg which is free to move vertically (see diagram). When in the equilibrium position, $A B = 0.24 \mathrm {~m}$. Show that $\lambda = 25$. The tray is pushed down by 0.02 m to the point $C$ and released from rest. At time $t$ seconds after release the displacement of the tray from the equilibrium position is $x \mathrm {~m}$. Show that $$\ddot { x } = - 1000 x .$$ Find the time taken for the tray to move a distance of 0.03 m from $C$.

CAIE FP2 2010 June Q4

4 A small ball $P$, of mass 40 grams, is dropped from rest at a point $A$ which is 10 m above a fixed horizontal plane. At the same instant an identical ball $Q$ is dropped from rest at the point $B$, which is vertically below $A$ and at a height of 5 m above the plane. The coefficient of restitution between $Q$ and the plane is $\frac { 1 } { 2 }$. Find the magnitude of the impulse exerted on $Q$ by the plane. The balls collide after $Q$ rebounds from the plane and before $Q$ hits the plane again. Find the height above the plane of the point at which the collision occurs.

CAIE FP2 2010 June Q5

3 marks

5
\includegraphics[max width=\textwidth, alt={}, center]{f6887893-66c5-40df-ba8d-9439a5c268eb-3_531_908_1674_616} A rectangular pool table $K L M N$ has $K L = a$ and $K N = 2 a$. A ball lies at rest on the table just outside the pocket at $L$ and is projected along the table with speed $u$ in a direction making an angle $\theta$ with the edge $L M$. The ball hits the edge $K N$ at $Y$, rebounds to hit the edge $L M$ at $X$ and then rebounds into the pocket at $N$. Angle $L X Y$ is denoted by $\phi$ (see diagram). The coefficient of restitution between the ball and an edge is $\frac { 3 } { 4 }$, and all resistances to motion may be neglected. Show that $\tan \phi = \frac { 3 } { 4 } \tan \theta$. [3] Show that $X M = \left( 2 - \frac { 7 } { 3 } \cot \theta \right) a$, and find the value of $\theta$. Find the speed with which the ball reaches $N$, giving the answer in the form $k u$, where $k$ is correct to 3 significant figures.

CAIE FP2 2010 June Q6

6 The amount of caffeine in a randomly selected cup of coffee dispensed by a machine has a normal distribution. The amount of caffeine in each of a random sample of 25 cups was measured. The sample mean was 110.4 mg and the unbiased estimate of the population variance was $50.42 \mathrm { mg } ^ { 2 }$. Calculate a 90\% confidence interval for the mean amount of caffeine dispensed.

CAIE FP2 2010 June Q7

7 Benford's Law states that, in many tables containing large numbers of numerical values, the probability distribution of the leading non-zero digit $D$ is given by $$\mathrm { P } ( D = d ) = \log _ { 10 } \left( \frac { d + 1 } { d } \right) , \quad d = 1,2 , \ldots , 9 .$$ The following table shows a summary of a random sample of 100 non-zero leading digits taken from a table of cumulative probabilities for the Poisson distribution.

Leading digit	1	2	3	4	5	$\geqslant 6$
Frequency	22	21	13	11	11	22

Carry out a suitable goodness of fit test at the 10\% significance level.

CAIE FP2 2010 June Q8

8 A certain mechanical component has a lifetime, $T$ months, which has a negative exponential distribution with mean 2.5.

A machine is fitted with 5 of these components which function independently.
1. Find the probability that all 5 components are operating 3 months after being fitted.
2. Find also the probability that exactly two components fail within one month of being fitted.
Show that the probability that $n$ independent components are all operating $c$ months after being fitted is equal to the probability that a single component is operating $n c$ months after being fitted.

CAIE FP2 2010 June Q9

The following are values of the product moment correlation coefficient between the $x$ and $y$ values of three different large samples of bivariate data. State what each indicates about the appearance of a scatter diagram illustrating the data.
1. - 1 ,
2. 0.02 ,
3. 0.92 .
In 1852 Dr William Farr published data on deaths due to cholera during an outbreak of the disease in London. The table shows the altitude (in feet, above the level of the river Thames) at which people lived and the corresponding number of deaths from cholera per 10000 people.
Altitude, $x$ 10 30 50 70 90 100 350
Number of deaths, $y$ 102 65 34 27 22 17 8
$$\left[ \Sigma x = 700 , \Sigma x ^ { 2 } = 149000 , \Sigma y = 275 , \Sigma y ^ { 2 } = 17351 , \Sigma x y = 13040 . \right]$$
1. Calculate the product moment correlation coefficient.
2. Test, at the $5 \%$ significance level, whether there is evidence of negative correlation.

CAIE FP2 2010 June Q10

10 Carpal Tunnel syndrome is a condition which affects a person's ability to grip with their hands. Researchers tested a treatment for this syndrome which was applied to 8 randomly chosen patients. A pre-treatment and a post-treatment test of grip was given to each patient, with the following results, measured in kg.

Patient	1	2	3	4	5	6	7	8
Pre-treatment grip	24.3	29.5	28.0	28.5	21.5	28.7	25.1	26.3
Post-treatment grip	28.3	34.6	30.3	31.6	21.5	29.8	26.0	27.5

Stating any required assumption, test, at the $1 \%$ significance level, whether the mean grip of people with the syndrome increases after undergoing the treatment. It is given that there is evidence at the $10 \%$ significance level that the mean grip increases by more than $w \mathrm {~kg}$. Find an inequality for $w$.

CAIE FP2 2010 June Q11 EITHER

\includegraphics[max width=\textwidth, alt={}]{f6887893-66c5-40df-ba8d-9439a5c268eb-5_456_615_1210_765}

Two uniform rods $A B$ and $A C$ have lengths $2 a$ and $4 a$ and weights $W$ and $2 W$ respectively. They are freely hinged together at $A$ and rest in equilibrium in a vertical plane with $B$ and $C$ in contact with two rough parallel vertical walls. The plane containing the rods is perpendicular to the walls. The rods $A B$ and $A C$ each make an angle $\beta$ with the vertical (see diagram). Show that the magnitude of the frictional force acting on $A B$ at $B$ is $\frac { 5 } { 4 } W$. Given that the coefficient of friction at $B$ and at $C$ is $\mu$, find the set of possible values of $\mu$ in terms of $\beta$.

Altitude, \(x\)	10	30	50	70	90	100	350
Number of deaths, \(y\)	102	65	34	27	22	17	8

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