Questions - CAIE - A-Level Maths

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

Standard +0.8

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

Moderate -0.3

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

Standard +0.3

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

Challenging +1.2

\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 $\beta$.

CAIE FP2 2010 June Q11 OR

Challenging +1.2

Aram is a packer at a supermarket checkout and the time he takes to pack a randomly chosen item has mean 1.5 s and standard deviation 0.4 s . Justifying any approximation that you make, find the probability that Aram will pack 50 randomly chosen items in less than 70 s . Find the greatest number of items that Aram could pack within 70 s with probability at least $90 \%$. Huldu is also a packer at the supermarket. The time that she takes to pack a randomly chosen item has mean 1.3 s and standard deviation 0.5 s . Aram and Huldu each have 50 items to pack. Find the probability that Huldu takes a shorter time than Aram.

CAIE FP2 2011 June Q1

Standard +0.3

1 A particle oscillates in simple harmonic motion with centre $O$. When its distance from $O$ is 3 m its speed is $16 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, and when its distance from $O$ is 4 m its speed is $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the period and amplitude of the motion.

CAIE FP2 2011 June Q2

Challenging +1.2

2 A particle of mass $m$ is attached to the mid-point of a light elastic string. The string is stretched between two points $A$ and $B$ on a smooth horizontal surface, where $A B = 2 a$. The string has modulus of elasticity $\lambda$ and natural length $2 l$, where $l < a$. The particle is in motion on the surface along a line passing through the mid-point of $A B$ and perpendicular to $A B$. When the displacement of the particle from $A B$ is $x$, the tension in the string is $T$. Given that $x$ is small enough for $x ^ { 2 }$ to be neglected, show that $$T = \frac { \lambda } { l } ( a - l )$$ The particle is slightly disturbed from its equilibrium position. Show that it will perform approximate simple harmonic motion and find the period of the motion.

CAIE FP2 2011 June Q3

Challenging +1.8

3
\includegraphics[max width=\textwidth, alt={}, center]{e8a16ec8-b6b7-4b0c-b0c1-8f5f7a9e4fa6-2_355_695_1073_726} A uniform solid hemisphere, of radius $a$ and mass $M$, is placed with its curved surface in contact with a rough plane that is inclined at an angle $\alpha$ to the horizontal. A particle $P$ of mass $m$ is attached to the rim of the hemisphere. The system rests in equilibrium with the rim of the hemisphere horizontal and $P$ at the point on the rim that is closest to the inclined plane (see diagram). Given that the coefficient of friction between the plane and the hemisphere is $\frac { 1 } { 2 }$, show that

$\tan \alpha \leqslant \frac { 1 } { 2 }$,
$m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }$.

CAIE FP2 2011 June Q4

Standard +0.8

4 Two uniform spheres $A$ and $B$, of equal radius, are at rest on a smooth horizontal table. Sphere $A$ has mass $3 m$ and sphere $B$ has mass $m$. Sphere $A$ is projected directly towards $B$, with speed $u$. The coefficient of restitution between the spheres is 0.6 . Find the speeds of $A$ and $B$ after they collide. Sphere $B$ now strikes a wall that is perpendicular to its path, rebounds and collides with $A$ again. The coefficient of restitution between $B$ and the wall is $e$. Given that the second collision between $A$ and $B$ brings $A$ to rest, find $e$.

CAIE FP2 2011 June Q5

Challenging +1.2

5 A particle $P$ of mass $m$ is placed at the point $Q$ on the outer surface of a fixed smooth sphere with centre $O$ and radius $a$. The acute angle between $O Q$ and the upward vertical is $\alpha$, where $\cos \alpha = \frac { 9 } { 10 }$. The particle is released from rest and begins to move in a vertical circle on the surface of the sphere. Show that $P$ loses contact with the sphere when $O P$ makes an angle $\theta$ with the upward vertical, where $\cos \theta = \frac { 3 } { 5 }$, and find the speed of $P$ at this instant. Show that, in the subsequent motion, when $P$ is at a distance $\frac { 7 } { 5 } a$ from the vertical diameter through $O$, its distance below the horizontal through $O$ is $\frac { 31 } { 30 } a$.

CAIE FP2 2011 June Q6

Standard +0.3

6 A random sample of residents in a town took part in a survey. They were asked whether they would prefer the local council to spend money on improving the local bus service or on improving the quality of road surfaces. The responses are shown in the following table, classified according to the area of the town in which the residents live.

	Area 1	Area 2	Area 3
Local bus service	73	36	30
Road surfaces	47	44	20

Using a $5 \%$ significance level, test whether there is an association between the area lived in and preference for improving the local bus service or improving the quality of road surfaces.

CAIE FP2 2011 June Q7

Standard +0.3

7 A greengrocer claims that his cabbages have a mean mass of more than 1.2 kg . In order to check his claim, he weighs 10 cabbages, chosen at random from his stock. The masses, in kg, are as follows. $$\begin{array} { l l l l l l l l l l } 1.26 & 1.24 & 1.17 & 1.23 & 1.18 & 1.25 & 1.19 & 1.20 & 1.21 & 1.18 \end{array}$$ Stating any assumption that you make, test at the $10 \%$ significance level whether the greengrocer's claim is supported by this evidence.

CAIE FP2 2011 June Q8

Standard +0.3

8 In a crossword competition the times, $x$ minutes, taken by a random sample of 6 entrants to complete a crossword are summarised as follows. $$\Sigma x = 210.9 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 151.2$$ The time to complete a crossword has a normal distribution with mean $\mu$ minutes. Calculate a $95 \%$ confidence interval for $\mu$. Assume now that the standard deviation of the population is known to be 5.6 minutes. Find the smallest sample size that would lead to a $95 \%$ confidence interval for $\mu$ of width at most 5 minutes.

CAIE FP2 2011 June Q9

Challenging +1.2

9 Mr Lee asserts that boys are slower than girls at completing a particular mathematical puzzle. In order to test his assertion, a random sample of 40 boys and a random sample of 60 girls are selected from a large group of students who attempted the puzzle. The times taken by the boys, $b$ minutes, and the times taken by the girls, $g$ minutes, are summarised as follows. $$\Sigma b = 92.0 \quad \Sigma b ^ { 2 } = 216.5 \quad \Sigma g = 129.8 \quad \Sigma g ^ { 2 } = 288.8$$ Test at the $2.5 \%$ significance level whether this evidence supports Mr Lee's assertion.

CAIE FP2 2011 June Q10

Standard +0.3

10 The mid-day temperature, $x ^ { \circ } \mathrm { C }$, and the amount of sunshine, $y$ hours, were recorded at a winter holiday resort on each of 12 days, chosen at random during the winter season. The results are summarised as follows. $$\Sigma x = 18.7 \quad \Sigma x ^ { 2 } = 106.43 \quad \Sigma y = 34.7 \quad \Sigma y ^ { 2 } = 133.43 \quad \Sigma x y = 92.01$$

Find the product moment correlation coefficient for the data.
Stating your hypotheses, test at the $1 \%$ significance level whether there is a non-zero correlation between mid-day temperature and amount of sunshine.
Use the equation of a suitable regression line to estimate the number of hours of sunshine on a day when the mid-day temperature is $2 ^ { \circ } \mathrm { C }$.