Questions FP2 - A-Level Maths

exactly 4 throws are required to obtain a pair of heads,
fewer than 6 throws are required to obtain a pair of heads.

CAIE FP2 2014 June Q7

Standard +0.3

7 The random variable $T$ is the lifetime, in hours, of a randomly chosen decorative light bulb of a particular type. It is given that $T$ has a negative exponential distribution with mean 1000 hours.

Write down the probability density function of $T$.
Find the probability that a randomly chosen bulb of this type has a lifetime of more than 2000 hours. A display uses 10 randomly chosen bulbs of this type, and they are all switched on simultaneously. Find the greatest value of $t$ such that the probability that they are all alight at time $t$ hours is at least 0.9 .

CAIE FP2 2014 June Q8

Standard +0.3

8 Weekly expenses claimed by employees at two different branches, $A$ and $B$, of a large company are being compared. Expenses claimed by an employee at branch $A$ and by an employee at branch $B$ are denoted by $\$ x$ and $\$ y$ respectively. A random sample of 60 employees from branch $A$ and a random sample of 50 employees from branch $B$ give the following summarised data. $$\Sigma x = 6060 \quad \Sigma x ^ { 2 } = 626220 \quad \Sigma y = 4750 \quad \Sigma y ^ { 2 } = 464500$$ Using a $2 \%$ significance level, test whether, on average, employees from branch $A$ claim the same as employees from branch $B$.

CAIE FP2 2014 June Q9

Standard +0.8

9 A random sample of 200 observations of the continuous random variable $X$ was taken and the values are summarised in the following table.

Interval	$1 \leqslant x < 2$	$2 \leqslant x < 3$	$3 \leqslant x < 4$	$4 \leqslant x < 5$	$5 \leqslant x < 6$	$6 \leqslant x < 7$	$7 \leqslant x < 8$
Observed frequency	63	45	32	25	22	7	6

It is required to test the goodness of fit of the distribution with probability density function $f$ given by $$f ( x ) = \begin{cases} \frac { 1 } { x \ln 8 } & 1 \leqslant x < 8 \\ 0 & \text { otherwise } \end{cases}$$ The relevant expected frequencies, correct to 2 decimal places, are given in the following table.

Interval	$1 \leqslant x < 2$	$2 \leqslant x < 3$	$3 \leqslant x < 4$	$4 \leqslant x < 5$	$5 \leqslant x < 6$	$6 \leqslant x < 7$	$7 \leqslant x < 8$
Expected frequency	66.67	$p$	27.67	$q$	17.54	14.83	12.84

Show that $p = 39.00$, correct to 2 decimal places, and find the value of $q$. Carry out a goodness of fit test at the 5\% significance level.

CAIE FP2 2014 June Q10

Standard +0.3

10 Samples of rock from a number of geological sites were analysed for the quantities of two types, $X$ and $Y$, of rare minerals. The results, in milligrams, for 10 randomly chosen samples, each of 10 kg , are summarised as follows. $$\Sigma x = 866 \quad \Sigma x ^ { 2 } = 121276 \quad \Sigma y = 639 \quad \Sigma y ^ { 2 } = 55991 \quad \Sigma x y = 73527$$ Find the product moment correlation coefficient. Stating your hypotheses, test at the $5 \%$ significance level whether there is non-zero correlation between quantities of the two rare minerals. Find the equation of the regression line of $x$ on $y$ in the form $x = p y + q$, where $p$ and $q$ are constants to be determined.

CAIE FP2 2014 June Q11 EITHER

Challenging +1.8

\includegraphics[max width=\textwidth, alt={}]{ab5f2781-e5ce-4fce-bc95-9d7f55ea66d9-5_869_621_370_762}

The points $C$ and $D$ are at a distance $( 2 \sqrt { } 3 ) a$ apart on a horizontal surface. A rough peg $A$ is fixed at a vertical distance $6 a$ above $C$ and a smooth peg $B$ is fixed at a vertical distance $4 a$ above $D$. A uniform rectangular frame $P Q R S$, with $P Q = 3 a$ and $Q R = 6 a$, is made of rigid thin wire and has weight $W$. It rests in equilibrium in a vertical plane with $P S$ on $A$ and $S R$ on $B$, and with angle $S A C = 30 ^ { \circ }$ (see diagram).

Show that $A B = 4 a$ and that angle $S A B = 30 ^ { \circ }$.
Show that the normal reaction at $A$ is $\frac { 1 } { 2 } W$.
Find the frictional force at $A$.

CAIE FP2 2014 June Q11 OR

Challenging +1.2

The time taken for a randomly chosen student at College $P$ to complete a particular puzzle has a normal distribution with mean $\mu$ minutes. The times, $x$ minutes, are recorded for a random sample of 8 students chosen from the college. The results are summarised as follows. $$\Sigma x = 42.8 \quad \Sigma x ^ { 2 } = 236.0$$ Find a 95\% confidence interval for $\mu$. A test is carried out on this sample data, at the $10 \%$ significance level. The test supports the claim that $\mu > k$. Find the greatest possible value of $k$. A random sample, of size 12, is taken from the students at College $Q$. Their times to complete the puzzle give a sample mean of 4.60 minutes and an unbiased variance estimate of 1.962 minutes ${ } ^ { 2 }$. Use a 2 -sample test at the $10 \%$ significance level to test whether the mean time for students at College $Q$ to complete the puzzle is less than the mean time for students at College $P$ to complete the puzzle. You should state any assumptions necessary for the test to be valid.

CAIE FP2 2014 June Q4

Challenging +1.2

4
\includegraphics[max width=\textwidth, alt={}, center]{ae8d874a-5c1d-45bb-b853-d12006004b7f-2_519_583_1384_781} A smooth wire is in the form of an $\operatorname { arc } A B$ of a circle, of radius $a$, that subtends an obtuse angle $\pi - \theta$ at the centre $O$ of the circle. It is given that $\sin \theta = \frac { 1 } { 4 }$. The wire is fixed in a vertical plane, with $A O$ horizontal and $B$ below the level of $O$ (see diagram). A small bead of mass $m$ is threaded on the wire and projected vertically downwards from $A$ with speed $\sqrt { } \left( \frac { 3 } { 10 } g a \right)$.

Find the reaction between the bead and the wire when the bead is vertically below $O$.
Find the speed of the bead as it leaves the wire at $B$.
Show that the greatest height reached by the bead is $\frac { 1 } { 8 } a$ above the level of $O$.

CAIE FP2 2014 June Q11 EITHER

Challenging +1.8

\includegraphics[max width=\textwidth, alt={}]{ae8d874a-5c1d-45bb-b853-d12006004b7f-5_871_621_370_762}

Show that $A B = 4 a$ and that angle $S A B = 30 ^ { \circ }$.
Show that the normal reaction at $A$ is $\frac { 1 } { 2 } W$.
Find the frictional force at $A$.

CAIE FP2 2015 June Q1

Standard +0.3

1 A particle $P$ is moving in a circle of radius 0.25 m . At time $t$ seconds, its velocity is $\left( 2 t ^ { 2 } - 4 t + 3 \right) \mathrm { m } \mathrm { s } ^ { - 1 }$. At a particular time $T$ seconds, where $T > 0$, the magnitude of the transverse component of the acceleration of $P$ is $6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Find the magnitude of the radial component of the acceleration of $P$ at this instant.

CAIE FP2 2015 June Q2

Challenging +1.2

2 A particle $P$ moves on a straight line $A O B$ in simple harmonic motion. The centre of the motion is $O$, and $P$ is instantaneously at rest at $A$ and $B$. The point $C$ is on the line $A O B$, between $A$ and $O$, and $C O = 10 \mathrm {~m}$. When $P$ is at $C$, the magnitude of its acceleration is $0.625 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and it is moving towards $O$ with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find

the period of the motion, in terms of $\pi$,
the amplitude of the motion. The point $M$ is the mid-point of $O B$. Find the time that $P$ takes to travel directly from $C$ to $M$.

CAIE FP2 2015 June Q3

Standard +0.8

3 A particle $P$, of mass $m$, is placed at the highest point of a fixed solid smooth sphere with centre $O$ and radius $a$. The particle $P$ is given a horizontal speed $u$ and it moves in part of a vertical circle, with centre $O$, on the surface of the sphere. When $O P$ makes an angle $\theta$ with the upward vertical, and $P$ is still in contact with the surface of the sphere, the speed of $P$ is $v$ and the reaction of the sphere on $P$ has magnitude $R$. Show that $R = m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }$. The particle loses contact with the sphere at the instant when $v = 2 u$. Find $u$ in terms of $a$ and $g$.

CAIE FP2 2015 June Q4

Challenging +1.2

4
\includegraphics[max width=\textwidth, alt={}, center]{baea9836-ea05-442f-9e87-a2a1480dc74c-2_338_957_1482_593} A uniform rod $B C$ of length $2 a$ and weight $W$ is hinged to a fixed point at $B$. A particle of weight $3 W$ is attached to the rod at $C$. The system is held in equilibrium by a light elastic string of natural length $\frac { 3 } { 5 } a$ in the same vertical plane as the rod. One end of the elastic string is attached to the rod at $C$ and the other end is attached to a fixed point $A$ which is at the same horizontal level as $B$. The rod and the string each make an angle of $30 ^ { \circ }$ with the horizontal (see diagram). Find

the modulus of elasticity of the string,
the magnitude and direction of the force acting on the rod at $B$.

CAIE FP2 2015 June Q5

Challenging +1.2

5 Three uniform small smooth spheres $A , B$ and $C$ have equal radii and masses $3 m , 2 m$ and $m$ respectively. The spheres are at rest in a straight line on a smooth horizontal surface, with $B$ between $A$ and $C$. The coefficient of restitution between $A$ and $B$ is $e$ and the coefficient of restitution between $B$ and $C$ is $e ^ { \prime }$. Sphere $A$ is projected directly towards $B$ with speed $u$. Show that, after the collision between $B$ and $C$, the speed of $C$ is $\frac { 2 } { 5 } u ( 1 + e ) \left( 1 + e ^ { \prime } \right)$ and find the corresponding speed of $B$. After this collision between $B$ and $C$ it is found that each of the three spheres has the same momentum. Find the values of $e$ and $e ^ { \prime }$.

CAIE FP2 2015 June Q6

Standard +0.8

6 The independent random variables $X$ and $Y$ have distributions with the same variance $\sigma ^ { 2 }$. Random samples of $N$ observations of $X$ and 10 observations of $Y$ are taken, and the results are summarised by $$\Sigma x = 5 , \quad \Sigma x ^ { 2 } = 11 , \quad \Sigma y = 10 , \quad \Sigma y ^ { 2 } = 160 .$$ These data give a pooled estimate of 12 for $\sigma ^ { 2 }$. Find $N$.

CAIE FP2 2015 June Q7

Challenging +1.2

7 A random sample of 8 sunflower plants is taken from the large number grown by a gardener, and the heights of the plants are measured. A 95\% confidence interval for the population mean, $\mu$ metres, is calculated from the sample data as $1.17 < \mu < 2.03$. Given that the height of a sunflower plant is denoted by $x$ metres, find the values of $\Sigma x$ and $\Sigma x ^ { 2 }$ for this sample of 8 plants.

CAIE FP2 2015 June Q8

Standard +0.3

For a random sample of ten pairs of values of $x$ and $y$ taken from a bivariate distribution, the equations of the regression lines of $y$ on $x$ and of $x$ on $y$ are, respectively, $$y = 0.38 x + 1.41 \quad \text { and } \quad x = 0.96 y + 7.47$$
1. Find the value of the product moment correlation coefficient for this sample.
2. Using a $5 \%$ significance level, test whether there is positive correlation between the variables.
For a random sample of $n$ pairs of values of $u$ and $v$ taken from another bivariate distribution, the value of the product moment correlation coefficient is 0.507 . Using a test at the $5 \%$ significance level, there is evidence of non-zero correlation between the variables. Find the least possible value of $n$.

CAIE FP2 2015 June Q9

Standard +0.8

9 Cotton cloth is sold from long rolls of cloth. The number of flaws on a randomly chosen piece of cloth of length $a$ metres has a Poisson distribution with mean $0.8 a$. The random variable $X$ is the length of cloth, in metres, between two successive flaws.

Explain why, for $x \geqslant 0 , \mathrm { P } ( X > x ) = \mathrm { e } ^ { - 0.8 x }$.
Find the probability that there is at least one flaw in a 4 metre length of cloth.
Find
(a) the distribution function of $X$,
(b) the probability density function of $X$,
(c) the interquartile range of $X$.

CAIE FP2 2015 June Q10

Standard +0.3

10 Young children at a primary school are learning to throw a ball as far as they can. The distance thrown at the beginning of the school year and the distance thrown at the end of the same school year are recorded for each child. The distance thrown, in metres, at the beginning of the year is denoted by $x$; the distance thrown, in metres, at the end of the year is denoted by $y$. For a random sample of 10 children, the results are shown in the following table.

Child	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
$x$	5.2	4.1	3.7	5.4	7.6	6.1	3.2	4.0	3.5	8.0
$y$	6.2	4.8	5.0	5.6	7.7	7.0	4.0	4.5	3.6	8.5

$$\left[ \Sigma x = 50.8 , \quad \Sigma x ^ { 2 } = 284.16 , \quad \Sigma y = 56.9 , \quad \Sigma y ^ { 2 } = 347.59 , \quad \Sigma x y = 313.28 . \right]$$ A particular child threw the ball a distance of 7.0 metres at the beginning of the year, but he could not throw at the end of the year because he had broken his arm. By finding the equation of an appropriate regression line, estimate the distance this child would have thrown at the end of the year. The teacher suspects that, on average, the distance thrown by a child increases between the two throws by more than 0.4 metres. Stating suitable hypotheses and assuming a normal distribution, test the teacher's suspicion at the $5 \%$ significance level.

CAIE FP2 2015 June Q11 EITHER

Hard +2.3

\includegraphics[max width=\textwidth, alt={}]{baea9836-ea05-442f-9e87-a2a1480dc74c-5_691_698_440_721}

A uniform disc, with centre $O$ and radius $a$, is surrounded by a uniform concentric ring with radius $3 a$. The ring is rigidly attached to the rim of the disc by four symmetrically positioned uniform rods, each of mass $\frac { 3 } { 2 } m$ and length $2 a$. The disc and the ring each have mass $2 m$. The rods meet the ring at the points $A , B , C$ and $D$. The disc, the ring and the rods are all in the same plane (see diagram). Show that the moment of inertia of this object about an axis through $O$ perpendicular to the plane of the object is $45 m a ^ { 2 }$. Find the moment of inertia of the object about an axis $l$ through $A$ in the plane of the object and tangential to the ring. A particle of mass $3 m$ is now attached to the object at $C$. The object, including the additional particle, is suspended from the point $A$ and hangs in equilibrium. It is free to rotate about the axis $l$. The centre of the disc is given a horizontal speed $u$. When, in the subsequent motion, the object comes to instantaneous rest, $C$ is below the level of $A$ and $A C$ makes an angle $\sin ^ { - 1 } \left( \frac { 1 } { 4 } \right)$ with the horizontal. Find $u$ in terms of $a$ and $g$.

CAIE FP2 2015 June Q11 OR

Challenging +1.2

Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws, $x$, needed is recorded and the results are summarised in the following table.

$x$	1	2	3	4	5	6	$\geqslant 7$
Frequency	126	43	22	3	5	1	0

State a type of distribution that could be used to fit the data given in the table above. Fit a distribution of this type in which the probability of throwing an even number for each die is 0.6 and carry out a goodness of fit test at the 5\% significance level. For each of these dice, it is known that the probability of obtaining a 6 when it is thrown is 0.25 . Ten of these dice are each thrown 5 times. Find the probability that at least one 6 is obtained on exactly 4 of the 10 dice.

CAIE FP2 2015 June Q4

Challenging +1.2

4
\includegraphics[max width=\textwidth, alt={}, center]{eb3dccaf-d151-472d-82f3-6ba215b0b7f0-2_339_957_1482_593} A uniform rod $B C$ of length $2 a$ and weight $W$ is hinged to a fixed point at $B$. A particle of weight $3 W$ is attached to the rod at $C$. The system is held in equilibrium by a light elastic string of natural length $\frac { 3 } { 5 } a$ in the same vertical plane as the rod. One end of the elastic string is attached to the rod at $C$ and the other end is attached to a fixed point $A$ which is at the same horizontal level as $B$. The rod and the string each make an angle of $30 ^ { \circ }$ with the horizontal (see diagram). Find

the modulus of elasticity of the string,
the magnitude and direction of the force acting on the rod at $B$.

CAIE FP2 2015 June Q11 EITHER

Hard +2.3

\includegraphics[max width=\textwidth, alt={}]{eb3dccaf-d151-472d-82f3-6ba215b0b7f0-5_691_698_440_721}

CAIE FP2 2015 June Q1

Standard +0.3

1 Two uniform small smooth spheres, $A$ and $B$, of equal radii and masses 2 kg and 3 kg respectively, are at rest and not in contact on a smooth horizontal plane. Sphere $A$ receives an impulse of magnitude 8 N s in the direction $A B$. The coefficient of restitution between the spheres is $e$. Find, in terms of $e$, the speeds of $A$ and $B$ after $A$ collides with $B$. Given that the spheres move in opposite directions after the collision, show that $e > \frac { 2 } { 3 }$.

CAIE FP2 2015 June Q2

7 marks Standard +0.8

2
\includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-2_547_811_625_667} A uniform sphere $P$ of mass $m$ is at rest on a smooth horizontal table. The sphere is projected along the table with speed $u$ and strikes a smooth vertical barrier $A$ at an acute angle $\alpha$. It then strikes another smooth vertical barrier $B$ which is at right angles to $A$ (see diagram). The coefficient of restitution between $P$ and each of the barriers is $e$. Show that the final direction of motion of $P$ makes an angle $\frac { 1 } { 2 } \pi - \alpha$ with the barrier $B$ and find the total loss in kinetic energy as a result of the two impacts. [7]

Interval	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)	\(5 \leqslant x < 6\)	\(6 \leqslant x < 7\)	\(7 \leqslant x < 8\)
Expected frequency	66.67	\(p\)	27.67	\(q\)	17.54	14.83	12.84

Child	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
\(x\)	5.2	4.1	3.7	5.4	7.6	6.1	3.2	4.0	3.5	8.0
\(y\)	6.2	4.8	5.0	5.6	7.7	7.0	4.0	4.5	3.6	8.5

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