Questions - CAIE FP2 - A-Level Maths

Show that the probability that Jill scores on her 5th kick is $\frac { 1 } { 48 }$.
Show that the probability that Kate wins the game on her $n$th kick is $\frac { 1 } { 3 \times 2 ^ { n } }$.
Find the probability that Jill wins the game.
Find the probability that the game is a draw.

CAIE FP2 2013 June Q11 EITHER

A uniform rod $A B$ rests in limiting equilibrium in a vertical plane with the end $A$ on rough horizontal ground and the end $B$ against a rough vertical wall that is perpendicular to the plane of the rod. The angle between the rod and the ground is $\theta$. The coefficient of friction between the rod and the wall is $\mu$, and the coefficient of friction between the rod and the ground is $2 \mu$. Show that $\tan \theta = \frac { 1 - 2 \mu ^ { 2 } } { 4 \mu }$. Given that $\theta \leqslant 45 ^ { \circ }$, find the set of possible values of $\mu$.

CAIE FP2 2013 June Q11 OR

A researcher is investigating the relationship between the political allegiance of university students and their childhood environment. He chooses a random sample of 100 students and finds that 60 have political allegiance to the Alliance party. He also classifies their childhood environment as rural or urban, and finds that 45 had a rural childhood. The researcher carries out a test, at the $10 \%$ significance level, on this data and finds that political allegiance is independent of childhood environment. Given that $A$ is the number of students in the sample who both support the Alliance party and have a rural childhood, find the greatest and least possible values of $A$. A second random sample of size $100 N$, where $N$ is an integer, is taken from the university student population. It is found that the proportions supporting the Alliance party from urban and rural childhoods are the same as in the first sample. Given that the value of $A$ in the first sample was 29, find the greatest possible value of $N$ that would lead to the same conclusion (that political allegiance is independent of childhood environment) from a test, at the $10 \%$ significance level, on this second set of data.

CAIE FP2 2014 June Q1

1 A small smooth sphere $P$ of mass $2 m$ is at rest on a smooth horizontal surface. A horizontal impulse of magnitude $8 m u$ is given to $P$. Subsequently $P$ collides directly with a fixed smooth vertical barrier at right angles to $P$ 's direction of motion. Given that the coefficient of restitution between $P$ and the barrier is 0.75 , find the speed of $P$ after the collision.

CAIE FP2 2014 June Q2

2 The point $O$ is on the fixed line $l$. Points $A$ and $B$ on $l$ are such that $O A = 0.5 \mathrm {~m}$ and $O B = 0.75 \mathrm {~m}$, with $A$ between $O$ and $B$. A particle $P$ of mass $m$ oscillates on $l$ in simple harmonic motion with centre $O$. The ratio of the kinetic energy of $P$ when it is at $A$ to its kinetic energy when it is at $B$ is $12 : 11$. Find the amplitude of the motion. Given that the greatest speed of $P$ is $0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the time taken by $P$ to travel directly from $A$ to $B$.

CAIE FP2 2014 June Q3

3 Three small smooth spheres $A , B$ and $C$ have equal radii and have masses $m , 9 m$ and $k m$ respectively. They are at rest on a smooth horizontal table and lie in a straight line with $B$ between $A$ and $C$. The coefficient of restitution between any pair of the spheres is $e$. Sphere $A$ is projected directly towards $B$ with speed $u$. Given that half of the total kinetic energy is lost as result of the collision between $A$ and $B$, find the value of $e$. After $B$ and $C$ collide they move in the same direction and the speed of $C$ is twice the speed of $B$. Find the value of $k$.

CAIE FP2 2014 June Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{ab5f2781-e5ce-4fce-bc95-9d7f55ea66d9-2_515_583_1388_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 Q5

5 A uniform rectangular thin sheet of glass $A B C D$, in which $A B = 8 a$ and $B C = 6 a$, has mass $\frac { 3 } { 5 } M$. Each of the edges $A B , B C , C D$ and $D A$ has a thin strip of metal attached to it, as a border to the glass. The strips along $A B$ and $C D$ each have mass $M$, and the strips along $B C$ and $D A$ each have mass $\frac { 1 } { 3 } M$. Show that the moment of inertia of the whole object (glass and metal strips) about an axis through $A$ perpendicular to the plane of the object is $128 M a ^ { 2 }$. The object is free to rotate about this axis, which is fixed and smooth. The object hangs in equilibrium with $C$ vertically below $A$. It is displaced through a small angle and released from rest. Show that it will move in approximate simple harmonic motion and state the period of the motion.

CAIE FP2 2014 June Q6

6 A pair of coins is thrown repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable $X$. State the expected value of $X$. Find the probability that

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

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

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

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

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

\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

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

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

\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

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

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

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

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

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

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

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

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 $n$.

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

Questions — CAIE FP2 (474 questions)

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