Questions - CAIE FP2 - A-Level Maths

Number of successful applicants	0	1	2	3	4	5	6
Number of groups	1	3	25	51	38	30	2

Test, at the $5 \%$ significance level, the goodness of fit of the distribution $\mathbf { B } ( 6,0.6 )$ for the number of successful applicants in a group.

CAIE FP2 2016 June Q10

10 For a random sample of 6 observations of pairs of values $( x , y )$, where $0 < x < 21$ and $0 < y < 14$, the following results are obtained. $$\Sigma x ^ { 2 } = 844.20 \quad \Sigma y ^ { 2 } = 481.50 \quad \Sigma x y = 625.59$$ It is also found that the variance of the $x$-values is 36.66 and the variance of the $y$-values is 9.69 .

Find the product moment correlation coefficient for the sample.
Find the equations of the regression lines of $y$ on $x$ and $x$ on $y$.
Use the appropriate regression line to estimate the value of $x$ when $y = 6.4$ and comment on the reliability of your estimate.

CAIE FP2 2016 June Q11 EITHER

\includegraphics[max width=\textwidth, alt={}]{3e224c82-68df-427e-a59b-7dc2bfd716a2-5_732_609_431_769}

The end $A$ of a uniform rod $A B$, of length $2 a$ and weight $W$, is freely hinged to a vertical wall. The end $B$ of the rod is attached to a light elastic string of natural length $\frac { 3 } { 2 } a$ and modulus of elasticity $3 W$. The other end of the string is attached to the point $C$ on the wall, where $C$ is vertically above $A$ and $A C = 2 a$. A particle of weight $2 W$ is attached to the rod at the point $D$, where $D B = \frac { 1 } { 2 } a$. The angle $A B C$ is equal to $\theta$ (see diagram). Show that $\cos \theta = \frac { 3 } { 4 }$ and find the tension in the string in terms of $W$. Find the magnitude of the reaction force at the hinge.

CAIE FP2 2016 June Q11 OR

Petra is studying a particular species of bird. She takes a random sample of 12 birds from nature reserve $A$ and measures the wing span, $x \mathrm {~cm}$, for each bird. She then calculates a $95 \%$ confidence interval for the population mean wing span, $\mu \mathrm { cm }$, for birds of this species, assuming that wing spans are normally distributed. Later, she is not able to find the summary of the results for the sample, but she knows that the $95 \%$ confidence interval is $25.17 \leqslant \mu \leqslant 26.83$. Find the values of $\sum x$ and $\sum x ^ { 2 }$ for this sample. Petra also measures the wing spans of a random sample of 7 birds from nature reserve $B$. Their wing spans, $y \mathrm {~cm}$, are as follows. $$\begin{array} { l l l l l l l } 23.2 & 22.4 & 27.6 & 25.3 & 28.4 & 26.5 & 23.6 \end{array}$$ She believes that the mean wing span of birds found in nature reserve $A$ is greater than the mean wing span of birds found in nature reserve $B$. Assuming that this second sample also comes from a normal distribution, with variance the same as the first distribution, test, at the $10 \%$ significance level, whether there is evidence to support Petra's belief.

CAIE FP2 2017 June Q1

3 marks

1 A bullet of mass 0.08 kg is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed $300 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and emerges horizontally after 0.02 s . There is a constant horizontal resisting force of magnitude 1000 N . Find the speed with which the bullet emerges from the barrier.
[0pt] [3]
\includegraphics[max width=\textwidth, alt={}, center]{1dba0ab0-f3a4-4e7e-a67a-00fd37223cc7-04_748_561_260_794} A uniform smooth disc with centre $O$ and radius $a$ is fixed at the point $D$ on a horizontal surface. A uniform rod of length $3 a$ and weight $W$ rests on the disc with its end $A$ in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point $E$ such that $A E = a$ and $E D = \frac { 5 } { 4 } a$. A particle of weight $k W$ is hung from the rod at $B$ (see diagram). The coefficient of friction between the rod and the wall is $\frac { 1 } { 8 }$ and the system is in limiting equilibrium. Find the value of $k$.

CAIE FP2 2017 June Q3

3 Two uniform small smooth spheres $A$ and $B$ have equal radii and masses $3 m$ and $m$ respectively. Sphere $A$ is moving with speed $u$ on a smooth horizontal surface when it collides directly with sphere $B$ which is at rest. The coefficient of restitution between the spheres is $e$.

Find, in terms of $u$ and $e$, expressions for the velocities of $A$ and $B$ after the collision.
Sphere $B$ continues to move until it strikes a fixed smooth vertical barrier which is perpendicular to the direction of motion of $B$. The coefficient of restitution between $B$ and the barrier is $\frac { 3 } { 4 }$. When the spheres subsequently collide, $A$ is brought to rest.
Find the value of $e$.
\includegraphics[max width=\textwidth, alt={}, center]{1dba0ab0-f3a4-4e7e-a67a-00fd37223cc7-08_608_652_258_744} Three identical uniform discs, $A , B$ and $C$, each have mass $m$ and radius $a$. They are joined together by uniform rods, each of which has mass $\frac { 1 } { 3 } m$ and length $2 a$. The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side $4 a$. Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

CAIE FP2 2017 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{1dba0ab0-f3a4-4e7e-a67a-00fd37223cc7-10_445_735_264_696} A particle of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The point $A$ is such that $O A = a$ and $O A$ makes an angle $\alpha$ with the upward vertical through $O$. The particle is held at $A$ and then projected downwards with speed $\sqrt { } ( a g )$ so that it begins to move in a vertical circle with centre $O$. There is a small smooth peg at the point $B$ which is at the same horizontal level as $O$ and at a distance $\frac { 1 } { 3 } a$ from $O$ on the opposite side of $O$ to $A$ (see diagram).

Show that, when the string first makes contact with the peg, the speed of the particle is $\sqrt { } ( \operatorname { ag } ( 1 + 2 \cos \alpha ) )$.
The particle now begins to move in a vertical circle with centre $B$. When the particle is at the point $C$ where angle $C B O = 150 ^ { \circ }$, the tension in the string is the same as it was when the particle was at the point $A$.
Find the value of $\cos \alpha$.

CAIE FP2 2017 June Q6

6 A fair die is thrown repeatedly until a 6 is obtained.

Find the probability that obtaining a 6 takes no more than four throws.
Find the least integer $N$ such that the probability of obtaining a 6 before the $N$ th throw is more than 0.95.

CAIE FP2 2017 June Q7

7 A farmer grows a particular type of fruit tree. On average, the mass of fruit produced per tree has been 6.2 kg . He has developed a new kind of soil and claims that the mean mass of fruit produced per tree when growing in this new soil has increased. A random sample of 10 trees grown in the new soil is chosen. The masses, $x \mathrm {~kg}$, of fruit produced are summarised as follows. $$\Sigma x = 72.0 \quad \Sigma x ^ { 2 } = 542.0$$ Test at the $5 \%$ significance level whether the farmer's claim is justified, assuming a normal distribution.

CAIE FP2 2017 June Q8

8 The continuous random variable $X$ has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( x - 1 ) & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

Find the distribution function of $X$.
The random variable $Y$ is defined by $Y = ( X - 1 ) ^ { 3 }$.
Find the probability density function of $Y$.
Find the median value of $Y$.

CAIE FP2 2017 June Q9

10 marks

9 Two fish farmers $X$ and $Y$ produce a particular type of fish. Farmer $X$ chooses a random sample of 8 of his fish and records the masses, $x \mathrm {~kg}$, as follows. $$\begin{array} { l l l l l l l l } 1.2 & 1.4 & 0.8 & 2.1 & 1.8 & 2.6 & 1.5 & 2.0 \end{array}$$ Farmer $Y$ chooses a random sample of 10 of his fish and summarises the masses, $y \mathrm {~kg}$, as follows. $$\Sigma y = 20.2 \quad \Sigma y ^ { 2 } = 44.6$$ You should assume that both distributions are normal with equal variances. Test at the $10 \%$ significance level whether the mean mass of fish produced by farmer $X$ differs from the mean mass of fish produced by farmer $Y$.
[0pt] [10]

CAIE FP2 2017 June Q10

10 A random sample of 5 pairs of values $( x , y )$ is given in the following table.

$x$	1	2	4	5	8
$y$	7	5	8	6	4

Find, showing all necessary working, the equation of the regression line of $y$ on $x$.
Find, showing all necessary working, the value of the product moment correlation coefficient for this sample.
Test, at the $10 \%$ significance level, whether there is evidence of non-zero correlation between the variables.

CAIE FP2 2017 June Q11 EITHER

A particle $P$ of mass $3 m$ is attached to one end of a light elastic spring of natural length $a$ and modulus of elasticity $k m g$. The other end of the spring is attached to a fixed point $O$ on a smooth plane that is inclined to the horizontal at an angle $\alpha$, where $\sin \alpha = \frac { 2 } { 3 }$. The system rests in equilibrium with $P$ on the plane at the point $E$. The length of the spring in this position is $\frac { 5 } { 4 } a$.

Find the value of $k$.
The particle $P$ is now replaced by a particle $Q$ of mass $2 m$ and $Q$ is released from rest at the point $E$.
Show that, in the resulting motion, $Q$ performs simple harmonic motion. State the centre and the period of the motion.
Find the least tension in the spring and the maximum acceleration of $Q$ during the motion.

CAIE FP2 2017 June Q11 OR

A shop is supplied with large quantities of plant pots in packs of six. These pots can be damaged easily if they are not packed carefully. The manager of the shop is a statistician and he believes that the number of damaged pots in a pack of six has a binomial distribution. He chooses a random sample of 250 packs and records the numbers of damaged pots per pack. His results are shown in the following table.

Number of damaged

pots per pack $( x )$

Frequency

Show that the mean number of damaged pots per pack in this sample is 1.656 .
The following table shows some of the expected frequencies, correct to 2 decimal places, using an appropriate binomial distribution.
Number of damaged
pots per pack $( x )$
0 1 2 3 4 5 6
Expected frequency 36.01 82.36 $a$ 39.89 $b$ 1.74 0.11
Find the values of $a$ and $b$, correct to 2 decimal places
Use a goodness-of-fit test at the $1 \%$ significance level to determine whether the manager's belief is justified.

CAIE FP2 2017 June Q1

3 marks

1 A bullet of mass 0.08 kg is fired horizontally into a fixed vertical barrier. It enters the barrier horizontally with speed $300 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and emerges horizontally after 0.02 s . There is a constant horizontal resisting force of magnitude 1000 N . Find the speed with which the bullet emerges from the barrier.
[0pt] [3]
\includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-04_748_561_260_794} A uniform smooth disc with centre $O$ and radius $a$ is fixed at the point $D$ on a horizontal surface. A uniform rod of length $3 a$ and weight $W$ rests on the disc with its end $A$ in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point $E$ such that $A E = a$ and $E D = \frac { 5 } { 4 } a$. A particle of weight $k W$ is hung from the rod at $B$ (see diagram). The coefficient of friction between the rod and the wall is $\frac { 1 } { 8 }$ and the system is in limiting equilibrium. Find the value of $k$.

CAIE FP2 2017 June Q3

Find, in terms of $u$ and $e$, expressions for the velocities of $A$ and $B$ after the collision.
Sphere $B$ continues to move until it strikes a fixed smooth vertical barrier which is perpendicular to the direction of motion of $B$. The coefficient of restitution between $B$ and the barrier is $\frac { 3 } { 4 }$. When the spheres subsequently collide, $A$ is brought to rest.
Find the value of $e$.
\includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-08_608_652_258_744} Three identical uniform discs, $A , B$ and $C$, each have mass $m$ and radius $a$. They are joined together by uniform rods, each of which has mass $\frac { 1 } { 3 } m$ and length $2 a$. The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side $4 a$. Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

CAIE FP2 2017 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{c437c752-5518-4185-b02f-74206dc4b13c-10_445_735_264_696} A particle of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The point $A$ is such that $O A = a$ and $O A$ makes an angle $\alpha$ with the upward vertical through $O$. The particle is held at $A$ and then projected downwards with speed $\sqrt { } ( a g )$ so that it begins to move in a vertical circle with centre $O$. There is a small smooth peg at the point $B$ which is at the same horizontal level as $O$ and at a distance $\frac { 1 } { 3 } a$ from $O$ on the opposite side of $O$ to $A$ (see diagram).

Show that, when the string first makes contact with the peg, the speed of the particle is $\sqrt { } ( \operatorname { ag } ( 1 + 2 \cos \alpha ) )$.
The particle now begins to move in a vertical circle with centre $B$. When the particle is at the point $C$ where angle $C B O = 150 ^ { \circ }$, the tension in the string is the same as it was when the particle was at the point $A$.
Find the value of $\cos \alpha$.

CAIE FP2 2017 June Q2

2 A particle $P$ moves on a straight line in simple harmonic motion. The centre of the motion is $O$, and the amplitude of the motion is 2.5 m . The points $L$ and $M$ are on the line, on opposite sides of $O$, with $O L = 1.5 \mathrm {~m}$. The magnitudes of the accelerations of $P$ at $L$ and at $M$ are in the ratio 3:4.

Find the distance $O M$.
................................................................................................................................. .
The time taken by $P$ to travel directly from $L$ to $M$ is 2 s .
Find the period of the motion.
Find the speed of $P$ when it passes through $L$.

CAIE FP2 2017 June Q3

3 Two uniform small smooth spheres $A$ and $B$ have equal radii and each has mass $m$. Sphere $A$ is moving with speed $u$ on a smooth horizontal surface when it collides directly with sphere $B$ which is at rest. The coefficient of restitution between the spheres is $\frac { 2 } { 3 }$. Sphere $B$ is initially at a distance $d$ from a fixed smooth vertical wall which is perpendicular to the direction of motion of $A$. The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 3 }$.

Show that the speed of $B$ after its collision with the wall is $\frac { 5 } { 18 } u$.
Find the distance of $B$ from the wall when it collides with $A$ for the second time.
\includegraphics[max width=\textwidth, alt={}, center]{b10d2991-abff-4d2b-b470-1df844d1c7ee-08_743_673_258_737} A uniform rod $A B$ of length $3 a$ and weight $W$ is freely hinged to a fixed point at the end $A$. The end $B$ is below the level of $A$ and is attached to one end of a light elastic string of natural length 4a. The other end of the string is attached to a point $O$ on a vertical wall. The horizontal distance between $A$ and the wall is $5 a$. The string and the rod make angles $\theta$ and $2 \theta$ respectively with the horizontal (see diagram). The system is in equilibrium with the rod and the string in the same vertical plane. It is given that $\sin \theta = \frac { 3 } { 5 }$ and you may use the fact that $\cos 2 \theta = \frac { 7 } { 25 }$.

CAIE FP2 2017 June Q6

6 The independent variables $X$ and $Y$ have distributions with the same variance $\sigma ^ { 2 }$. Random samples of $N$ observations of $X$ and $2 N$ observations of $Y$ are taken, and the results are summarised by $$\Sigma x = 4 , \quad \Sigma x ^ { 2 } = 10 , \quad \Sigma y = 8 , \quad \Sigma y ^ { 2 } = 102 .$$ These data give a pooled estimate of 10 for $\sigma ^ { 2 }$. Find $N$.

CAIE FP2 2017 June Q7

7 A random sample of twelve pairs of values of $x$ and $y$ is taken from a bivariate distribution. The equations of the regression lines of $y$ on $x$ and of $x$ on $y$ are respectively $$y = 0.46 x + 1.62 \quad \text { and } \quad x = 0.93 y + 8.24$$

Find the value of the product moment correlation coefficient for this sample.
Using a $5 \%$ significance level, test whether there is non-zero correlation between the variables.

CAIE FP2 2017 June Q8

8 The number, $x$, of beech trees was counted in each of 50 randomly chosen regions of equal size in beech forests in country $A$. The number, $y$, of beech trees was counted in each of 40 randomly chosen regions of the same equal size in beech forests in country $B$. The results are summarised as follows. $$\Sigma x = 1416 \quad \Sigma x ^ { 2 } = 41100 \quad \Sigma y = 888 \quad \Sigma y ^ { 2 } = 20140$$ Find a 95\% confidence interval for the difference between the mean number of beech trees in regions of this size in country $A$ and in country $B$.

CAIE FP2 2017 June Q9

9 The continuous random variable $X$ has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 0 ,
a \mathrm { e } ^ { - x \ln 2 } & x \geqslant 0 , \end{cases}$$ where $a$ is a positive constant.

Find the value of $a$.
State the value of $\mathrm { E } ( X )$.
Find the interquartile range of $X$.
The variable $Y$ is related to $X$ by $Y = 2 ^ { X }$.
Find the probability density function of $Y$.

CAIE FP2 2017 June Q10

10 Roberto owns a small hotel and offers accommodation to guests. Over a period of 100 nights, the numbers of rooms, $x$, that are occupied each night at Roberto's hotel and the corresponding frequencies are shown in the following table.

Show that the mean number of rooms that are occupied each night is 3.25 .
The following table shows most of the corresponding expected frequencies, correct to 2 decimal places, using a Poisson distribution with mean 3.25.
Number of rooms
occupied $( x )$
0 1 2 3 4 5 6 $\geqslant 7$
Observed frequency 4 9 18 26 20 16 7 0
Expected frequency 3.88 12.60 20.48 22.18 18.02 11.72
Show how the expected value of 22.18 , for $x = 3$, is obtained and find the expected values for $x = 6$ and for $x \geqslant 7$.
Use a goodness-of-fit test at the $5 \%$ significance level to determine whether the Poisson distribution is a suitable model for the number of rooms occupied each night at Roberto's hotel.

CAIE FP2 2017 June Q11 EITHER

\includegraphics[max width=\textwidth, alt={}]{b10d2991-abff-4d2b-b470-1df844d1c7ee-20_312_787_440_678}

The diagram shows a uniform thin rod $A B$ of length $3 a$ and mass $8 m$. The end $A$ is rigidly attached to the surface of a sphere with centre $O$ and radius $a$. The rod is perpendicular to the surface of the sphere. The sphere consists of two parts: an inner uniform solid sphere of mass $\frac { 3 } { 2 } m$ and radius $a$ surrounded by a thin uniform spherical shell of mass $m$ and also of radius $a$. The horizontal axis $l$ is perpendicular to the rod and passes through the point $C$ on the rod where $A C = a$.

Show that the moment of inertia of the object, consisting of rod, shell and inner sphere, about the axis $l$ is $\frac { 289 } { 15 } m a ^ { 2 }$.
The object is free to rotate about the axis $l$. The object is held so that $C A$ makes an angle $\alpha$ with the downward vertical and is released from rest.
Given that $\cos \alpha = \frac { 1 } { 6 }$, find the greatest speed achieved by the centre of the sphere in the subsequent motion.

\(x\)	1	2	4	5	8
\(y\)	7	5	8	6	4

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