Questions - CAIE FP2 - A-Level Maths

Member	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
$x$	24.2	23.8	22.8	25.1	24.5	24.0	23.8	22.8
$y$	23.9	23.6	22.8	24.5	24.2	23.5	23.6	22.7

$$\left[ \Sigma x = 191 , \quad \Sigma x ^ { 2 } = 4564.46 , \quad \Sigma y = 188.8 , \quad \Sigma y ^ { 2 } = 4458.4 , \quad \Sigma x y = 4510.99 . \right]$$

Find, showing all necessary working, the equation of the regression line of $y$ on $x$.
The athletics coach believes that, on average, the time taken by an athlete to run 200 metres decreases between the beginning and the end of the year by more than 0.2 seconds.
Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the $10 \%$ significance level.

CAIE FP2 2018 June Q1

1 A bullet of mass $m \mathrm {~kg}$ is fired horizontally into a fixed vertical block of material. It enters the block horizontally with speed $250 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and emerges horizontally with speed $70 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ after 0.04 s . The block offers a constant horizontal resisting force of magnitude 450 N . Find the value of $m$.

CAIE FP2 2018 June Q2

2 A particle $P$ moves on a straight line in simple harmonic motion. The centre of the motion is $O$. The points $A$ and $B$ are on the line, on opposite sides of $O$, with $O A = 1.6 \mathrm {~m}$ and $O B = 1.2 \mathrm {~m}$. The ratio of the speed of $P$ at $A$ to its speed at $B$ is $3 : 4$.

Find the amplitude of the motion.
The maximum speed of $P$ during its motion is $\frac { 1 } { 3 } \pi \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the period of the motion.
Find the time taken for $P$ to travel directly from $A$ to $B$.

CAIE FP2 2018 June Q3

3 Two identical uniform small spheres $A$ and $B$, each of mass $m$, are moving towards each other in a straight line on a smooth horizontal surface. Their speeds are $u$ and $k u$ respectively, and they collide directly. The coefficient of restitution between the spheres is $e$. Sphere $B$ is brought to rest by the collision.

Show that $e = \frac { k - 1 } { k + 1 }$.
Given that $60 \%$ of the total initial kinetic energy is lost in the collision, find the values of $k$ and $e$.

CAIE FP2 2018 June Q4

3 marks

4 A uniform $\operatorname { rod } A B$ has length $2 a$ and weight $W$. The end $A$ rests on rough horizontal ground and the end $B$ rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is $\theta$. A particle of weight $5 W$ hangs from the rod at the point $C$, with $A C = x a$, where $0 < x < 1$.

By taking moments about $A$, show that the magnitude of the normal reaction at $B$ is $\frac { W ( 5 x + 1 ) } { 2 \tan \theta }$.
[0pt] [3]
The particle of weight $5 W$ is now moved a distance $a$ up the rod, so that $A C = ( x + 1 ) a$. This results in the magnitude of the normal reaction at $B$ being double its previous value. The system remains in equilibrium with the rod at angle $\theta$ with the horizontal.
Show that $x = \frac { 4 } { 5 }$.
The coefficient of friction between the rod and the ground is $\frac { 2 } { 3 }$.
Given that the rod is about to slip when the particle of weight $5 W$ is in its second position, find the value of $\tan \theta$.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Axis $l$} \includegraphics[alt={},max width=\textwidth]{0eb3892f-628f-449a-b022-b38170754d89-08_462_693_301_731}
\end{figure} Three thin uniform rings $A , B$ and $C$ are joined together, so that each ring is in contact with each of the other two rings. Ring $A$ has radius $2 a$ and mass $3 M$; rings $B$ and $C$ each have radius $3 a$ and mass $2 M$. The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis $l$ which is tangential to ring $A$, in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).

CAIE FP2 2018 June Q6

6 The continuous random variable $X$ has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.4 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$

Find $\mathrm { P } ( X > 2 )$.
Find the interquartile range of $X$.

CAIE FP2 2018 June Q7

7 A large number of athletes are taking part in a competition. The masses, in kg , of a random sample of 7 athletes are as follows. $$\begin{array} { l l l l l l l } 98.1 & 105.0 & 92.2 & 89.8 & 99.9 & 95.4 & 101.2 \end{array}$$ Assuming that masses are normally distributed, test, at the $10 \%$ significance level, whether the mean mass of athletes in this competition is equal to 94 kg .

CAIE FP2 2018 June Q8

8 A manufacturer produces three types of car: hatchbacks, saloons and estates. Each type of car is available in one of three colours: silver, blue and red. The manufacturer wants to know whether the popularity of the colour of the car is related to the type of car. A random sample of 300 cars chosen by customers gives the information summarised in the following table.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Colour of car
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Silver	Blue	Red
\multirow{3}{*}{Type of car}	Hatchback	53	36	41
\cline { 2 - 5 }	Saloon	29	40	31
\cline { 2 - 5 }	Estate	28	24	18

Test at the $10 \%$ significance level whether the colour of car chosen by customers is independent of the type of car.

CAIE FP2 2018 June Q9

9 At a ski resort, the probability of snow on any particular day is constant and equal to $p$. The skiing season begins on 1 November. The random variable $X$ denotes the day of the skiing season on which the first snowfall occurs. (For example, if the first snowfall is on 5 November, then $X = 5$.) The variance of $X$ is $\frac { 4 } { 9 }$.

Show that $4 p ^ { 2 } + 9 p - 9 = 0$ and hence find the value of $p$.
Find the probability that the first snowfall will be on 3 November.
Find the probability that the first snowfall will not be before 4 November.
Find the least integer $N$ so that the probability of the first snowfall being on or before the $N$ th day of November is more than 0.999 .

CAIE FP2 2018 June Q10

10 The times taken to run 400 metres by students at two large colleges $P$ and $Q$ are being compared. There is no evidence that the population variances are equal. The time taken by a student at college $P$ and the time taken by a student at college $Q$ are denoted by $x$ seconds and $y$ seconds respectively. A random sample of 50 students from college $P$ and a random sample of 60 students from college $Q$ give the following summarised data. $$\Sigma x = 2620 \quad \Sigma x ^ { 2 } = 138200 \quad \Sigma y = 3060 \quad \Sigma y ^ { 2 } = 157000$$

Using a 10\% significance level, test whether, on average, students from college $P$ take longer to run 400 metres than students from college $Q$.
Find a $90 \%$ confidence interval for the difference in the mean times taken to run 400 metres by students from colleges $P$ and $Q$.

CAIE FP2 2018 June Q11 EITHER

A particle $P$ 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 particle is held so that the string is taut, with $O P$ horizontal. The particle is projected downwards with speed $\sqrt { } \left( \frac { 2 } { 5 } a g \right)$ and begins to move in a vertical circle. The string breaks when its tension is equal to $\frac { 11 } { 5 } m g$.

Show that the string breaks when $O P$ makes an angle $\theta$ with the downward vertical through $O$, where $\cos \theta = \frac { 3 } { 5 }$. Find the speed of $P$ at this instant.
For the subsequent motion after the string breaks, find the distance $O P$ when the particle $P$ is vertically below $O$.

CAIE FP2 2018 June Q11 OR

The regression line of $y$ on $x$, obtained from a random sample of 6 pairs of values of $x$ and $y$, has equation $$y = 0.25 x + k$$ where $k$ is a constant. The values from the sample are shown in the following table.

$x$	4	5	7	8	10	14
$y$	5	8	$p$	7	$p$	9

Find the value of $p$ and the value of $k$.
Find the product moment correlation coefficient for the data.
Test, at the $5 \%$ significance level, whether there is evidence of positive correlation between the variables.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE FP2 2018 June Q4

3 marks

By taking moments about $A$, show that the magnitude of the normal reaction at $B$ is $\frac { W ( 5 x + 1 ) } { 2 \tan \theta }$.
[0pt] [3]
The particle of weight $5 W$ is now moved a distance $a$ up the rod, so that $A C = ( x + 1 ) a$. This results in the magnitude of the normal reaction at $B$ being double its previous value. The system remains in equilibrium with the rod at angle $\theta$ with the horizontal.
Show that $x = \frac { 4 } { 5 }$.
The coefficient of friction between the rod and the ground is $\frac { 2 } { 3 }$.
Given that the rod is about to slip when the particle of weight $5 W$ is in its second position, find the value of $\tan \theta$.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Axis $l$} \includegraphics[alt={},max width=\textwidth]{c6c8e0fd-6af2-40c9-9513-6581e26e2aec-08_462_693_301_731}
\end{figure} Three thin uniform rings $A , B$ and $C$ are joined together, so that each ring is in contact with each of the other two rings. Ring $A$ has radius $2 a$ and mass $3 M$; rings $B$ and $C$ each have radius $3 a$ and mass $2 M$. The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis $l$ which is tangential to ring $A$, in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).

CAIE FP2 2018 June Q1

1 A particle $P$ is moving in a fixed circle of radius 0.8 m . At time $t \mathrm {~s}$ its velocity is $\left( t ^ { 2 } - t + 2 \right) \mathrm { m } \mathrm { s } ^ { - 1 }$. Find the magnitudes of the radial and the transverse components of the acceleration of $P$ when $t = 2$. Radial component
Transverse component $\_\_\_\_$

CAIE FP2 2018 June Q2

2 Two uniform small spheres $A$ and $B$ have equal radii and masses $4 m$ 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 $e$.

Show that after the collision $A$ moves with speed $\frac { 1 } { 5 } u ( 4 - e )$ and find the speed of $B$.
Sphere $B$ continues to move until it collides with 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 $\frac { 3 } { 4 } e$. After this collision, the speeds of $A$ and $B$ are equal.
Find the value of $e$.
The spheres $A$ and $B$ now collide directly again.
Determine whether sphere $B$ collides with the barrier for a second time.

CAIE FP2 2018 June Q3

3 A particle $P$ moves on the positive $x$-axis in simple harmonic motion. The centre of the motion is a distance $d \mathrm {~m}$ from the origin $O$, where $0 < d < 6.5$. The points $A$ and $B$ are on the positive $x$-axis, with $O A = 6.5 \mathrm {~m}$ and $O B = 7.5 \mathrm {~m}$. The magnitude of the acceleration of $P$ when it is at $B$ is twice the magnitude of the acceleration of $P$ when it is at $A$.

Find $d$.
The period of the motion is $\pi \mathrm { s }$ and the maximum acceleration of $P$ during the motion is $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Find the speed of $P$ when it is 7 m from $O$.
Find the time taken by $P$ to travel directly from $A$ to $B$.

CAIE FP2 2018 June Q4

4 A uniform $\operatorname { rod } A B$ has length $2 a$ and weight $W$. The end $A$ rests on rough horizontal ground and the end $B$ rests against a smooth vertical wall. The angle between the rod and the horizontal is $\theta$, where $\tan \theta = \frac { 4 } { 3 }$. One end of a light inextensible rope is attached to a point $C$ on the rod. The other end is attached to a point where the vertical wall and the horizontal ground meet. The rope is taut and perpendicular to the rod. The rope and rod are in a vertical plane perpendicular to the wall.

Show that $A C = \frac { 18 } { 25 } a$.
The magnitude of the frictional force at $A$ is equal to one quarter of the magnitude of the normal reaction force at $A$.
Show that the tension in the rope is $\frac { 1 } { 4 } W$.
Find expressions, in terms of $W$, for the magnitudes of the normal reaction forces at $A$ and $B$.

CAIE FP2 2018 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-08_323_515_260_813} A particle $P$ 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 particle is held with the string taut and horizontal. It is projected downwards with speed $\sqrt { } ( 12 a g )$. At the lowest point of its motion, $P$ collides directly with a particle $Q$ of mass $k m$ which is at rest (see diagram). In the collision, $P$ and $Q$ coalesce. The tension in the string immediately after the collision is half of its value immediately before the collision. Find the possible values of $k$.

CAIE FP2 2018 June Q6

6 A random sample of 15 observations of pairs of values of two variables gives a product moment correlation coefficient of 0.430 .

Test at the $10 \%$ significance level whether there is evidence of non-zero correlation between the variables.
A second random sample of $N$ observations gives a product moment correlation coefficient of 0.615 . Using a 5\% significance level, there is evidence of positive correlation between the variables.
Find the least possible value of $N$, justifying your answer.

CAIE FP2 2018 June Q7

7 The probability that a driver passes an advanced driving test has a fixed value $p$ for each attempt. A driver keeps taking the test until he passes. The random variable $X$ denotes the number of attempts required for the driver to pass. The variance of $X$ is 3.75 .

Show that $15 p ^ { 2 } + 4 p - 4 = 0$ and hence find the value of $p$.
Find $\mathrm { P } ( X = 5 )$.
Find $\mathrm { P } ( 3 \leqslant X \leqslant 7 )$.

CAIE FP2 2018 June Q8

8 For a random sample of 6 observations of pairs of values $( x , y )$, the equation of the regression line of $y$ on $x$ is $y = b x + 1.306$, where $b$ is a constant. The corresponding equation of the regression line of $x$ on $y$ is $x = 0.6331 y + d$, where $d$ is a constant. The values of $x$ from the sample are $$\begin{array} { l l l l l l } 2.3 & 2.8 & 3.7 & p & 6.1 & 6.4 \end{array}$$ and the sum of the values of $y$ is 46.5 . The product moment correlation coefficient is 0.9797 .

Find the value of $b$ correct to 3 decimal places.
Find the value of $p$.
Use the equation of the regression line of $x$ on $y$ to estimate the value of $x$ when $y = 8.5$.

CAIE FP2 2018 June Q9

9 The continuous random variable $X$ has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 20 } \left( 3 - \frac { 1 } { \sqrt { } x } \right) & 1 \leqslant x \leqslant 9
0 & \text { otherwise } \end{cases}$$ The random variable $Y$ is defined by $Y = \sqrt { } X$.

Show that the probability density function of $Y$ is given by $$\operatorname { g } ( y ) = \begin{cases} \frac { 1 } { 10 } ( 3 y - 1 ) & 1 \leqslant y \leqslant 3
0 & \text { otherwise } \end{cases}$$
Find the mean value of $Y$.

CAIE FP2 2018 June Q10

10 During the summer months, all members of a large swimming club take part in intensive training. The times taken to swim 50 metres at the beginning of the summer and at the end of the summer are recorded for each member of the club. The time taken, in seconds, at the beginning of the summer is denoted by $x$ and the time taken at the end of the summer is denoted by $y$. For a random sample of 9 members the results are shown in the following table.

Member	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$
$x$	38.5	40.2	32.3	35.1	36.2	41.4	32.0	38.2	38.2
$y$	37.4	38.1	31.6	34.7	34.2	38.6	31.8	36.3	36.8

The swimming coach believes that, on average, the time taken by a swimmer to swim 50 metres will decrease by more than one second as a result of the intensive training.

Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the $10 \%$ significance level.
Find a 95\% confidence interval for the population mean time taken to swim 50 metres after the intensive training, assuming a normal distribution.

CAIE FP2 2018 June Q11 EITHER

\includegraphics[max width=\textwidth, alt={}]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-18_588_609_438_769}

An object is formed from a square frame $A B C D$ with a square lamina attached in one corner of the frame. The frame consists of four identical thin rods, each of mass $M$ and length $2 a$. The lamina has mass $k M$ and edges of length $a$. It has one vertex at $C$ and adjacent sides in contact with $C B$ and $C D$ (see diagram).

Show that the moment of inertia of the object about an axis $l$ through $A$ perpendicular to the plane of the object is $\frac { 2 } { 3 } M a ^ { 2 } ( 7 k + 20 )$.
The object is released from rest with the edge $A B$ horizontal and $D$ vertically above $A$. The object rotates freely about the fixed axis $l$. The angular speed of the object is $\frac { 1 } { 2 } \sqrt { } \left( \frac { 5 g } { a } \right)$ when $D$ is first vertically below $A$.
Find the value of $k$.

CAIE FP2 2018 June Q11 OR

A scientist carries out an experiment to investigate the quantity $X$, which takes the values $0,1,2,3,4$, 5 or 6 . He believes that the values taken by $X$ follow a binomial distribution. He conducts 250 trials. His results are summarised in the following table.

$x$	0	1	2	3	4	5	6
Observed frequency	22	83	72	53	17	3	0

Show that unbiased estimates of the mean and variance for these results are 1.876 and 1.266 respectively, correct to 3 decimal places. By evaluating the mean and variance of the distribution B(6, 0.313), explain why $X$ could have this distribution.
The expected frequencies corresponding to the distribution $\mathrm { B } ( 6,0.313 )$ are shown in the following table.
$x$ 0 1 2 3 4 5 6
Observed frequency 22 83 72 53 17 3 0
Expected frequency 26.3 71.9 81.8 49.7 17.0 3.1 0.2
Show how the expected frequency for $x = 4$ is calculated.
Test at the $5 \%$ significance level whether the scientist's belief is correct.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

\(x\)	4	5	7	8	10	14
\(y\)	5	8	\(p\)	7	\(p\)	9

Member	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
\(x\)	24.2	23.8	22.8	25.1	24.5	24.0	23.8	22.8
\(y\)	23.9	23.6	22.8	24.5	24.2	23.5	23.6	22.7

\(x\)	0	1	2	3	4	5	6
Observed frequency	22	83	72	53	17	3	0
Expected frequency	26.3	71.9	81.8	49.7	17.0	3.1	0.2

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