Questions - CAIE FP2 - A-Level Maths

Use an energy method to show that $k = \frac { 16 u ^ { 2 } } { a g }$.
Show that $P$ performs simple harmonic motion and find the period of this motion, giving your answer in terms of $u$ and $a$.
Find, in terms of $u$ and $a$, the time that elapses before $P$ first loses $25 \%$ of its initial kinetic energy.

CAIE FP2 2019 June Q11 OR

A company produces packets of sweets. Two different machines, $A$ and $B$, are used to fill the packets. The manager decides to assess the performance of the two machines. He selects a random sample of 50 packets filled by machine $A$ and a random sample of 60 packets filled by machine $B$. The masses of sweets, $x \mathrm {~kg}$, in packets filled by machine $A$ and the masses of sweets, $y \mathrm {~kg}$, in packets filled by machine $B$ are summarised as follows. $$\Sigma x = 22.4 \quad \Sigma x ^ { 2 } = 10.1 \quad \Sigma y = 28.8 \quad \Sigma y ^ { 2 } = 16.3$$ A test at the $\alpha \%$ significance level provides evidence that the mean mass of sweets in packets filled by machine $A$ is less than the mean mass of sweets in packets filled by machine $B$. Find the set of possible values of $\alpha$.
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 2008 November Q1

1 A uniform wire, of length $24 a$ and mass $m$, is bent into the form of a triangle $A B C$ with angle $A B C = 90 ^ { \circ }$, $A B = 6 a$ and $B C = 8 a$ (see diagram). Find the moment of inertia of the wire about an axis through $A$ perpendicular to the plane of the wire.

CAIE FP2 2008 November Q2

2 A small bead $B$ of mass $m$ is threaded on a smooth wire fixed in a vertical plane. The wire forms a circle of radius $a$ and centre $O$. The highest point of the circle is $A$. The bead is slightly displaced from rest at $A$. When angle $A O B = \theta$, where $\theta < \cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)$, the force exerted on the bead by the wire has magnitude $R _ { 1 }$. When angle $A O B = \pi + \theta$, the force exerted on the bead by the wire has magnitude $R _ { 2 }$. Show that $R _ { 2 } - R _ { 1 } = 4 m g$.

CAIE FP2 2008 November Q3

9 marks

3
\includegraphics[max width=\textwidth, alt={}, center]{28e7fb78-e2b6-4f6e-92dc-a06eb87fe1ef-2_582_513_1292_815} A uniform disc, of mass $m$ and radius $a$, is free to rotate without resistance in a vertical plane about a horizontal axis through its centre. A light inextensible string has one end fixed to the rim of the disc, and is wrapped round the rim. A block of mass $2 m$ is attached to the other end of the string (see diagram). The system is released from rest with the block hanging vertically. While the block moves it experiences a constant resistance to motion of magnitude $\frac { 1 } { 10 } m g$. Find the angular acceleration of the disc, and find also the angular speed of the disc when it has turned through one complete revolution.
[0pt] [9]

CAIE FP2 2008 November Q4

4 Two smooth spheres $A$ and $B$, of equal radii, have masses 0.1 kg and $m \mathrm {~kg}$ respectively. They are moving towards each other in a straight line on a smooth horizontal table and collide directly. Immediately before collision the speed of $A$ is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the speed of $B$ is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Assume that in the collision $A$ does not change direction. The speeds of $A$ and $B$ after the collision are $v _ { A } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $v _ { B } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively. Express $m$ in terms of $v _ { A }$ and $v _ { B }$, and hence show that $m < 0.25$.
Assume instead that $m = 0.2$ and that the coefficient of restitution between the spheres is $\frac { 1 } { 2 }$. Find the magnitude of the impulse acting on $A$ in the collision.

CAIE FP2 2008 November Q5

5 A particle of mass $m$ moves in a straight line $A B$ of length $2 a$. When the particle is at a general point $P$ there are two forces acting, one in the direction $\overrightarrow { P A }$ with magnitude $m g \left( \frac { P A } { a } \right) ^ { - \frac { 1 } { 4 } }$ and the other in the direction $\overrightarrow { P B }$ with magnitude $m g \left( \frac { P B } { a } \right) ^ { \frac { 1 } { 2 } }$. At time $t = 0$ the particle is released from rest at the point $C$, where $A C = 1.04 a$. At time $t$ the distance $A P$ is $a + x$. Show that the particle moves in approximate simple harmonic motion. Using the approximate simple harmonic motion, find the speed of $P$ when it first reaches the mid-point of $A B$ and the time taken for $P$ to first reach half of this speed.

CAIE FP2 2008 November Q6

6 The independent random variables $X$ and $Y$ have normal distributions with the same variance $\sigma ^ { 2 }$. Samples of 5 observations of $X$ and 10 observations of $Y$ are made, and the results are summarised by $\Sigma x = 15 , \Sigma x ^ { 2 } = 128 , \Sigma y = 36$ and $\Sigma y ^ { 2 } = 980$. Find a pooled estimate of $\sigma ^ { 2 }$.

CAIE FP2 2008 November Q7

7 The pulse rate of each member of a random sample of 25 adult UK males who exercise for a given period each week is measured in beats per minute. A $98 \%$ confidence interval for the mean pulse rate, $\mu$ beats per minute, for all such UK males was calculated as $61.21 < \mu < 64.39$, based on a $t$-distribution.

Calculate the sample mean pulse rate and the standard deviation used in the calculation.
State an assumption necessary for the validity of the confidence interval.
The mean pulse rate for all UK males is 72 beats per minute. State, giving a reason, if it can be concluded that, on average, UK males who exercise have a reduced pulse rate.

CAIE FP2 2008 November Q8

8 The equations of the regression lines for a random sample of 25 pairs of data $( x , y )$ from a bivariate population are $$\begin{array} { c c } y \text { on } x : & y = 1.28 - 0.425 x ,
x \text { on } y : & x = 1.05 - 0.516 y . \end{array}$$

Find the sample means, $\bar { x }$ and $\bar { y }$.
Find the product moment correlation coefficient for the sample.
Test at the $5 \%$ significance level whether the population correlation coefficient differs from zero.

CAIE FP2 2008 November Q9

9 A sample of 100 observations of the continuous random variable $T$ was obtained and the values are summarised in the following table.

Interval	$1 \leqslant t < 1.5$	$1.5 \leqslant t < 2$	$2 \leqslant t < 2.5$	$2.5 \leqslant t < 3$
Frequency	64	17	16	3

It is required to test the goodness of fit of the distribution with probability density function given by $$f ( t ) = \begin{cases} \frac { 9 } { 4 t ^ { 3 } } & 1 \leqslant t < 3
0 & \text { otherwise } \end{cases}$$ The relevant expected values are as follows.

Interval	$1 \leqslant t < 1.5$	$1.5 \leqslant t < 2$	$2 \leqslant t < 2.5$	$2.5 \leqslant t < 3$
Expected frequency	62.5	21.875	10.125	5.5

Show how the expected value 10.125 is obtained. Carry out the test, at the $10 \%$ significance level.

CAIE FP2 2008 November Q10

10 The continuous random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 0 ,
\frac { a } { 2 ^ { x } } & x \geqslant 0 , \end{cases}$$ where $a$ is a positive constant. By expressing $2 ^ { x }$ in the form $\mathrm { e } ^ { k x }$, where $k$ is a constant, show that $X$ has a negative exponential distribution, and find the value of $a$. State the value of $\mathrm { E } ( X )$. The variable $Y$ is related to $X$ by $Y = 2 ^ { X }$. Find the distribution function of $Y$ and hence find its probability density function.

CAIE FP2 2008 November Q11 EITHER

\includegraphics[max width=\textwidth, alt={}]{28e7fb78-e2b6-4f6e-92dc-a06eb87fe1ef-5_976_1043_434_550}

The diagram shows a central cross-section $C D E F$ of a uniform solid cube of weight $k W$ with edges of length 4a. The cube rests on a rough horizontal floor. One of the vertical faces of the cube is parallel to a smooth vertical wall and at a distance $5 a$ from it. A uniform ladder, of length $10 a$ and weight $W$, is represented by $A B$. The ladder rests in equilibrium with $A$ in contact with the rough top surface of the cube and $B$ in contact with the wall. The distance $A C$ is $a$ and the vertical plane containing $A B$ is perpendicular to the wall. The coefficients of friction between the ladder and the cube, and between the cube and the floor, are both equal to $\mu$. A small dog of weight $\frac { 1 } { 4 } W$ climbs the ladder and reaches the top without the ladder sliding or the cube turning about the edge through $D$. Show that $\mu \geqslant \frac { 4 } { 5 }$. Show that the cube does not slide whatever the value of $k$. Find the smallest possible value of $k$ for which equilibrium is not broken.

CAIE FP2 2008 November Q11 OR

A perfume manufacturer had one bottle-filling machine, but because of increased sales a second machine was obtained. In order to compare the performance of the two machines, a random sample of 50 bottles filled by the first machine and a random sample of 60 bottles filled by the second machine were checked. The volumes of the contents from the first machine, $x _ { 1 } \mathrm { ml }$, and from the second machine, $x _ { 2 } \mathrm { ml }$, are summarised by $$\Sigma x _ { 1 } = 1492.0 , \quad \Sigma x _ { 1 } ^ { 2 } = 44529.52 , \quad \Sigma x _ { 2 } = 1803.6 , \quad \Sigma x _ { 2 } ^ { 2 } = 54220.84 .$$ The volumes have distributions with means $\mu _ { 1 } \mathrm { ml }$ and $\mu _ { 2 } \mathrm { ml }$ for the first and second machines respectively. Test, at the $2 \%$ significance level, whether $\mu _ { 2 }$ is greater than $\mu _ { 1 }$. Find the set of values of $\alpha$ for which there would be evidence at the $\alpha \%$ significance level that $\mu _ { 2 } - \mu _ { 1 } > 0.1$.

CAIE FP2 2009 November Q1

1 A particle of mass $m$ is attached to one end $A$ of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$ and the particle hangs in equilibrium under gravity. The particle is projected horizontally so that it starts to move in a vertical circle. The string slackens after turning through an angle of $120 ^ { \circ }$. Show that the speed of the particle is then $\sqrt { } \left( \frac { 1 } { 2 } g a \right)$ and find the initial speed of projection.

CAIE FP2 2009 November Q2

2 A circular wheel is modelled as a uniform disc of mass 6 kg and radius 0.25 m . It is rotating with angular speed $2 \mathrm { rad } \mathrm { s } ^ { - 1 }$ about a fixed smooth axis perpendicular to its plane and passing through its centre. A braking force of constant magnitude is applied tangentially to the rim of the wheel. The wheel comes to rest 5 s after the braking force is applied. Find the magnitude of the braking force and the angle turned through by the wheel while the braking force acts.

CAIE FP2 2009 November Q3

3 Two small smooth spheres $A$ and $B$ of equal radius have masses $m$ and $3 m$ respectively. They lie at rest on a smooth horizontal plane with their line of centres perpendicular to a smooth fixed vertical barrier with $B$ between $A$ and the barrier. The coefficient of restitution between $A$ and $B$, and between $B$ and the barrier, is $e$, where $e > \frac { 1 } { 3 }$. Sphere $A$ is projected directly towards $B$ with speed $u$. Show that after colliding with $B$ the direction of motion of $A$ is reversed. After the impact, $B$ hits the barrier and rebounds. Show that $B$ will subsequently collide with $A$ again unless $e = 1$.

CAIE FP2 2009 November Q4

4 A uniform rod $A B$, of length $2 a$ and mass $2 m$, can rotate freely in a vertical plane about a smooth horizontal axis through $A$. A small rough ring of mass $m$ is threaded on the rod. The rod is held in a horizontal position with the ring at rest at the mid-point of the rod. The rod is released from rest. Using energy considerations, show that, until the ring slides, $$a \dot { \theta } ^ { 2 } = \frac { 18 } { 11 } g \sin \theta$$ where $\theta$ is the angle turned through by the rod. Show that, until the ring slides, the magnitudes of the friction force and normal contact force acting on the ring are $\frac { 29 } { 11 } m g \sin \theta$ and $\frac { 2 } { 11 } m g \cos \theta$ respectively. The coefficient of friction between the ring and the rod is $\mu$. Find, in terms of $\mu$, the value of $\theta$ when the ring starts to slide.

CAIE FP2 2009 November Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{20dd0f90-8c10-4a7b-a383-4ba89d167cd6-3_716_549_269_799} Two uniform rods, $A B$ and $B C$, each have length $2 a$ and weight $W$. They are smoothly jointed at $B$, and $A$ is attached to a smooth fixed pivot. A light inextensible string of length ( $2 \sqrt { } 2$ ) $a$ joins $A$ to $C$ so that angle $A B C = 90 ^ { \circ }$. The system hangs in equilibrium, with $A B$ making an angle $\alpha$ with the vertical (see diagram). By taking moments about $A$ for the system, or otherwise, show that $\alpha = 18.4 ^ { \circ }$, correct to the nearest $0.1 ^ { \circ }$. Find the tension in the string in the form $k W$, giving the value of $k$ correct to 3 significant figures. Find, in terms of $W$, the magnitude of the force acting on the $\operatorname { rod } B C$ at $B$.

CAIE FP2 2009 November Q6

6 A machine produces metal discs whose diameters have a normal distribution. The mean of this distribution is intended to be 10 cm . Accuracy is checked by measuring the diameters of a random sample of six discs. The diameters, in cm, are as follows. $$\begin{array} { l l l l l l } 10.03 & 10.02 & 9.98 & 10.06 & 10.08 & 10.01 \end{array}$$ Calculate a 99\% confidence interval for the mean diameter of all discs produced by the machine. Deduce a 99\% confidence interval for the mean circumference of all discs produced by the machine.

CAIE FP2 2009 November Q7

7 A continuous random variable $X$ has cumulative distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < - 1 ,
\frac { 1 } { 2 } \left( x ^ { 3 } + 1 \right) & - 1 \leqslant x \leqslant 1 ,
1 & x > 1 . \end{cases}$$ Find $\mathrm { P } \left( X \geqslant \frac { 3 } { 4 } \right)$, and state what can be deduced about the upper quartile of $X$. Obtain the cumulative distribution function of $Y$, where $Y = X ^ { 2 }$. 8150 sheep, chosen from a large flock of sheep, were divided into two groups of 75 . Over a fixed period, one group had their grazing controlled and the other group grazed freely. The gains in weight, in kg, were recorded for each animal and the table below shows the sample means and the unbiased estimates of the population variances for the two samples.

It is required to test whether the population mean for sheep having their grazing controlled exceeds the population mean for sheep grazing freely by less than 5 kg . State, giving a reason, if it is necessary for the validity of the test to assume that the two population variances are equal. Stating any other assumption, carry out the test at the 5\% significance level.

CAIE FP2 2009 November Q9

9 It has been found that $60 \%$ of the computer chips produced in a factory are faulty. As part of quality control, 100 samples of 4 chips are selected at random, and each chip is tested. The number of faulty chips in each sample is recorded, with the results given in the following table.

Number of faulty chips	0	1	2	3	4
Number of samples	2	12	27	49	10

The expected values for a binomial distribution with parameters $n = 4$ and $p = 0.6$ are given in the following table.

Number of faulty chips	0	1	2	3	4
Expected value	2.56	15.36	34.56	34.56	12.96

Show how the expected value 34.56 corresponding to 2 faulty chips is obtained. Carry out a goodness of fit test at the 5\% significance level, and state what can be deduced from the outcome of the test.

CAIE FP2 2009 November Q10

10 An archer shoots at a target. It may be assumed that each shot is independent of all other shots and that, on average, she hits the bull's-eye with 3 shots in 20 . Find the probability that she requires at least 6 shots to hit the bull's-eye. When she hits the bull's-eye for the third time her total number of shots is $Y$. Show that $$\mathrm { P } ( Y = r ) = \frac { 1 } { 2 } ( r - 1 ) ( r - 2 ) \left( \frac { 3 } { 20 } \right) ^ { 3 } \left( \frac { 17 } { 20 } \right) ^ { r - 3 } .$$ Simplify $\frac { \mathrm { P } ( Y = r + 1 ) } { \mathrm { P } ( Y = r ) }$, and hence find the set of values of $r$ for which $\mathrm { P } ( Y = r + 1 ) < \mathrm { P } ( Y = r )$. Deduce the most probable value of $Y$.

CAIE FP2 2009 November Q11 EITHER

A light elastic string, of natural length $l$ and modulus of elasticity $4 m g$, is attached at one end to a fixed point and has a particle $P$ of mass $m$ attached to the other end. When $P$ is hanging in equilibrium under gravity it is given a velocity $\sqrt { } ( g l )$ vertically downwards. At time $t$ the downward displacement of $P$ from its equilibrium position is $x$. Show that, while the string is taut, $$\ddot { x } = - \frac { 4 g } { l } x .$$ Find the speed of $P$ when the length of the string is $l$. Show that the time taken for $P$ to move from the lowest point to the highest point of its motion is $$\left( \frac { \pi } { 3 } + \frac { \sqrt { } 3 } { 2 } \right) \sqrt { } \left( \frac { l } { g } \right)$$

CAIE FP2 2009 November Q11 OR

\includegraphics[max width=\textwidth, alt={}]{20dd0f90-8c10-4a7b-a383-4ba89d167cd6-5_374_569_1123_788}

The scatter diagram shows a sample of size 5 of bivariate data, together with the regression line of $y$ on $x$. State what is minimised in obtaining this regression line, illustrating your answer on a copy of this diagram. State, giving a reason, whether, for the data shown, the regression line of $y$ on $x$ is the same as the regression line of $x$ on $y$. A car is travelling along a stretch of road with speed $v \mathrm {~km} \mathrm {~h} ^ { - 1 }$ when the brakes are applied. The car comes to rest after travelling a further distance of $z \mathrm {~m}$. The values of $z$ (and $\sqrt { } z$ ) for 8 different values of $v$ are given in the table, correct to 2 decimal places.

$v$	25	30	35	40	45	50	55	60
$z$	2.83	4.63	4.84	5.29	9.73	10.30	14.82	15.21
$\sqrt { } z$	1.68	2.15	2.20	2.30	3.12	3.21	3.85	3.90

$$\left[ \Sigma v = 340 , \Sigma v ^ { 2 } = 15500 , \Sigma \sqrt { } z = 22.41 , \Sigma z = 67.65 , \Sigma v \sqrt { } z = 1022.15 . \right]$$

Calculate the product moment correlation coefficient between $v$ and $\sqrt { } z$. What does this indicate about the scatter diagram of the points $( v , \sqrt { } z )$ ?
Given that the product moment correlation coefficient between $v$ and $z$ is 0.965 , correct to 3 decimal places, state why the regression line of $\sqrt { } z$ on $v$ is more suitable than the regression line of $z$ on $v$, and find the equation of the regression line of $\sqrt { } z$ on $v$.
Comment, in the context of the question, on the value of the constant term in the equation of the regression line of $\sqrt { } z$ on $v$.

Interval	\(1 \leqslant t < 1.5\)	\(1.5 \leqslant t < 2\)	\(2 \leqslant t < 2.5\)	\(2.5 \leqslant t < 3\)
Frequency	64	17	16	3

\(v\)	25	30	35	40	45	50	55	60
\(z\)	2.83	4.63	4.84	5.29	9.73	10.30	14.82	15.21
\(\sqrt { } z\)	1.68	2.15	2.20	2.30	3.12	3.21	3.85	3.90

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