Questions - CAIE - A-Level Maths

$\cos \alpha$,
the vertical component of the velocity of $P$ as it strikes the horizontal plane.

CAIE FP2 2012 June Q5

Challenging +1.8

5
\includegraphics[max width=\textwidth, alt={}, center]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-3_319_794_255_678} A uniform rod $A B$, of mass $m$ and length $6 a$, is rigidly attached at $B$ to a point on the circumference of a uniform circular lamina of mass $m$, radius $2 a$ and centre $O$. The lamina and the rod are in the same vertical plane, and $A B O$ is a straight line (see diagram). Show that the moment of inertia of the system about an axis $l$ through $A$ perpendicular to the plane of the lamina is $78 m a ^ { 2 }$. A particle of mass $2 m$ is now attached at $B$ and the system is free to rotate in a vertical plane about the fixed axis $l$ which is horizontal. Initially $A B$ is horizontal, with $O$ moving downwards and the system having angular velocity $\frac { 3 } { 5 } \sqrt { } \left( \frac { g } { a } \right)$. At time $t , A B$ makes an angle $\theta$ with the downward vertical through $A$.

Find, in terms of $a , g$ and $\theta$, an expression for $\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } }$.
Find the angular velocity of the system when $B$ is vertically below $A$.

CAIE FP2 2012 June Q6

Standard +0.3

6 A random sample of 10 observations of a normal random variable $X$ has mean $\bar { x }$, where $$\bar { x } = 8.254 , \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.912 .$$ Using a $5 \%$ significance level, test whether the mean of $X$ is greater than 8.05.

CAIE FP2 2012 June Q7

Standard +0.3

7 The waiting time, $T$ minutes, before a customer is served in a restaurant has distribution function F given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \\ 0 & t < 0 \end{cases}$$ where $\lambda$ is a positive constant. The standard deviation of $T$ is 8 . Find

the value of $\lambda$,
the probability that a customer has to wait between 5 and 10 minutes before being served,
the median value of $T$.

CAIE FP2 2012 June Q8

Standard +0.3

8 Residents of three towns $A , B$ and $C$ were asked to grade the reliability of their digital television signal as good, satisfactory or poor. A random sample of responses from each town is taken and the numbers in each category are given in the following table.

	Good	Satisfactory	Poor
Town $A$	24	34	14
Town $B$	58	60	26
Town $C$	20	34	30

Test, at the 2.5\% significance level, whether grade of reliability is independent of town. Identify which town makes the greatest contribution to the test statistic and relate your answer to the context of the question.

CAIE FP2 2012 June Q9

Standard +0.3

9 The continuous random variable $X$ has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 a } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where $a$ is a positive constant. Find the distribution function of $X$. The random variable $Y$ is defined by $Y = \mathrm { e } ^ { X }$. Find the distribution function of $Y$. Given that $a = 4$, find the value of $k$ for which $\mathrm { P } ( Y \geqslant k ) = 0.25$.

CAIE FP2 2012 June Q10

Standard +0.3

10 Engineers are investigating the speed of the internet connection received by households in two towns $P$ and $Q$. The speeds, in suitable units, in $P$ and $Q$ are denoted by $x$ and $y$ respectively. For a random sample of 50 houses in town $P$ and a random sample of 40 houses in town $Q$ the results are summarised as follows. $$\Sigma x = 240 \quad \Sigma x ^ { 2 } = 1224 \quad \Sigma y = 168 \quad \Sigma y ^ { 2 } = 754$$ Calculate a $95 \%$ confidence interval for $\mu _ { P } - \mu _ { Q }$, where $\mu _ { P }$ and $\mu _ { Q }$ are the population mean speeds for $P$ and $Q$. Test, at the $1 \%$ significance level, whether $\mu _ { P }$ is greater than $\mu _ { Q }$.

CAIE FP2 2012 June Q11 EITHER

Challenging +1.8

\includegraphics[max width=\textwidth, alt={}]{71a3b842-9d31-4c25-b894-ca6d1f47d84b-5_474_796_479_676}

The diagram shows a uniform rod $A B$, of length $4 a$ and weight $W$, resting in equilibrium with its end $A$ on rough horizontal ground. The rod rests at $C$ on the surface of a smooth cylinder whose axis is horizontal. The cylinder rests on the ground and is fixed to it. The rod is in a vertical plane perpendicular to the axis of the cylinder and is inclined at an angle $\theta$ to the horizontal, where $\cos \theta = \frac { 3 } { 5 }$. A particle of weight $k W$ is attached to the rod at $B$. Given that $A C = 3 a$, show that the least possible value of the coefficient of friction $\mu$ between the rod and the ground is $\frac { 8 ( 2 k + 1 ) } { 13 k + 19 }$. Given that $\mu = \frac { 9 } { 10 }$, find the set of values of $k$ for which equilibrium is possible.

CAIE FP2 2012 June Q11 OR

Challenging +1.2

For a random sample of 5 pairs of values of $x$ and $y$, the equations of the regression lines of $y$ on $x$ and $x$ on $y$ are respectively $$y = - 0.5 x + 5 \quad \text { and } \quad x = - 1.2 y + 7.6$$ Find the value of the product moment correlation coefficient for this sample. Test, at the $5 \%$ significance level, whether the population product moment correlation coefficient differs from zero. The following table shows the sample data.

$x$	1	2	5	5	$p$
$y$	5	3	4	2	$q$

Find the values of $p$ and $q$.

CAIE FP2 2013 June Q1

Challenging +1.2

1
\includegraphics[max width=\textwidth, alt={}, center]{137d2806-f45c-4121-8ee9-bf89580e1cca-2_684_714_246_717} A uniform $\operatorname { rod } A B$, of mass $m$ and length $4 a$, rests with the end $A$ on rough horizontal ground. The point $C$ on $A B$ is such that $A C = 3 a$. A light inextensible string has one end attached to the point $P$ which is at a distance $5 a$ vertically above $A$, and the other end attached to $C$. The rod and the string are in the same vertical plane and the system is in equilibrium with angle $A C P$ equal to $90 ^ { \circ }$ (see diagram). The coefficient of friction between the rod and the ground is $\mu$. Show that the least possible value of $\mu$ is $\frac { 24 } { 43 }$.

CAIE FP2 2013 June Q2

Challenging +1.2

2 Three uniform small smooth spheres, $A , B$ and $C$, have equal radii. Their masses are $4 m , 2 m$ and $m$ respectively. They lie in a straight line on a smooth horizontal surface with $B$ between $A$ and $C$. Initially $A$ is moving towards $B$ with speed $u , B$ is at rest and $C$ is moving in the same direction as $A$ with speed $\frac { 1 } { 2 } u$. The coefficient of restitution between any two of the spheres is $e$. The first collision is between $A$ and $B$. In this collision sphere $A$ loses three-quarters of its kinetic energy. Show that $e = \frac { 1 } { 2 }$. Find the speed of $B$ after its collision with $C$ and deduce that there are no further collisions between the spheres.

CAIE FP2 2013 June Q3

Challenging +1.2

3 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 $O$. When $P$ is hanging vertically below $O$, it is given a horizontal speed $u$. In the subsequent motion, $P$ moves in a complete circle. When $O P$ makes an angle $\theta$ with the downward vertical, the tension in the string is $T$. Show that $$T = \frac { m u ^ { 2 } } { a } + m g ( 3 \cos \theta - 2 )$$ Given that the ratio of the maximum value of $T$ to the minimum value of $T$ is $3 : 1$, find $u$ in terms of $a$ and $g$. Assuming this value of $u$, find the value of $\cos \theta$ when the tension is half of its maximum value.

CAIE FP2 2013 June Q4

Challenging +1.8

4
\includegraphics[max width=\textwidth, alt={}, center]{137d2806-f45c-4121-8ee9-bf89580e1cca-3_906_1538_248_301} The end $A$ of a uniform $\operatorname { rod } A B$, of mass $4 m$ and length $3 a$, is rigidly attached to a point on a uniform spherical shell, of mass $\lambda m$ and radius $3 a$. The end $B$ of the rod is rigidly attached to a point on a uniform ring. The ring has centre $O$, mass $4 m$ and radius $\frac { 1 } { 2 } a$. The ring and the rod are in the same vertical plane. The line $O B A$, extended, passes through the centre of the spherical shell. $B C$ is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through $C$ perpendicular to the plane of the ring, is $( 30 + 55 \lambda ) m a ^ { 2 }$. Given that the system performs small oscillations of period $2 \pi \sqrt { } \left( \frac { 5 a } { g } \right)$ about this axis, find the value of $\lambda$.

CAIE FP2 2013 June Q5

Standard +0.3

5 For a random sample of 12 observations of pairs of values $( x , y )$, the product moment correlation coefficient is - 0.456 . Test, at the $5 \%$ significance level, whether there is evidence of negative correlation between the variables.

CAIE FP2 2013 June Q6

Moderate -0.8

6 The random variable $X$ has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.6 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$ Identify the distribution of $X$ and state its mean. Find

$\mathrm { P } ( X > 4 )$,
the median of $X$.

CAIE FP2 2013 June Q7

Standard +0.8

7 A random sample of 80 observations of the continuous random variable $X$ was taken and the values are summarised in the following table.

Interval	$2 \leqslant x < 3$	$3 \leqslant x < 4$	$4 \leqslant x < 5$	$5 \leqslant x < 6$
Observed frequency	36	29	9	6

It is required to test the goodness of fit of the distribution having probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { x ^ { 2 } } & 2 \leqslant x < 6 \\ 0 & \text { otherwise. } \end{cases}$$ Show that the expected frequency for the interval $2 \leqslant x < 3$ is 40 and calculate the remaining expected frequencies. Carry out a goodness of fit test, at the $10 \%$ significance level.

CAIE FP2 2013 June Q8

Standard +0.8

8 The continuous random variable $X$ has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The random variable $Y$ is defined by $Y = X ^ { 3 }$. Show that $Y$ has probability density function g given by $$g ( y ) = \begin{cases} \frac { 1 } { 18 } y ^ { - \frac { 1 } { 3 } } & 8 \leqslant y \leqslant 64 \\ 0 & \text { otherwise } \end{cases}$$ Find $\mathrm { E } ( Y )$.

CAIE FP2 2013 June Q9

Challenging +1.2

9 A gardener $P$ claims that a new type of fruit tree produces a higher annual mass of fruit than the type that he has previously grown. The old type of tree produced 5.2 kg of fruit per tree, on average. A random sample of 10 trees of the new type is chosen. The masses, $x \mathrm {~kg}$, of fruit produced are summarised as follows. $$\Sigma x = 61.0 \quad \Sigma x ^ { 2 } = 384.0$$ Test, at the $5 \%$ significance level, whether gardener $P$ 's claim is justified, assuming a normal distribution. Another gardener $Q$ has his own type of fruit tree. The masses, $y \mathrm {~kg}$, of fruit produced by a random sample of 10 trees grown by gardener $Q$ are summarised as follows. $$\Sigma y = 70.0 \quad \Sigma y ^ { 2 } = 500.6$$ Test, at the $5 \%$ significance level, whether the mean mass of fruit produced by gardener $Q$ 's trees is greater than the mean mass of fruit produced by gardener $P$ 's trees. You may assume that both distributions are normal and you should state any additional assumption.

CAIE FP2 2013 June Q10 EITHER

Challenging +1.8

A light elastic string has modulus of elasticity $\frac { 3 } { 2 } m g$ and natural length $a$. A particle of mass $m$ is attached to one end of the string. The other end of the string is attached to a fixed point $A$. The particle is released from rest at $A$. Show that when the particle has fallen a distance $k a$ from $A$, where $k > 1$, its kinetic energy is $$\frac { 1 } { 4 } m g a \left( 10 k - 3 - 3 k ^ { 2 } \right) .$$ Show that the particle first comes to instantaneous rest at the point $B$ which is at a distance $3 a$ vertically below $A$. Show that the time taken by the particle to travel from $A$ to $B$ is $$\sqrt { } \left( \frac { 2 a } { g } \right) + \frac { 2 \pi } { 3 } \sqrt { } \left( \frac { 2 a } { 3 g } \right)$$

CAIE FP2 2013 June Q10 OR

Standard +0.8

The regression line of $y$ on $x$, obtained from a random sample of five pairs of values of $x$ and $y$, has equation $$y = x + k$$ where $k$ is a constant. The following table shows the data.

$x$	2	3	3	4	$p$
$y$	4	5	8	4	2

Find the two possible values of $p$. For the smaller of these two values of $p$, find

the product moment correlation coefficient,
the equation of the regression line of $x$ on $y$.

CAIE FP2 2013 June Q1

Standard +0.8

1
\includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-2_684_714_246_717} A uniform $\operatorname { rod } A B$, of mass $m$ and length $4 a$, rests with the end $A$ on rough horizontal ground. The point $C$ on $A B$ is such that $A C = 3 a$. A light inextensible string has one end attached to the point $P$ which is at a distance $5 a$ vertically above $A$, and the other end attached to $C$. The rod and the string are in the same vertical plane and the system is in equilibrium with angle $A C P$ equal to $90 ^ { \circ }$ (see diagram). The coefficient of friction between the rod and the ground is $\mu$. Show that the least possible value of $\mu$ is $\frac { 24 } { 43 }$.

CAIE FP2 2013 June Q4

Challenging +1.8

4
\includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-3_906_1538_248_301} The end $A$ of a uniform $\operatorname { rod } A B$, of mass $4 m$ and length $3 a$, is rigidly attached to a point on a uniform spherical shell, of mass $\lambda m$ and radius $3 a$. The end $B$ of the rod is rigidly attached to a point on a uniform ring. The ring has centre $O$, mass $4 m$ and radius $\frac { 1 } { 2 } a$. The ring and the rod are in the same vertical plane. The line $O B A$, extended, passes through the centre of the spherical shell. $B C$ is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through $C$ perpendicular to the plane of the ring, is $( 30 + 55 \lambda ) m a ^ { 2 }$. Given that the system performs small oscillations of period $2 \pi \sqrt { } \left( \frac { 5 a } { g } \right)$ about this axis, find the value of $\lambda$.

\(x\)	1	2	5	5	\(p\)
\(y\)	5	3	4	2	\(q\)

Interval	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)	\(5 \leqslant x < 6\)
Observed frequency	36	29	9	6

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