Questions - OCR MEI - A-Level Maths

Determine the magnitude and the direction of $\mathbf { F }$.
Determine the magnitude and direction of the additional couple that must be applied to the lamina in order to keep it in equilibrium.

OCR MEI Further Mechanics Major 2020 November Q4

4 A particle P moves so that its position vector $\mathbf { r }$ at time $t$ is given by $\mathbf { r } = ( 5 + 20 t ) \mathbf { i } + \left( 95 + 10 t - 5 t ^ { 2 } \right) \mathbf { j }$.

Determine the initial velocity of P . At time $t = T , \mathrm { P }$ is moving in a direction perpendicular to its initial direction of motion.
Determine the value of $T$.
Determine the distance of P from its initial position at time $T$.

OCR MEI Further Mechanics Major 2020 November Q5

5 A car of mass 900 kg moves along a straight level road. The power developed by the car is constant and equal to 60 kW . The resistance to the motion of the car is constant and equal to 1500 N . At time $t$ seconds the velocity of the car is denoted by $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Initially the car is at rest.

Show that $\frac { 3 v } { 5 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = 40 - v$.
Verify that $t = 24 \ln \left( \frac { 40 } { 40 - v } \right) - \frac { 3 } { 5 } v$.

OCR MEI Further Mechanics Major 2020 November Q6

6 A small ball of mass $m \mathrm {~kg}$ is held at a height of 78.4 m above horizontal ground. The ball is released from rest, falls vertically and rebounds from the ground. The coefficient of restitution between the ball and ground is $e$. The ball continues to bounce until it comes to rest after 6 seconds.

Determine the value of $e$.
Given that the magnitude of the impulse that the ground exerts on the ball at the first bounce is 23.52 Ns , determine the value of $m$.

OCR MEI Further Mechanics Major 2020 November Q7

7 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-04_483_988_989_251} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure} A particle P of mass $m$ is attached to one end of a light elastic string of natural length $6 a$ and modulus of elasticity 3 mg . The other end of the string is fixed to a point O on a smooth plane, which is inclined at an angle of $30 ^ { \circ }$ to the horizontal. The string lies along a line of greatest slope of the plane and P rests in equilibrium on the inclined plane at a point A , as shown in Fig. 7. P is now pulled a further distance $2 a$ down the line of greatest slope through A and released from rest. At time $t$ later, the displacement of P from A is $x$, where the positive direction of $x$ is down the plane.

Show that, until the string slackens, $x$ satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + \frac { g x } { 2 a } = 0$$
Determine, in terms of $a$ and $g$, the time at which the string slackens.
Find, in terms of $a$ and $g$, the speed of P when the string slackens.

OCR MEI Further Mechanics Major 2020 November Q8

8 [In this question, you may use the fact that the volume of a right circular cone of base radius $r$ and height $h$ is $\frac { 1 } { 3 } \pi r ^ { 2 } h$.]

By using integration, show that the centre of mass of a uniform solid right circular cone of height $h$ and base radius $r$ is at a distance $\frac { 3 } { 4 } h$ from the vertex. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-05_929_679_504_333} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Fig. 8 shows the side view of a toy formed by joining a uniform solid circular cylinder of radius $r$ and height $2 r$ to a uniform solid right circular cone, made of the same material as the cylinder, of radius $r$ and height $r$. The toy is placed on a horizontal floor with the curved surface of the cone in contact with the floor.
Determine whether the toy will topple.
Explain why it is not necessary to know whether the floor is rough or smooth in answering part (b). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-06_397_1036_264_255} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure} Fig. 9 shows a uniform rod AB of length $2 a$ and weight $8 W$ which is smoothly hinged at the end A to a point on a fixed horizontal rough bar. A small ring of weight $W$ is threaded on the bar and is connected to the rod at B by a light inextensible string of length $2 a$. The system is in equilibrium with the rod inclined at an angle $\theta$ to the horizontal.

OCR MEI Further Mechanics Major 2020 November Q11

11 Two uniform small smooth spheres A and B have equal radii and equal masses. The spheres are on a smooth horizontal surface. Sphere A is moving at an acute angle $\alpha$ to the line of centres, when it collides with B, which is stationary. After the impact A is moving at an acute angle $\beta$ to the line of centres. The coefficient of restitution between A and B is $\frac { 1 } { 3 }$.

Show that $\tan \beta = 3 \tan \alpha$.
Explain why the assumption that the contact between the spheres is smooth is needed in answering part (a). It is given that A is deflected through an angle $\gamma$.
Determine, in terms of $\alpha$, an expression for $\tan \gamma$.
Determine the maximum value of $\gamma$. You do not need to justify that this value is a maximum. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-09_488_903_264_258} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius $r$. The bowl is fixed with its rim horizontal and uppermost. A particle P , of mass $m$, is connected by a light inextensible string of length $l$ to the lowest point A on the bowl and describes a horizontal circle with constant angular speed $\omega$ on the smooth inner surface of the bowl. The string is taut, and AP makes an angle $\alpha$ with the vertical.
Show that the normal contact force between P and the bowl is of magnitude $m g + 2 m r \omega ^ { 2 } \cos ^ { 2 } \alpha$.
Deduce that $g < r \omega ^ { 2 } \left( k _ { 1 } + k _ { 2 } \cos ^ { 2 } \alpha \right)$, stating the value of the constants $k _ { 1 }$ and $k _ { 2 }$.

OCR MEI Further Mechanics Major 2021 November Q1

1 A small ball of mass 0.25 kg is held above a horizontal floor. The ball is released from rest and hits the floor with a speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It rebounds from the floor with a speed of $4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The situation is modelled by assuming that the ball is in contact with the floor for 0.02 s and during this time the normal contact force the floor exerts on the ball is constant. Determine the magnitude of the normal contact force that the floor exerts on the ball.

OCR MEI Further Mechanics Major 2021 November Q2

2 The diagram shows a system of three particles of masses $3 m , 5 m$ and $2 m$ situated in the $x - y$ plane at the points $\mathrm { A } ( 1,2 ) , \mathrm { B } ( 2 , - 2 )$ and $\mathrm { C } ( 5,3 )$ respectively.
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-02_789_744_1085_239} Determine the coordinates of the centre of mass of the system of particles.

OCR MEI Further Mechanics Major 2021 November Q3

3 One end of a light elastic spring of natural length 0.3 m is attached to a fixed point. A mass of 4 kg is attached to the other end of the spring. When the spring hangs vertically in equilibrium the extension of the spring is 0.02 m .

Determine the modulus of elasticity of the spring. A student calculates that if the mass of 4 kg is removed and replaced with a mass of 20 kg the extension of the spring will be 0.1 m .
Suggest a reason why this extension may not be 0.1 m .

OCR MEI Further Mechanics Major 2021 November Q4

4 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_646_812_312_242} The diagram shows parts of the curves $y = 3 \sqrt { x }$ and $y = 4 - x ^ { 2 }$, which intersect at the point ( 1,3 ). The shaded region, bounded by the two curves and the $y$-axis, is occupied by a uniform lamina. Determine the exact $x$-coordinate of the centre of mass of the lamina.

OCR MEI Further Mechanics Major 2021 November Q5

5 Two small uniform smooth spheres A and B , of equal radius, have masses 2 kg and 4 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, A has speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving along the line of centres, and B has speed $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving along a line which is perpendicular to the line of centres (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_389_764_1592_244} The direction of motion of B after the collision makes an angle of $45 ^ { \circ }$ with the line of centres. Determine the coefficient of restitution between A and B .

OCR MEI Further Mechanics Major 2021 November Q6

Write down the dimensions of force. The force $F$ of gravitational attraction between two objects with masses $m _ { 1 }$ and $m _ { 2 }$, at a distance $d$ apart, is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where $G$ is the universal gravitational constant.
In SI units the value of $G$ is $6.67 \times 10 ^ { - 11 } \mathrm {~kg} ^ { - 1 } \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 2 }$.
Write down the dimensions of $G$.
Determine the value of $G$ in imperial units based on pounds, feet, and seconds. Use the facts that 1 pound $= 0.454 \mathrm {~kg}$ and 1 foot $= 0.305 \mathrm {~m}$. For a planet of mass $M$ and radius $r$, it is suggested that the velocity $v$ needed for an object to escape the gravitational pull of the planet, the 'escape velocity', is given by the following formula.
$\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }$,
where $k$ is a dimensionless constant.
Show that this formula is dimensionally consistent. Information regarding the planets Earth and Mars can be found in the table below.
Earth Mars
Radius (m) 6371000 3389500
Mass (kg) $5.97 \times 10 ^ { 24 }$ $6.39 \times 10 ^ { 23 }$
Escape velocity ( $\mathrm { m } \mathrm { s } ^ { - 1 }$ ) 11186
Using the formula $\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }$, determine the escape velocity for planet Mars.

OCR MEI Further Mechanics Major 2021 November Q7

7 A box B of mass $m \mathrm {~kg}$ is raised vertically by an engine working at a constant rate of $k m g \mathrm {~W}$. Initially B is at rest. The speed of B when it has been raised a distance $x \mathrm {~m}$ is denoted by $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $v ^ { 2 } \frac { d v } { d x } = ( k - v ) g$.
Verify that $\mathrm { gx } = \mathrm { k } ^ { 2 } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { v } } \right) - \mathrm { kv } - \frac { 1 } { 2 } \mathrm { v } ^ { 2 }$.
By using the work-energy principle, show that the time taken for B to reach a speed $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from rest is given by
$\frac { \mathrm { k } } { \mathrm { g } } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { V } } \right) - \frac { \mathrm { V } } { \mathrm { g } }$.

OCR MEI Further Mechanics Major 2021 November Q8

8 A capsule consists of a uniform hollow right circular cylinder of radius $r$ and length $2 h$ attached to two uniform hollow hemispheres of radius $r$.
The centres of the plane faces of the hemispheres coincide with the centres, A and B , of the ends of the cylinder. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{17e92314-d7df-49b8-a441-8d18c91dbbb0-06_702_684_445_244} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure} Fig. 8 represents a vertical cross-section through a plane of symmetry of the capsule as it rests in limiting equilibrium with a point C of one hemisphere on a rough horizontal floor and a point D of the other hemisphere against a rough vertical wall. The total weight of the capsule is $W$ and acts at a point midway between A and B . The plane containing $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D is vertical, with AB making an acute angle $\theta$ with the downward vertical.

Complete the copy of Fig. 8 in the Printed Answer Booklet to show all the remaining forces acting on the capsule. The coefficient of friction at each point of contact is $\frac { 1 } { 3 }$.
By resolving vertically and horizontally, determine the magnitude of the normal contact force between the floor and the capsule in terms of $W$.
By determining an expression for $r$ in terms of $h$ and $\theta$, show that $\tan \theta > \frac { 3 } { 4 }$.

OCR MEI Further Mechanics Major 2021 November Q9

9 A small ball P is projected with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of elevation of $( \alpha + \theta )$ from a point O at the bottom of a plane inclined at $\alpha$ to the horizontal. P subsequently hits the plane at a point R , where OR is a line of greatest slope, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-07_456_862_406_242}

By deriving an expression, in terms of $\theta$, $\alpha$ and $g$, for the time of flight of P , show that the distance OR, in metres, is $$\frac { 50 \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
By using the identity $2 \sin \mathrm {~A} \cos \mathrm {~B} \equiv \sin ( \mathrm {~A} + \mathrm { B } ) - \sin ( \mathrm { B } - \mathrm { A } )$, determine, in terms of $g$ and $\sin \alpha$, an expression for the maximum range of P up the plane, as $\theta$ varies.
Given that OR is the maximum range of P up the plane and is equal to 1.8 m , determine the value of $\theta$.
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-08_625_1180_255_239} A rigid wire ABC is fixed in a vertical plane. The section AB of the wire, of length $b$, is straight and horizontal. The section BC of the wire is smooth and in the form of a circular arc of radius $a$ and length $\frac { 1 } { 2 } a \pi$. The centre of the arc is O , which is vertically above B . A bead P of mass $m$ is threaded on the wire and projected from B with speed $u$ towards C . The angle BOP when P is between B and C is denoted by $\theta$, as shown in the diagram.

OCR MEI Further Mechanics Major 2021 November Q11

11 Two small uniform smooth spheres A and B , of equal radius, have masses 4 kg and 3 kg respectively. The spheres are placed in a smooth horizontal circular groove. The coefficient of restitution between the spheres is $e$, where $e > \frac { 2 } { 5 }$. At a given instant B is at rest and A is set moving along the groove with speed $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It may be assumed that in the subsequent motion the two spheres do not leave the groove.

Determine, in terms of $e$ and $V$, the speeds of A and B immediately after the first collision.
Show that the arc through which A moves between the first and second collisions subtends an angle at the centre of the circular groove of $$\frac { 2 \pi ( 4 - 3 e ) } { 7 e } \text { radians. }$$
1. Determine, in terms of $e$ and $V$, the speed of B immediately after the second collision.
2. What can be said about the motion of A and B if the collisions between A and B are perfectly elastic?

OCR MEI Further Mechanics Major 2021 November Q12

12 A particle P of mass $m$ is fixed to one end of a light elastic string of natural length $l$ and modulus of elasticity 12 mg . The other end of the string is attached to a fixed point O . Particle P is held next to O and then released from rest.

Show that P next comes instantaneously to rest when the length of the string is $\frac { 3 } { 2 } l$. The string first becomes taut at time $t = 0$. At time $t \geqslant 0$, the length of the string is $l + x$, where $x$ is the extension in the string.
Show that when the string is taut, $x$ satisfies the differential equation $$\ddot { \mathrm { x } } + \omega ^ { 2 } \mathrm { x } = \mathrm { g } \text {, where } \omega ^ { 2 } = \frac { 12 \mathrm {~g} } { \mathrm { I } } \text {. }$$
By using the substitution $x = y + \frac { g } { \omega ^ { 2 } }$, solve the differential equation to show that the time when the string first becomes slack satisfies the equation $$\cos \omega \mathrm { t } - \sqrt { \mathrm { k } } \sin \omega \mathrm { t } = 1$$ where $k$ is an integer to be determined.

OCR MEI Further Statistics Minor 2019 June Q1

1 In a game at a charity fair, a spinner is spun 4 times.
On each spin the chance that the spinner lands on a score of 5 is 0.2 .
The random variable $X$ represents the number of spins on which the spinner lands on a score of 5 .

Find $\mathrm { P } ( X = 3 )$.
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
One game costs $\pounds 1$ to play and, for each spin that lands on a score of 5 , the player receives 50 pence.
1. Find the expected total amount of money gained by a player in one game.
2. Find the standard deviation of the total amount of money gained by a player in one game.

OCR MEI Further Statistics Minor 2019 June Q2

2 A market researcher wants to interview people who watched a particular television programme. Audience research data used by the broadcaster indicates that $12 \%$ of the adult population watched this programme. This figure is used to model the situation.
The researcher asks people in a shopping centre, one at a time, if they watched the programme. You should assume that these people form a random sample of the adult population.

Find the probability that the fifth person the researcher asks is the first to have watched the programme.
Find the probability that the researcher has to ask at least 10 people in order to find one who watched the programme.
Find the probability that the twentieth person the researcher asks is the third to have watched the programme.
Find how many people the researcher would have to ask to ensure that there is a probability of at least 0.95 that at least one of them watched the programme.

OCR MEI Further Statistics Minor 2019 June Q3

3 A company has been commissioned to make 50 very expensive titanium components.
A sample of the components needs to be tested to ensure that they are sufficiently strong. However, this is a test to destruction, so the components which are tested can no longer be used.

Explain why it would not be appropriate to use a census in these circumstances. A manager suggests that the first 5 components to be manufactured should be tested.
Explain why this would not be a sensible method of selecting the sample. A statistician advises the manager that the sample selected should be a random sample.
Give two desirable features (other than randomness) that the sample should have.

OCR MEI Further Statistics Minor 2019 June Q4

4 Zara uses a metal detector to search for coins on a beach.
She wonders if the numbers of coins that she finds in an area of $10 \mathrm {~m} ^ { 2 }$ can be modelled by a Poisson distribution. The table below shows the numbers of coins that she finds in randomly chosen areas of $10 \mathrm {~m} ^ { 2 }$ over a period of months.

Number of coins found	0	1	2	3	4	5	6	$> 6$
Frequency	13	28	30	14	10	2	3	0

Software gives the sample mean as 1.98 and the sample standard deviation as 1.4212. Explain how these values suggest that a Poisson distribution may be an appropriate model for the numbers of coins found. Zara decides to carry out a chi-squared test to investigate whether a Poisson distribution is an appropriate model.
Fig. 4 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]
A B C D
1 Number of coins found Observed frequency Expected frequency Chi-squared contribution
2 0 13 13.8069 0.0472
3 1 28
4 2 30 27.0643 0.3184
5 3 14 17.8625 0.8352
6 4 10 8.8419 0.1517
7 $\geqslant 5$ 5 0.0015
\captionsetup{labelformat=empty} \caption{Fig. 4}
\end{table}
Showing your calculations, find the missing values in each of the following cells.
- C3
- C7
- D3
- Explain why the numbers for 5, 6 and more than 6 coins found have been combined into the single category of at least 5 coins found, as shown in the spreadsheet.
- Complete the hypothesis test at the $5 \%$ level of significance.
For the rest of this question, you should assume that the number of coins that Zara finds in an area of $10 \mathrm {~m} ^ { 2 }$ can be modelled by a Poisson distribution with mean 1.98.
Zara also finds pieces of jewellery independently of the coins she finds. The number of pieces of jewellery that she finds per $10 \mathrm {~m} ^ { 2 }$ area is modelled by a Poisson distribution with mean 0.42 .
Find the probability that Zara finds a total of exactly 3 items (coins and/or jewellery) in an area of $10 \mathrm {~m} ^ { 2 }$.
Find the probability that Zara finds a total of at least 30 items (coins and/or jewellery) in an area of $100 \mathrm {~m} ^ { 2 }$.

OCR MEI Further Statistics Minor 2019 June Q5

5 A student wants to know if there is a positive correlation between the amounts of two pollutants, sulphur dioxide and PM10 particulates, on different days in the area of London in which he lives; these amounts, measured in suitable units, are denoted by $s$ and $p$ respectively.
He uses a government website to obtain data for a random sample of 15 days on which the amounts of these pollutants were measured simultaneously. Fig. 5.1 is a scatter diagram showing the data. Summary statistics for these 15 values of $s$ and $p$ are as follows.
$\sum s _ { 1 } = 155.4 \quad \sum p = 518.9 \quad \sum s ^ { 2 } = 2322.7 \quad \sum p ^ { 2 } = 21270.5 \quad \sum s p = 6009.1$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4a4d5816-5b53-49a1-b72f-f8bcf3b4e8bc-4_935_1134_683_260} \captionsetup{labelformat=empty} \caption{Fig. 5.1}

\end{figure}

Explain why the student might come to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.
Find the value of Pearson's product moment correlation coefficient.
Carry out a test at the $5 \%$ significance level to investigate whether there is positive correlation between the amounts of sulphur dioxide and PM10 particulates.
Explain why the student made sure that the sample chosen was a random sample. The student also wishes to model the relationship between the amounts of nitrogen dioxide $n$ and PM10 particulates $p$.
He takes a random sample of 54 values of the two variables, both measured at the same times. Fig. 5.2 is a scatter diagram which shows the data, together with the regression line of $n$ on $p$, the equation of the regression line and the value of $r ^ { 2 }$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4a4d5816-5b53-49a1-b72f-f8bcf3b4e8bc-5_824_1230_495_258} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
\end{figure}
Predict the value of $n$ for $p = 150$.
Discuss the reliability of your prediction in part (e).

OCR MEI Further Statistics Minor 2019 June Q6

6 The discrete random variable $X$ has a uniform distribution over $\{ n , n + 1 , \ldots , 2 n \}$.

Given that $n$ is odd, find $\mathrm { P } \left( X < \frac { 3 } { 2 } n \right)$.
Given instead that $n$ is even, find $\mathrm { P } \left( X < \frac { 3 } { 2 } n \right)$, giving your answer as a single algebraic fraction.
The sum of 6 independent values of $X$ is denoted by $Y$. Find $\operatorname { Var } ( Y )$.

OCR MEI Further Statistics Minor 2022 June Q1

1 In a quiz a contestant is asked up to four questions. The contestant's turn ends once the contestant gets a question wrong or has answered all four questions. The probability that a particular contestant gets any question correct is 0.6 , independently of other questions. The discrete random variable $X$ models the number of questions which the contestant gets correct in a turn.

Show that $\mathrm { P } ( X = 4 ) = 0.1296$. The probability distribution of $X$ is shown in Fig. 1.1. \begin{table}[h]
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ 0.4 0.24 0.144 0.0864 0.1296
\captionsetup{labelformat=empty} \caption{Fig. 1.1}
\end{table}
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
The number of points that a contestant scores is as shown in Fig. 1.2. \begin{table}[h]
Number of
questions correct
Number of
points scored
0 or 1 0
2 2
3 3
4 5
\captionsetup{labelformat=empty} \caption{Fig. 1.2}
\end{table} The discrete random variable $Y$ models the number of points which the contestant scores.
Without doing any working, explain whether each of the following will be less than, equal to or greater than the corresponding value for $X$.
- $\mathrm { E } ( Y )$
- $\operatorname { Var } ( Y )$

	Earth	Mars
Radius (m)	6371000	3389500
Mass (kg)	\(5.97 \times 10 ^ { 24 }\)	\(6.39 \times 10 ^ { 23 }\)
Escape velocity ( \(\mathrm { m } \mathrm { s } ^ { - 1 }\) )	11186

	A	B	C	D
1	Number of coins found	Observed frequency	Expected frequency	Chi-squared contribution
2	0	13	13.8069	0.0472
3	1	28
4	2	30	27.0643	0.3184
5	3	14	17.8625	0.8352
6	4	10	8.8419	0.1517
7	\(\geqslant 5\)	5		0.0015

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	0.4	0.24	0.144	0.0864	0.1296

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