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OCR MEI S4 2016 June Q1

24 marks Hard +2.3

1 The random variable $X$ has a Cauchy distribution centred on $m$. Its probability density function ( pdf ) is $\mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \frac { 1 } { \pi } \frac { 1 } { 1 + ( x - m ) ^ { 2 } } , \quad \text { for } - \infty < x < \infty$$

Sketch the pdf. Show that the mode and median are at $x = m$.
A sample of size 1 , consisting of the observation $x _ { 1 }$, is taken from this distribution. Show that the maximum likelihood estimate (MLE) of $m$ is $x _ { 1 }$.
Now suppose that a sample of size 2 , consisting of observations $x _ { 1 }$ and $x _ { 2 }$, is taken from the distribution. By considering the logarithm of the likelihood function or otherwise, show that the MLE, $\hat { m }$, satisfies the cubic equation $$\left( 2 \hat { m } - \left( x _ { 1 } + x _ { 2 } \right) \right) \left( \hat { m } ^ { 2 } - \left( x _ { 1 } + x _ { 2 } \right) \hat { m } + 1 + x _ { 1 } x _ { 2 } \right) = 0$$
Obtain expressions for the three roots of this equation. Show that if $\left| x _ { 1 } - x _ { 2 } \right| < 2$ then only one root is real. How do you know, without doing further calculations, that in this case the real root will be the MLE of $m$ ?
Obtain the three possible values of $\hat { m }$ in the case $x _ { 1 } = - 2$ and $x _ { 2 } = 2$. Evaluate the likelihood function for each value of $\hat { m }$ and comment on your answer.

OCR MEI S4 2016 June Q2

24 marks Challenging +1.2

2 The random variable $X$ has probability density function $\mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x } , \quad x > 0 .$$

Obtain the moment generating function (mgf) of $X$.
Use the mgf to find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$. The random variable $Y$ is defined as follows: $$Y = X _ { 1 } + \ldots + X _ { n } ,$$ where the $X _ { i }$ are independently and identically distributed as $X$.
Write down expressions for $\mathrm { E } ( Y )$ and $\operatorname { Var } ( Y )$. Obtain the $\operatorname { mgf }$ of $Y$.
Find the $\operatorname { mgf }$ of $Z$ where $Z = \frac { Y - \frac { n } { \lambda } } { \frac { \sqrt { n } } { \lambda } }$.
By considering the logarithm of the mgf of $Z$, show that the distribution of $Z$ tends to the standard Normal distribution as $n$ tends to infinity.

OCR MEI S4 2016 June Q3

24 marks Standard +0.3

3 A large department in a university wished to compare the standards of literacy and numeracy of its students. A random sample of 24 students was taken and sub-divided, randomly, into two groups of 12 . The students in one group took a literacy assessment (scores denoted by $x$ ); the students in the other group took a numeracy assessment (scores denoted by $y$ ). The two assessments were designed to give the same distributions of scores when taken by random samples from the general population. The scores obtained by the students on the two assessments are shown in the table.

$x$	23	42	43	46	48	48	50	54	58	59	62	65
$y$	44	36	63	55	53	58	63	80	61	57	83	54

$$\sum x = 598 \quad \sum x ^ { 2 } = 31196 \quad \sum y = 707 \quad \sum y ^ { 2 } = 43543$$

Carry out an appropriate $t$ test, at the $5 \%$ level of significance, to compare the standards of literacy and numeracy.
State the distributional assumptions required for the $t$ test to be valid. Name the test that you would use if the assumptions required for the $t$ test are thought not to hold. State the hypotheses for this new test. Explain, in general terms, which of the two tests is more powerful, and why. A statistician at the university looked at the data and commented that a paired sample design would have been better.
Explain how a paired sample design would be applied in this context, and how the data would be analysed. Explain also why it would be better than the design used.

OCR MEI C2 Q7

Moderate -0.8

7 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-003_305_897_310_795} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle $\theta$ radians. The sector has area $30 \mathrm {~cm} ^ { 2 }$.

Find $\theta$.
Hence find the perimeter of the sector.

OCR MEI C2 Q9

Standard +0.3

A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where $x$ and $y$ are horizontal and vertical distances in metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
\end{figure} Using this model,
(A) find the greatest height of the tunnel,
(B) explain why $100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x$ gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e43ddbe-ae95-467b-a527-351ab8a4c4fe-004_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
\end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
Hence estimate the volume of earth removed when the tunnel is re-shaped.

OCR MEI C2 Q11

Moderate -0.3

11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.

Time (minutes)	10	20	30	40	50
Temperature $\left( { } ^ { \circ } \mathrm { C } \right)$	68	53	42	36	31

The room temperature is $22 ^ { \circ } \mathrm { C }$. The difference between the temperature of the drink and room temperature at time $t$ minutes is $z ^ { \circ } \mathrm { C }$. The relationship between $z$ and $t$ is modelled by $$z = z _ { 0 } 10 ^ { - k t }$$ where $z _ { 0 }$ and $k$ are positive constants.

Give a physical interpretation for the constant $z _ { 0 }$.
Show that $\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }$.
On the insert, complete the table and draw the graph of $\log _ { 10 } z$ against $t$. Use your graph to estimate the values of $k$ and $z _ { 0 }$.
Hence estimate the temperature of the drink 70 minutes after it is made. \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Concepts for Advanced Mathematics (C2)
INSERT
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the length of BC ,
the area of triangle ABC .

OCR MEI C2 2005 January Q5

5 marks Moderate -0.8

5 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2.
Find also the sum to infinity of this progression.

OCR MEI C2 2005 January Q6

5 marks Easy -1.2

6 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.

OCR MEI C2 2005 January Q7

5 marks Moderate -0.8

7 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-4_305_897_310_795} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure} Fig. 7 shows a sector of a circle of radius 5 cm which has angle $\theta$ radians. The sector has area $30 \mathrm {~cm} ^ { 2 }$.

Find $\theta$.
Hence find the perimeter of the sector.

OCR MEI C2 2005 January Q8

5 marks Moderate -0.8

Solve the equation $10 ^ { x } = 316$.
Simplify $\log _ { a } \left( a ^ { 2 } \right) - 4 \log _ { a } \left( \frac { 1 } { a } \right)$.

OCR MEI C2 2005 January Q9

12 marks Moderate -0.3

A tunnel is 100 m long. Its cross-section, shown in Fig. 9.1, is modelled by the curve $$y = \frac { 1 } { 4 } \left( 10 x - x ^ { 2 } \right) ,$$ where $x$ and $y$ are horizontal and vertical distances in metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_506_812_676_653} \captionsetup{labelformat=empty} \caption{Figure 9.1}
\end{figure} Using this model,
(A) find the greatest height of the tunnel,
(B) explain why $100 \int _ { 0 } ^ { 10 } y \mathrm {~d} x$ gives the volume, in cubic metres, of earth removed to make the tunnel. Calculate this volume.
The roof of the tunnel is re-shaped to allow for larger vehicles. Fig. 9.2 shows the new crosssection. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{72b4624f-e716-4a37-96f3-01b46e0bd0fd-5_513_1256_1894_575} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
\end{figure} Use the trapezium rule with 5 strips to estimate the new cross-sectional area.
Hence estimate the volume of earth removed when the tunnel is re-shaped.

OCR MEI C2 2005 January Q10

11 marks Moderate -0.8

10 A curve has equation $y = x ^ { 3 } - 6 x ^ { 2 } + 12$.

Use calculus to find the coordinates of the turning points of this curve. Determine also the nature of these turning points.
Find, in the form $y = m x + c$, the equation of the normal to the curve at the point $( 2 , - 4 )$.

OCR MEI C2 2005 January Q11

13 marks Moderate -0.3

11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.

Time (minutes)	10	20	30	40	50
Temperature $\left( { } ^ { \circ } \mathrm { C } \right)$	68	53	42	36	31

Give a physical interpretation for the constant $z _ { 0 }$.
Show that $\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }$.
On the insert, complete the table and draw the graph of $\log _ { 10 } z$ against $t$. Use your graph to estimate the values of $k$ and $z _ { 0 }$.
Hence estimate the temperature of the drink 70 minutes after it is made.

OCR MEI C2 2006 January Q1

2 marks Easy -1.8

1 Given that $140 ^ { \circ } = k \pi$ radians, find the exact value of $k$.

OCR MEI C2 2006 January Q2

2 marks Easy -1.8

2 Find the numerical value of $\sum _ { k = 2 } ^ { 5 } k ^ { 3 }$.

OCR MEI C2 2006 January Q3

3 marks Easy -1.8

3 Fig. 3 Beginning with the triangle shown in Fig. 3, prove that $\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$.

OCR MEI C2 2006 January Q4

5 marks Moderate -0.8

4 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-2_615_971_1457_539} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} Fig. 4 shows a curve which passes through the points shown in the following table.

$x$	1	1.5	2	2.5	3	3.5	4
$y$	8.2	6.4	5.5	5.0	4.7	4.4	4.2

Use the trapezium rule with 6 strips to estimate the area of the region bounded by the curve, the lines $x = 1$ and $x = 4$, and the $x$-axis. State, with a reason, whether the trapezium rule gives an overestimate or an underestimate of the area of this region.

OCR MEI C2 2006 January Q5

5 marks Moderate -0.8

Sketch the graph of $y = \tan x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Solve the equation $4 \sin x = 3 \cos x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.