Questions - OCR MEI - A-Level Maths

Find the acceptance region for the customary test based on the Normal distribution for the null hypothesis $\mu _ { 1 } = \mu _ { 2 }$, against the alternative hypothesis $\mu _ { 1 } \neq \mu _ { 2 }$, at the $5 \%$ level of significance.
The sample means are found to be 25.8 mm and 24.4 mm respectively. What is the result of the test? Provide a two-sided $99 \%$ confidence interval for $\mu _ { 1 } - \mu _ { 2 }$. The production lines are modified so that the diameters may be assumed to be of equal (but unknown) variance. However, they may no longer be Normally distributed. A two-sided test of the equality of the population medians is required, at the $5 \%$ significance level.
The diameters in independent random samples of sizes 6 and 8 are as follows, in mm .
First production line 25.9 25.8 25.3 24.7 24.4 25.4
Second production line 23.8 25.6 24.0 23.5 24.1 24.5 24.3 25.1
Use an appropriate procedure to carry out the test.

OCR MEI S4 2010 June Q4

Standard +0.3

4 At an agricultural research station, a trial is made of four varieties (A, B, C, D) of a certain crop in an experimental field. The varieties are grown on plots in the field and their yields are measured in a standard unit.

It is at first thought that there may be a consistent trend in the natural fertility of the soil in the field from the west side to the east, though no other trends are known. Name an experimental design that should be used in these circumstances and give an example of an experimental layout. Initial analysis suggests that any natural fertility trend may in fact be ignored, so the data from the trial are analysed by one-way analysis of variance.
The usual model for one-way analysis of variance of the yields $y _ { i j }$ may be written as $$y _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$ where the $e _ { i j }$ represent the experimental errors. Interpret the other terms in the model. State the usual distributional assumptions for the $e _ { i j }$.
The data for the yields are as follows, each variety having been used on 5 plots.
Variety
A B C D
12.3 14.2 14.1 13.6
11.9 13.1 13.2 12.8
12.8 13.1 14.6 13.3
12.2 12.5 13.7 14.3
13.5 12.7 13.4 13.8
$$\left[ \Sigma \Sigma y _ { i j } = 265.1 , \quad \Sigma \Sigma y _ { i j } ^ { 2 } = 3524.31 . \right]$$ Construct the usual one-way analysis of variance table and carry out the usual test, at the 5\% significance level. Report briefly on your conclusions. {www.ocr.org.uk}) after the live examination series.
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OCR MEI S4 2012 June Q1

Standard +0.3

1 In a certain country, any baby born is equally likely to be a boy or a girl, independently for all births. The birthweight of a baby boy is given by the continuous random variable $X _ { B }$ with probability density function (pdf) $\mathrm { f } _ { B } ( x )$ and cumulative distribution function (cdf) $\mathrm { F } _ { B } ( x )$. The birthweight of a baby girl is given by the continuous random variable $X _ { G }$ with pdf $\mathrm { f } _ { G } ( x )$ and cdf $\mathrm { F } _ { G } ( x )$. The continuous random variable $X$ denotes the birthweight of a baby selected at random.

By considering $$\mathrm { P } ( X \leqslant x ) = \mathrm { P } ( X \leqslant x \mid \text { boy } ) \mathrm { P } ( \text { boy } ) + \mathrm { P } ( X \leqslant x \mid \text { girl } ) \mathrm { P } ( \text { girl } ) ,$$ find the cdf of $X$ in terms of $\mathrm { F } _ { B } ( x )$ and $\mathrm { F } _ { G } ( x )$, and deduce that the pdf of $X$ is $$\mathrm { f } ( x ) = \frac { 1 } { 2 } \left\{ \mathrm { f } _ { B } ( x ) + \mathrm { f } _ { G } ( x ) \right\} .$$
The birthweights of baby boys and girls have means $\mu _ { B }$ and $\mu _ { G }$ respectively. Deduce that $$\mathrm { E } ( X ) = \frac { 1 } { 2 } \left( \mu _ { B } + \mu _ { G } \right) .$$
The birthweights of baby boys and girls have common variance $\sigma ^ { 2 }$. Find an expression for $\mathrm { E } \left( X ^ { 2 } \right)$ in terms of $\mu _ { B } , \mu _ { G }$ and $\sigma ^ { 2 }$, and deduce that $$\operatorname { Var } ( X ) = \sigma ^ { 2 } + \frac { 1 } { 4 } \left( \mu _ { B } - \mu _ { G } \right) ^ { 2 } .$$
A random sample of size $2 n$ is taken from all the babies born in a certain period. The mean birthweight of the babies in this sample is $\bar { X }$. Write down an approximation to the sampling distribution of $\bar { X }$ if $n$ is large.
Suppose instead that a stratified sample of size $2 n$ is taken by selecting $n$ baby boys at random and, independently, $n$ baby girls at random. The mean birthweight of the $2 n$ babies in this sample is $\bar { X } _ { s t }$. Write down the expected value of $\bar { X } _ { s t }$ and find the variance of $\bar { X } _ { s t }$.
Deduce that both $\bar { X }$ and $\bar { X } _ { s t }$ are unbiased estimators of the population mean birthweight. Find which is the more efficient.

OCR MEI S4 2012 June Q2

Standard +0.3

2 The random variable $X ( X = 1,2,3,4,5,6 )$ denotes the score when a fair six-sided die is rolled.

Write down the mean of $X$ and show that $\operatorname { Var } ( X ) = \frac { 35 } { 12 }$.
Show that $\mathrm { G } ( t )$, the probability generating function (pgf) of $X$, is given by $$\mathrm { G } ( t ) = \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) }$$ The random variable $N ( N = 0,1,2 , \ldots )$ denotes the number of heads obtained when an unbiased coin is tossed repeatedly until a tail is first obtained.
Show that $\mathrm { P } ( N = r ) = \left( \frac { 1 } { 2 } \right) ^ { r + 1 }$ for $r = 0,1,2 , \ldots$.
Hence show that $\mathrm { H } ( t )$, the pgf of $N$, is given by $\mathrm { H } ( t ) = ( 2 - t ) ^ { - 1 }$.
Use $\mathrm { H } ( t )$ to find the mean and variance of $N$. A game consists of tossing an unbiased coin repeatedly until a tail is first obtained and, each time a head is obtained in this sequence of tosses, rolling a fair six-sided die. The die is not rolled on the first occasion that a tail is obtained and the game ends at that point. The random variable $Q ( Q = 0,1,2 , \ldots )$ denotes the total score on all the rolls of the die. Thus, in the notation above, $Q = X _ { 1 } + X _ { 2 } + \ldots + X _ { N }$ where the $X _ { i }$ are independent random variables each distributed as $X$, with $Q = 0$ if $N = 0$. The pgf of $Q$ is denoted by $\mathrm { K } ( t )$. The familiar result that the pgf of a sum of independent random variables is the product of their pgfs does not apply to $\mathrm { K } ( t )$ because $N$ is a random variable and not a fixed number; you should instead use without proof the result that $\mathrm { K } ( t ) = \mathrm { H } ( \mathrm { G } ( t ) )$.
Show that $\mathrm { K } ( t ) = 6 \left( 12 - t - t ^ { 2 } - \ldots - t ^ { 6 } \right) ^ { - 1 }$.
[0pt] [Hint. $\left. \left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + \ldots + t ^ { 5 } \right) .\right]$
Use $\mathrm { K } ( t )$ to find the mean and variance of $Q$.
Using your results from parts (i), (v) and (vii), verify the result that (in the usual notation for means and variances) $$\sigma _ { Q } { } ^ { 2 } = \sigma _ { N } { } ^ { 2 } \mu _ { X } { } ^ { 2 } + \mu _ { N } \sigma _ { X } { } ^ { 2 } .$$

OCR MEI S4 2012 June Q3

Standard +0.3

3 At an agricultural research station, trials are being made of two fertilisers, A and B, to see whether they differ in their effects on the yield of a crop. Preliminary investigations have established that the underlying variances of the distributions of yields using the two fertilisers may be assumed equal. Scientific analysis of the fertilisers has suggested that fertiliser A may be inferior in that it leads, on the whole, to lower yield. A statistical analysis is being carried out to investigate this. The crop is grown in carefully controlled conditions in 14 experimental plots, 6 with fertiliser A and 8 with fertiliser B. The yields, in kg per plot, are as follows, arranged in ascending order for each fertiliser.

Fertiliser A	9.8	10.2	10.9	11.5	12.7	13.3
Fertiliser B	10.8	11.9	12.0	12.2	12.9	13.5	13.6	13.7

Carry out a Wilcoxon rank sum test at the $5 \%$ significance level to examine appropriate hypotheses.
Carry out a $t$ test at the $5 \%$ significance level to examine appropriate hypotheses.
Goodness of fit tests based on more extensive data sets from other trials with these fertilisers have failed to reject hypotheses of underlying Normal distributions. Discuss the relative merits of the analyses in parts (i) and (ii).

OCR MEI S4 2014 June Q2

Challenging +1.2

The probability density function of the random variable $X$ is $$\mathrm { f } ( x ) = \frac { x ^ { k - 1 } \mathrm { e } ^ { - x / \phi } } { \phi ^ { k } ( k - 1 ) ! } , x > 0$$ where $k$ is a known positive integer and $\phi$ is an unknown parameter ( $\phi > 0$ ). Show that the moment generating function (mgf) of $X$ is $$\mathrm { M } _ { X } ( \theta ) = ( 1 - \phi \theta ) ^ { - k }$$ for $\theta < \frac { 1 } { \phi }$.
Write down the mgf of the random variable $W = \sum _ { i = 1 } ^ { n } X _ { i }$ where $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are independent random variables each with the same distribution as $X$.
Write down the mgf of the random variable $Y = \frac { 2 W } { \phi }$. Given that the mgf of the random variable $V$ having the $\chi _ { m } ^ { 2 }$ distribution is $\mathrm { M } _ { V } ( \theta ) = ( 1 - 2 \theta ) ^ { - m / 2 }$ (for $\theta < \frac { 1 } { 2 }$ ), deduce the distribution of $Y$.
Deduce that $\mathrm { P } \left( l < \frac { 2 W } { \phi } < u \right) = 0.95$ where $l$ and $u$ are the lower and upper $2 \frac { 1 } { 2 } \%$ points of the $\chi _ { 2 n k } ^ { 2 }$ distribution. Hence deduce that a $95 \%$ confidence interval for $\phi$ is given by $\left( \frac { 2 w } { u } , \frac { 2 w } { l } \right)$ where $w$ is an observation on the random variable $W$.
For the case $k = 2$ and $n = 10$, use percentage points of the $\chi ^ { 2 }$ distribution to write down, in terms of $w$, an expression for a $95 \%$ confidence interval for $\phi$. By considering the $\operatorname { mgf }$ of $W$, find in terms of $\phi$ the expected length of this interval.

OCR MEI S4 2014 June Q3

Challenging +1.8

Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic, power.
A chemical manufacturer is endeavouring to reduce the amount of a certain impurity in one of its bulk products by improving the production process. The amount of impurity is measured in a convenient unit of concentration, and this is modelled by the Normally distributed random variable $X$. In the old production process, the mean of $X$, denoted by $\mu$, was 63 and the standard deviation of $X$ was 3.7. Experimental batches of the product are to be made using the new process, and it is desired to examine the hypotheses $\mathrm { H } _ { 0 } : \mu = 63$ and $\mathrm { H } _ { 1 } : \mu < 63$ for the new process. Investigation of the variability in the new process has established that the standard deviation may be assumed unchanged. The usual Normal test based on $\bar { X }$ is to be used, where $\bar { X }$ is the mean of $X$ over $n$ experimental batches (regarded as a random sample), with a critical value $c$ such that $\mathrm { H } _ { 0 }$ is rejected if the value of $\bar { X }$ is less than $c$. The following criteria are set out.
- If in fact $\mu = 63$, the probability of concluding that $\mu < 63$ must be only $1 \%$.
- If in fact $\mu = 60$, the probability of concluding that $\mu < 63$ must be $90 \%$.
Find $c$ and the smallest value of $n$ that is required. With these values, what is the power of the test if in fact $\mu = 58.5$ ?

OCR MEI S4 2015 June Q1

Standard +0.3

1 The random variable $X$ has the following probability density function, in which $a$ is a (positive) parameter. $$\mathrm { f } ( x ) = \frac { 2 } { a } x \mathrm { e } ^ { - x ^ { 2 } / a } , \quad x \geqslant 0 .$$

Verify that $\int _ { 0 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x = 1$.
Show that $\mathrm { E } \left( X ^ { 2 } \right) = a$ and $\mathrm { E } \left( X ^ { 4 } \right) = 2 a ^ { 2 }$. The parameter $a$ is to be estimated by maximum likelihood based on an independent random sample from the distribution, $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$.
Show that the logarithm of the likelihood function is $$n \ln 2 - n \ln a + \sum _ { i = 1 } ^ { n } \ln X _ { i } - \frac { 1 } { a } \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }$$ Hence obtain the maximum likelihood estimator, $\hat { a }$, for $a$.
[0pt] [You are not required to verify that any turning point you find is a maximum.]
Using the results from part (ii), show that $\hat { a }$ is unbiased for $a$ and find the variance of $\hat { a }$.
In a particular random sample from this distribution, $n = 100$ and $\sum x _ { i } ^ { 2 } = 147.1$. Obtain an approximate 95\% confidence interval for $a$. (You may assume that the Central Limit Theorem holds in this case.) Option 2: Generating Functions

OCR MEI S4 2015 June Q2

Challenging +1.8

2 The random variable $Z$ has the standard Normal distribution. The random variable $Y$ is defined by $Y = Z ^ { 2 }$.
You are given that $Y$ has the following probability density function. $$\mathrm { f } ( y ) = \frac { 1 } { \sqrt { 2 \pi y } } \mathrm { e } ^ { - \frac { 1 } { 2 } y } , \quad y > 0$$

Show that the moment generating function (mgf) of $Y$ is given by $$\mathrm { M } _ { Y } ( \theta ) = ( 1 - 2 \theta ) ^ { - \frac { 1 } { 2 } }$$
Use the mgf to obtain $\mathrm { E } ( Y )$ and $\operatorname { Var } ( Y )$. The random variable $U$ is defined by $$U = Z _ { 1 } ^ { 2 } + Z _ { 2 } ^ { 2 } + \ldots + Z _ { n } ^ { 2 } ,$$ where $Z _ { 1 } , Z _ { 2 } , \ldots , Z _ { n }$ are independent standard Normal random variables.
State an appropriate general theorem for mgfs and hence write down the mgf of $U$. State the values of $\mathrm { E } ( U )$ and $\operatorname { Var } ( U )$. The random variable $W$ is defined by $$W = \frac { U - n } { \sqrt { 2 n } }$$
Show that the logarithm of the $\operatorname { mgf }$ of $W$ is $$- \sqrt { \frac { n } { 2 } } \theta - \frac { n } { 2 } \ln \left( 1 - \sqrt { \frac { 2 } { n } } \theta \right) .$$ Use the series expansion of $\ln ( 1 - t )$ to show that, as $n \rightarrow \infty$, this expression tends to $\frac { 1 } { 2 } \theta ^ { 2 }$.
State what this implies about the distribution of $W$ for large $n$.

OCR MEI S4 2015 June Q3

Standard +0.3

3 At an agricultural research station, trials are being carried out to compare a standard variety of tomato with one that has been genetically modified (GM). The trials are concerned with the mean weight of the tomatoes and also with the aesthetic appearance of the tomatoes.

1. Tomatoes of the standard and GM varieties are grown under similar conditions. The tomatoes are weighed and the data are summarised as follows.
  Variety Sample size Sum of weights $( \mathrm { g } )$
  Sum of squares of
  weights $\left( \mathrm { g } ^ { 2 } \right)$
  Standard 30 3218.3 349257
  GM 26 2954.1 338691
  Carry out a test, using the Normal distribution, to investigate whether there is evidence, at the 5\% level of significance, that the two varieties of tomato differ in mean weight. State one assumption required for this test to be valid.
2. The data in part (i) could have been used to carry out a test for the equality of means based on the $t$ distribution. State two additional assumptions required for this test to be valid. Discuss briefly which test would be preferable in this case.
In order to judge whether, on the whole, GM tomatoes have a better aesthetic appearance than standard tomatoes, a trial is carried out as follows. 10 of each variety are chosen and consumer panel is asked to arrange the 20 tomatoes in order according to their appearance.
1. State two important features of the way in which this trial should be designed. Comment briefly on how reliable the evidence from the trial is likely to be.
2. The order in which the consumer panel arranges the tomatoes is as follows. The tomato with best appearance is listed first. $G$ and $S$ denote GM and standard tomatoes respectively. $$\begin{array} { c c c c c c c c c c c c c c c c c c c c } G & G & G & S & G & G & G & S & G & S & S & S & G & G & S & G & S & S & S & S \end{array}$$ Carry out an appropriate test at the $1 \%$ level of significance.

OCR MEI S4 2016 June Q1

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

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

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)
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OCR MEI C2 2005 January Q1

Easy -1.8

1 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $y = x ^ { 6 } + \sqrt { x }$.

OCR MEI C2 2005 January Q2

Easy -1.2

2 Find $\int \left( x ^ { 3 } + \frac { 1 } { x ^ { 3 } } \right) \mathrm { d } x$.

OCR MEI C2 2005 January Q3

Moderate -0.8

3 Sketch the graph of $y = \sin x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Solve the equation $\sin x = - 0.2$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.

OCR MEI C2 2005 January Q4

Moderate -0.8

4 Fig. 4 For triangle ABC shown in Fig. 4, calculate

the length of BC ,
the area of triangle ABC .

OCR MEI C2 2005 January Q5

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

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

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

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

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.

First production line	25.9	25.8	25.3	24.7	24.4	25.4
Second production line	23.8	25.6	24.0	23.5	24.1	24.5	24.3	25.1

Variety
A	B	C	D
12.3	14.2	14.1	13.6
11.9	13.1	13.2	12.8
12.8	13.1	14.6	13.3
12.2	12.5	13.7	14.3
13.5	12.7	13.4	13.8

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

\(x\)	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

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