Questions - OCR MEI - A-Level Maths

OCR MEI C4 2015 June Q6

18 marks Standard +0.3

6 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane $z = 0$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-03_785_1283_424_392} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure}

Verify that the equation of the plane through $\mathrm { A } , \mathrm { B }$ and E is $x + 6 y + 12 = 0$. Hence, given that F lies in this plane, show that $a = - 2 \frac { 1 } { 3 }$.
(A) Show that the vector $\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)$ is normal to the plane DHC.
(B) Hence find the cartesian equation of this plane.
(C) Given that G lies in the plane DHC , find $b$ and the length FG .
Find the angle EFB . A straight wire joins point H to a point P which is half way between E and F . Q is a point two-thirds of the way along this wire, so that $\mathrm { HQ } = 2 \mathrm { QP }$.
Find the height of Q above the ground. \section*{Question 7 begins on page 4.}

OCR MEI C4 2015 June Q7

18 marks Standard +0.3

7 A drug is administered by an intravenous drip. The concentration, $x$, of the drug in the blood is measured as a fraction of its maximum level. The drug concentration after $t$ hours is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k \left( 1 + x - 2 x ^ { 2 } \right) ,$$ where $0 \leqslant x < 1$, and $k$ is a positive constant. Initially, $x = 0$.

Express $\frac { 1 } { ( 1 + 2 x ) ( 1 - x ) }$ in partial fractions.
Hence solve the differential equation to show that $\frac { 1 + 2 x } { 1 - x } = \mathrm { e } ^ { 3 k t }$.
After 1 hour the drug concentration reaches $75 \%$ of its maximum value and so $x = 0.75$. Find the value of $k$, and the time taken for the drug concentration to reach $90 \%$ of its maximum value.
Rearrange the equation in part (ii) to show that $x = \frac { 1 - \mathrm { e } ^ { - 3 k t } } { 1 + 2 \mathrm { e } ^ { - 3 k t } }$. Verify that in the long term the drug concentration approaches its maximum value. \section*{END OF QUESTION PAPER} \section*{Tuesday 16 J une 2015 - Afternoon} \section*{A2 GCE MATHEMATICS (MEI)} 4754/01B Applications of Advanced Mathematics (C4) Paper B: Comprehension \section*{QUESTION PAPER} \section*{Candidates answer on the Question Paper.} \section*{OCR supplied materials:}
- Insert (inserted)
- MEI Examination Formulae and Tables (MF2)
\section*{Other materials required:}
- Scientific or graphical calculator
- Rough paper
Duration: Up to 1 hour \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-05_117_495_1014_1308} PLEASE DO NOT WRITE IN THIS SPACE 2 In line 79 it says "For most journeys, more than half the journey time is composed of load time and transfer time". For what percentage of the journey time for the round trip made by car A in Table 4 is the car stationary?
\includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_645_1746_388_164}
3 Using the expression on line 51, work out the answer to the question on lines 39 and 40 for the case where there are 10 upper floors and 7 people. Give your answer to 2 decimal places.
\includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-07_488_1746_1233_164}
4 In lines 89 and 90 it says "... on average there will be approximately 8 stops per trip. A round trip with 8 stops would take between 188 and 200 seconds". Explain how the figure of 188 seconds has been derived. 5
Referring to Strategy 3 and lines 99 to 101, complete the table below for car C .
Calculate the time car C will take to transport all the people who work on floors 7 and 8 , and return to the ground floor.
5
\includegraphics[max width=\textwidth, alt={}]{132ae754-bd4c-4819-80ef-4823ac2ead4f-08_1095_816_484_700}
68 people make independent visits to any one of the upper floors of a building with 10 upper floors. What is the probability that at least one of the visitors goes to the top floor?
6
7 On lines 94 and 95 it says "Table 4 gives the timings for round trips in which the cars are required to stop at every floor they serve; Table 2 suggests this is a common occurrence in this case". Explain how Table 2 is used to make this claim. \includegraphics[max width=\textwidth, alt={}, center]{132ae754-bd4c-4819-80ef-4823ac2ead4f-09_1093_1740_1238_166} END OF QUESTION PAPER

OCR MEI S1 2010 January Q1

8 marks Easy -1.3

1 A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$\begin{array} { l l l l l l l l l l l l l l l } 73 & 67 & 75 & 64 & 52 & 63 & 75 & 81 & 77 & 72 & 68 & 74 & 79 & 72 & 71 \end{array}$$

Construct a sorted stem and leaf diagram to represent these data, taking stem values of $50,60 , \ldots$.
Write down the median and midrange of the data.
Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer.

OCR MEI S1 2010 January Q2

8 marks Moderate -0.3

2 In her purse, Katharine has two $\pounds 5$ notes, two $\pounds 10$ notes and one $\pounds 20$ note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable $\pounds X$.

(A) Show that $\mathrm { P } ( X = 10 ) = 0.1$.
(B) Show that $\mathrm { P } ( X = 30 ) = 0.2$. The table shows the probability distribution of $X$.
$r$ 10 15 20 25 30
$\mathrm { P } ( X = r )$ 0.1 0.4 0.1 0.2 0.2
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2010 January Q3

8 marks Easy -1.2

3 In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random.

$G$ is the event that this person goes to the gym.
$R$ is the event that this person goes running.

You are given that $\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13$ and $\mathrm { P } ( G \cap R ) = 0.06$.

Draw a Venn diagram, showing the events $G$ and $R$, and fill in the probability corresponding to each of the four regions of your diagram.
Determine whether the events $G$ and $R$ are independent.
Find $\mathrm { P } ( R \mid G )$.

OCR MEI S1 2010 January Q4

5 marks Moderate -0.8

4 In a multiple-choice test there are 30 questions. For each question, there is a $60 \%$ chance that a randomly selected student answers correctly, independently of all other questions.

Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.

OCR MEI S1 2010 January Q6

4 marks Moderate -0.8

6 Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if

no student may win more than 1 prize,
no student may win all 3 prizes. Section B (36 marks)

OCR MEI S1 2010 January Q7

19 marks Moderate -0.8

7 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears. \includegraphics[max width=\textwidth, alt={}, center]{2f39c509-5429-4193-9526-15fb45b18a38-4_837_1651_466_246}

Calculate the number of pears which are between 90 and 100 mm long.
Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
Calculate an estimate of the standard deviation.
Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
Name the type of skewness of the distribution.
Illustrate the data using a cumulative frequency diagram.

OCR MEI S1 2010 January Q8

17 marks Standard +0.3

8 An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.

Pollution level	Low	Medium	High
Probability	0.5	0.35	0.15

Three days are chosen at random. Find the probability that the pollution level is
(A) low on all 3 days,
(B) low on at least one day,
(C) low on one day, medium on another day, and high on the other day.
Ten days are chosen at random. Find the probability that
(A) there are no days when the pollution level is high,
(B) there is exactly one day when the pollution level is high. The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
Carry out a test at the $5 \%$ level to determine if there is evidence to suggest that she is correct. Use hypotheses $\mathrm { H } _ { 0 } : p = 0.5 , \mathrm { H } _ { 1 } : p > 0.5$, where $p$ represents the probability that the pollution level in this street is low. Explain why $\mathrm { H } _ { 1 }$ has this form.

OCR MEI S1 2011 January Q1

3 marks Easy -1.3

1 The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas.

8	3	9
9	3	5	6	6	6	8	9	9
10	0	2	2	3	4	6	9
11	2	4	7	8
12	3	4	5
13	2

Key: 11 | 2 represents 1120 grams

Find the mode and the median of the data.
Identify the type of skewness of the distribution.

OCR MEI S1 2011 January Q2

4 marks Moderate -0.3

2 The table shows all the possible products of the scores on two fair four-sided dice.

\multirow{2}{*}{}		Score on second die
		1	2	3	4
\multirow{4}{*}{Score on first die}	1	1	2	3	4
	2	2	4	6	8
	3	3	6	9	12
	4	4	8	12	16

Find the probability that the product of the two scores is less than 10 .
Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent.

OCR MEI S1 2011 January Q3

6 marks Moderate -0.8

3 There are 13 men and 10 women in a running club. A committee of 3 men and 3 women is to be selected.

In how many different ways can the three men be selected?
In how many different ways can the whole committee be selected?
A random sample of 6 people is selected from the running club. Find the probability that this sample consists of 3 men and 3 women.

OCR MEI S1 2011 January Q5

8 marks Easy -1.3

5 Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics[max width=\textwidth, alt={}, center]{7df6dcad-790d-4d0e-b1a5-3371103997d9-3_732_1141_792_500} A day is selected at random. Find the probability that

the weather is wet and Andy travels by bus,
Andy walks or travels by bike,
the weather is dry given that Andy walks or travels by bike.

OCR MEI S1 2011 January Q6

8 marks Moderate -0.8

6 A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.

Reducing car use
Insulating their homes
Avoiding air travel

The numbers of citizens who have used each of these methods are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{7df6dcad-790d-4d0e-b1a5-3371103997d9-4_703_1087_712_529} One of the citizens is selected at random.

Find the probability that this citizen
(A) has avoided air travel,
(B) has used at least two of the three methods.
Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. Three of the citizens are selected at random.
Find the probability that none of them have avoided air travel. Section B (36 marks)

OCR MEI S1 2011 January Q7

19 marks Moderate -0.3

7 The incomes of a sample of 918 households on an island are given in the table below.

Income

$( x$ thousand pounds $)$

$0 \leqslant x \leqslant 20$

$20 < x \leqslant 40$

$40 < x \leqslant 60$

$60 < x \leqslant 100$

$100 < x \leqslant 200$

Frequency

238

365

142

128

45

Draw a histogram to illustrate the data.
Calculate an estimate of the mean income.
Calculate an estimate of the standard deviation of the incomes.
Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation.
The incomes were converted into another currency using the formula $y = 1.15 x$. Calculate estimates of the mean and variance of the incomes in the new currency.

OCR MEI S1 2011 January Q8

17 marks Standard +0.3

8 Mark is playing solitaire on his computer. The probability that he wins a game is 0.2 , independently of all other games that he plays.

Find the expected number of wins in 12 games.
Find the probability that
(A) he wins exactly 2 out of the next 12 games that he plays,
(B) he wins at least 2 out of the next 12 games that he plays.
Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the $5 \%$ level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test.

OCR MEI S1 2012 January Q1

7 marks Easy -1.8

1 The mean daily maximum temperatures at a research station over a 12-month period, measured to the nearest degree Celsius, are given below.

Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
8	15	25	29	31	31	34	36	34	26	15	8

Construct a sorted stem and leaf diagram to represent these data, taking stem values of $0,10 , \ldots$.
Write down the median of these data.
The mean of these data is 24.3 . Would the mean or the median be a better measure of central tendency of the data? Briefly explain your answer.

OCR MEI S1 2012 January Q2

7 marks Moderate -0.8

2 The hourly wages, $\pounds x$, of a random sample of 60 employees working for a company are summarised as follows. $$n = 60 \quad \sum x = 759.00 \quad \sum x ^ { 2 } = 11736.59$$

Calculate the mean and standard deviation of $x$.
The workers are offered a wage increase of $2 \%$. Use your answers to part (i) to deduce the new mean and standard deviation of the hourly wages after this increase.
As an alternative the workers are offered a wage increase of 25 p per hour. Write down the new mean and standard deviation of the hourly wages after this 25p increase.

OCR MEI S1 2012 January Q3

8 marks Standard +0.3

3 Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is

0.7 if he won the previous set,
0.4 if Alan won the previous set.

It is not possible to draw a set.

Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets.
Find the probability that Alan wins the match.
Find the probability that the match ends after exactly four sets have been played.

OCR MEI S1 2012 January Q4

6 marks Moderate -0.8

4 In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random.

$T$ is the event that this person likes tomato soup.
$M$ is the event that this person likes mushroom soup.

You are given that $\mathrm { P } ( T ) = 0.55 , \mathrm { P } ( M ) = 0.33$ and $\mathrm { P } ( T \mid M ) = 0.80$.

Use this information to show that the events $T$ and $M$ are not independent.
Find $\mathrm { P } ( T \cap M )$.
Draw a Venn diagram showing the events $T$ and $M$, and fill in the probability corresponding to each of the four regions of your diagram.

OCR MEI S1 2012 January Q5

8 marks Standard +0.3

5 A couple plan to have at least one child of each sex, after which they will have no more children. However, if they have four children of one sex, they will have no more children. You should assume that each child is equally likely to be of either sex, and that the sexes of the children are independent. The random variable $X$ represents the total number of girls the couple have.

Show that $\mathrm { P } ( X = 1 ) = \frac { 11 } { 16 }$. The table shows the probability distribution of $X$.
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ $\frac { 1 } { 16 }$ $\frac { 11 } { 16 }$ $\frac { 1 } { 8 }$ $\frac { 1 } { 16 }$ $\frac { 1 } { 16 }$
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2012 January Q6

17 marks Moderate -0.3

6 It is known that $25 \%$ of students in a particular city are smokers. A random sample of 20 of the students is selected.

(A) Find the probability that there are exactly 4 smokers in the sample.
(B) Find the probability that there are at least 3 but no more than 6 smokers in the sample.
(C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
(A) Write down suitable null and alternative hypotheses for the test.
(B) Explain why the alternative hypothesis has the form that it does.
Find the critical region for the test at the $5 \%$ level, showing all of your calculations.
In fact there are 3 smokers in the sample. Complete the test, stating your conclusion clearly.

OCR MEI S1 2012 January Q7

19 marks Moderate -0.3

7 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{4b259fe3-73ef-419f-85ad-1a3b1e6ea56e-4_917_1146_367_447}

Estimate the percentage of lambs with birth weight over 6 kg .
Estimate the median and interquartile range of the data.
Use your answers to part (ii) to show that there are very few, if any, outliers. Comment briefly on whether any outliers should be disregarded in analysing these data. The box and whisker plot shows the birth weights of 100 lambs from Welsh Mountain sheep. \includegraphics[max width=\textwidth, alt={}, center]{4b259fe3-73ef-419f-85ad-1a3b1e6ea56e-4_328_1616_1749_260}
Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.
The weight of the largest Welsh Mountain lamb was originally recorded as 6.5 kg , but then corrected. If this error had not been corrected, how would this have affected your answers to part (iv)? Briefly explain your answer.
One lamb of each type is selected at random. Estimate the probability that the birth weight of both lambs is at least 3.9 kg .

OCR MEI S1 2013 January Q2

8 marks Moderate -0.8

2 The probability distribution of the random variable $X$ is given by the formula $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } - 1 \right) \text { for } r = 2,3,4,5 .$$

Show the probability distribution in a table, and find the value of $k$.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2013 January Q3

8 marks Moderate -0.8

3 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.

$W$ is the event that Alan has to wait at the level crossing.
$L$ is the event that Alan is late for work.

You are given that $\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07$ and $\mathrm { P } ( L \cup W ) = 0.08$.

Calculate $\mathrm { P } ( L \cap W )$.
Draw a Venn diagram, showing the events $L$ and $W$. Fill in the probability corresponding to each of the four regions of your diagram.
Determine whether the events $L$ and $W$ are independent, explaining your method clearly.

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 16 }\)	\(\frac { 11 } { 16 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 1 } { 16 }\)

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