Questions - A-Level Maths

OCR MEI C4 2013 January Q3

7 marks Moderate -0.3

3 The parametric equations of a curve are $$x = \sin \theta , \quad y = \sin 2 \theta , \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi .$$

Find the exact value of the gradient of the curve at the point where $\theta = \frac { 1 } { 6 } \pi$.
Show that the cartesian equation of the curve is $y ^ { 2 } = 4 x ^ { 2 } - 4 x ^ { 4 }$.

OCR MEI C4 2013 January Q4

8 marks Standard +0.3

4 Fig. 4 shows the curve $y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }$, and the region between the curve, the $x$-axis, the $y$-axis and the line $x = 2$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9bceee25-35bd-448b-a4a2-1a5667be5f11-02_650_727_1176_653} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Find the exact volume of revolution when the shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis.
1. Complete the table of values, and use the trapezium rule with 4 strips to estimate the area of the shaded region.
  $x$ 0 0.5 1 1.5 2
  $y$ 1.9283 2.8964 4.5919
2. The trapezium rule for $\int _ { 0 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { 2 x } } \mathrm {~d} x$ with 8 and 16 strips gives 6.797 and 6.823, although not necessarily in that order. Without doing the calculations, say which result is which, explaining your reasoning.

OCR MEI C4 2013 January Q5

6 marks Moderate -0.3

5 Solve the equation $2 \sec ^ { 2 } \theta = 5 \tan \theta$, for $0 \leqslant \theta \leqslant \pi$.

OCR MEI C4 2013 January Q6

5 marks Moderate -0.3

6 In Fig. 6, $\mathrm { ABC } , \mathrm { ACD }$ and AED are right-angled triangles and $\mathrm { BC } = 1$ unit. Angles CAB and CAD are $\theta$ and $\phi$ respectively. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9bceee25-35bd-448b-a4a2-1a5667be5f11-03_440_524_504_753} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure}

Find AC and AD in terms of $\theta$ and $\phi$.
Hence show that $\mathrm { DE } = 1 + \frac { \tan \phi } { \tan \theta }$. Section B (36 marks)

OCR MEI C4 2013 January Q7

17 marks Standard +0.3

7 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The $\mathrm { O } x y$ plane is horizontal. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9bceee25-35bd-448b-a4a2-1a5667be5f11-03_547_987_1580_539} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure}

Find the length of the ridge of the tent DE , and the angle this makes with the horizontal.
Show that the vector $\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }$ is normal to the plane through $\mathrm { A } , \mathrm { D }$ and E . Hence find the equation of this plane. Given that B lies in this plane, find $a$.
Verify that the equation of the plane BCD is $x + z = 8$. Hence find the acute angle between the planes ABDE and BCD .

OCR MEI C4 2013 January Q8

19 marks Standard +0.3

8 The growth of a tree is modelled by the differential equation $$10 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 20 - h ,$$ where $h$ is its height in metres and the time $t$ is in years. It is assumed that the tree is grown from seed, so that $h = 0$ when $t = 0$.

Write down the value of $h$ for which $\frac { \mathrm { d } h } { \mathrm {~d} t } = 0$, and interpret this in terms of the growth of the tree.
Verify that $h = 20 \left( 1 - \mathrm { e } ^ { - 0.1 t } \right)$ satisfies this differential equation and its initial condition. The alternative differential equation $$200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 400 - h ^ { 2 }$$ is proposed to model the growth of the tree. As before, $h = 0$ when $t = 0$.
Using partial fractions, show by integration that the solution to the alternative differential equation is $$h = \frac { 20 \left( 1 - \mathrm { e } ^ { - 0.2 t } \right) } { 1 + \mathrm { e } ^ { - 0.2 t } } .$$
What does this solution indicate about the long-term height of the tree?
After a year, the tree has grown to a height of 2 m . Which model fits this information better?

OCR MEI C4 2015 June Q2

7 marks Moderate -0.3

2 Express $6 \cos 2 \theta + \sin \theta$ in terms of $\sin \theta$.
Hence solve the equation $6 \cos 2 \theta + \sin \theta = 0$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

OCR MEI C4 2015 June Q3

8 marks Standard +0.3

3

Find the first three terms of the binomial expansion of $\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }$. State the set of values of $x$ for which
the expansion is valid. the expansion is valid.
Hence find $a$ and $b$ such that $\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots$.

OCR MEI C4 2015 June Q4

8 marks Moderate -0.3

4 You are given that $\mathrm { f } ( x ) = \cos x + \lambda \sin x$ where $\lambda$ is a positive constant.

Express $\mathrm { f } ( x )$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$, giving $R$ and $\alpha$ in terms of $\lambda$.
Given that the maximum value (as $x$ varies) of $\mathrm { f } ( x )$ is 2 , find $R , \lambda$ and $\alpha$, giving your answers in exact form.

OCR MEI C4 2015 June Q5

8 marks Standard +0.3

5 A curve has parametric equations $x = \sec \theta , y = 2 \tan \theta$.

Given that the derivative of $\sec \theta$ is $\sec \theta \tan \theta$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \operatorname { cosec } \theta$.
Verify that the cartesian equation of the curve is $y ^ { 2 } = 4 x ^ { 2 } - 4$. Fig. 5 shows the region enclosed by the curve and the line $x = 2$. This region is rotated through $180 ^ { \circ }$ about the $x$-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-02_545_853_1738_607} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
Find the volume of revolution produced, giving your answer in exact form.

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:}
\section*{Other materials required:}
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 S1 2009 January Q1

8 marks Easy -1.2

1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table.

Number	Probability
0	0.7
1	0.2
2	0.1

The spinner is spun twice. The total of the two numbers on which it lands is denoted by $X$.

Show that $\mathrm { P } ( X = 2 ) = 0.18$. The probability distribution of $X$ is given in the table.
$x$ 0 1 2 3 4
$\mathrm { P } ( X = x )$ 0.49 0.28 0.18 0.04 0.01
Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR S1 2009 January Q2

8 marks Moderate -0.8

2 The table shows the age, $x$ years, and the mean diameter, $y \mathrm {~cm}$, of the trunk of each of seven randomly selected trees of a certain species.

Age $( x$ years $)$	11	12	20	28	35	45	51
Mean trunk diameter $( y \mathrm {~cm} )$	12.2	16.0	26.4	39.2	39.6	51.3	60.6

$$\left[ n = 7 , \Sigma x = 202 , \Sigma y = 245.3 , \Sigma x ^ { 2 } = 7300 , \Sigma y ^ { 2 } = 10510.65 , \Sigma x y = 8736.9 . \right]$$

(a) Use an appropriate formula to show that the gradient of the regression line of $y$ on $x$ is 1.13 , correct to 2 decimal places.
(b) Find the equation of the regression line of $y$ on $x$.
Use your equation to estimate the mean trunk diameter of a tree of this species with age
(a) 30 years,
(b) 100 years. It is given that the value of the product moment correlation coefficient for the data in the table is 0.988 , correct to 3 decimal places.
Comment on the reliability of each of your two estimates.

OCR S1 2009 January Q3

10 marks Moderate -0.8

3 Erika is a birdwatcher. The probability that she will see a woodpecker on any given day is $\frac { 1 } { 8 }$. It is assumed that this probability is unaffected by whether she has seen a woodpecker on any other day.

Calculate the probability that Erika first sees a woodpecker
1. on the third day,
2. after the third day.
3. Find the expectation of the number of days up to and including the first day on which she sees a woodpecker.
4. Calculate the probability that she sees a woodpecker on exactly 2 days in the first 15 days.

OCR S1 2009 January Q4

7 marks Moderate -0.3

4 Three tutors each marked the coursework of five students. The marks are given in the table.

Student	$A$	$B$	$C$	$D$	$E$
Tutor 1	73	67	60	48	39
Tutor 2	62	50	61	76	65
Tutor 3	42	50	63	54	71

Calculate Spearman's rank correlation coefficient, $r _ { \mathrm { s } }$, between the marks for tutors 1 and 2 .
The values of $r _ { \mathrm { s } }$ for the other pairs of tutors, are as follows. $$\begin{array} { c c } \text { Tutors } 1 \text { and 3: } & r _ { \mathrm { s } } = - 0.9 \\ \text { Tutors } 2 \text { and 3: } & r _ { \mathrm { s } } = 0.3 \end{array}$$ State which two tutors differ most widely in their judgements. Give your reason.

OCR S1 2009 January Q5

8 marks Easy -1.3

5 The stem-and-leaf diagram shows the masses, in grams, of 23 plums, measured correct to the nearest gram.

5	5	6	7	8	8	9
6	1	2	3	5	6	8	9
7	0	0	2	4	5	6	7	8
8	0
9	7
9

$\quad$ Key $: 6 \mid 2$ means 62

Find the median and interquartile range of these masses.
State one advantage of using the interquartile range rather than the standard deviation as a measure of the variation in these masses.
State one advantage and one disadvantage of using a stem-and-leaf diagram rather than a box-and-whisker plot to represent data.
James wished to calculate the mean and standard deviation of the given data. He first subtracted 5 from each of the digits to the left of the line in the stem-and-leaf diagram, giving the following.
0 5 6 7 8 8 9
1 1 2 3 5 6 8 9
2 0 0 2 4 5 6 7 8
3 0
4 7
The mean and standard deviation of the data in this diagram are 18.1 and 9.7 respectively, correct to 1 decimal place. Write down the mean and standard deviation of the data in the original diagram.

OCR S1 2009 January Q6

12 marks Standard +0.3

6 A test consists of 4 algebra questions, A, B, C and D, and 4 geometry questions, G, H, I and J.
The examiner plans to arrange all 8 questions in a random order, regardless of topic.

(a) How many different arrangements are possible?
(b) Find the probability that no two Algebra questions are next to each other and no two Geometry questions are next to each other. Later, the examiner decides that the questions should be arranged in two sections, Algebra followed by Geometry, with the questions in each section arranged in a random order.
(a) How many different arrangements are possible?
(b) Find the probability that questions A and H are next to each other.
(c) Find the probability that questions B and J are separated by more than four other questions.

OCR S1 2009 January Q7

12 marks Moderate -0.8

7 At a factory that makes crockery the quality control department has found that $10 \%$ of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by $X$.

State an appropriate distribution with which to model $X$. Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
Find
1. $\mathrm { P } ( X = 3 )$,
2. $\mathrm { P } ( X \geqslant 1 )$.
3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .

OCR S1 2009 January Q8

7 marks Moderate -0.3

8 A game uses an unbiased die with faces numbered 1 to 6 . The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.

Find the probability that the final score is 4 .
Given that the die is thrown only once, find the probability that the final score is 4 .
Given that the die is thrown twice, find the probability that the final score is 4 .

OCR S1 2011 January Q2

11 marks Moderate -0.8

2 The random variable $X$ has the distribution $\operatorname { Geo } ( 0.2 )$. Find

$\mathrm { P } ( X = 3 )$,
$\mathrm { P } ( 3 \leqslant X \leqslant 5 )$,
$\mathrm { P } ( X > 4 )$. Two independent values of $X$ are found.
Find the probability that the total of these two values is 3 .

OCR S1 2011 January Q3

12 marks Moderate -0.8

3 A firm wishes to assess whether there is a linear relationship between the annual amount spent on advertising, $\pounds x$ thousand, and the annual profit, $\pounds y$ thousand. A summary of the figures for 12 years is as follows. $$n = 12 \quad \Sigma x = 86.6 \quad \Sigma y = 943.8 \quad \Sigma x ^ { 2 } = 658.76 \quad \Sigma y ^ { 2 } = 83663.00 \quad \Sigma x y = 7351.12$$

Calculate the product moment correlation coefficient, showing that it is greater than 0.9 .
Comment briefly on this value in this context.
A manager claims that this result shows that spending more money on advertising in the future will result in greater profits. Make two criticisms of this claim.
Calculate the equation of the regression line of $y$ on $x$.
Estimate the annual profit during a year when $\pounds 7400$ was spent on advertising.

OCR S1 2011 January Q4

7 marks Moderate -0.8

4 Jenny and Omar are each allowed two attempts at a high jump.

The probability that Jenny will succeed on her first attempt is 0.6 . If she fails on her first attempt, the probability that she will succeed on her second attempt is 0.7 . Calculate the probability that Jenny will succeed.
The probability that Omar will succeed on his first attempt is $p$. If he fails on his first attempt, the probability that he will succeed on his second attempt is also $p$. The probability that he succeeds is 0.51 . Find $p$. $530 \%$ of packets of Natural Crunch Crisps contain a free gift. Jan buys 5 packets each week.

OCR S1 2011 January Q6

10 marks Moderate -0.8

6

The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number.
1 3 3 3 5 5 9
How many different 7 -digit numbers can be formed using these cards?
The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it. \includegraphics[max width=\textwidth, alt={}, center]{98ac515d-fd47-4864-afd6-321e9848d6cb-04_398_801_596_632} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
1. How many selections of seven cards are possible?
2. Find the probability that the seven cards include exactly one card showing the letter A .

OCR S1 2011 January Q7

5 marks Easy -1.2

7 The probability distribution of a discrete random variable, $X$, is shown below.

$x$	0	2
$\mathrm { P } ( X = x )$	$a$	$1 - a$

Find $\mathrm { E } ( X )$ in terms of $a$.
Show that $\operatorname { Var } ( X ) = 4 a ( 1 - a )$.

\(x\)	0	2
\(\mathrm { P } ( X = x )\)	\(a\)	\(1 - a\)

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OCR MEI C4 2013 January Q3

OCR MEI C4 2013 January Q4

OCR MEI C4 2013 January Q5

OCR MEI C4 2013 January Q6

OCR MEI C4 2013 January Q7

OCR MEI C4 2013 January Q8

OCR MEI C4 2015 June Q2

OCR MEI C4 2015 June Q3

OCR MEI C4 2015 June Q4

OCR MEI C4 2015 June Q5

OCR MEI C4 2015 June Q6

OCR MEI C4 2015 June Q7

OCR S1 2009 January Q1

OCR S1 2009 January Q2

OCR S1 2009 January Q3

OCR S1 2009 January Q4

OCR S1 2009 January Q5

OCR S1 2009 January Q6

OCR S1 2009 January Q7

OCR S1 2009 January Q8

OCR S1 2011 January Q2

OCR S1 2011 January Q3

OCR S1 2011 January Q4

OCR S1 2011 January Q6

OCR S1 2011 January Q7

Age \(( x\) years \()\)	11	12	20	28	35	45	51
Mean trunk diameter \(( y \mathrm {~cm} )\)	12.2	16.0	26.4	39.2	39.6	51.3	60.6

Student	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
Tutor 1	73	67	60	48	39
Tutor 2	62	50	61	76	65
Tutor 3	42	50	63	54	71

\(x\)	0	0.5	1	1.5	2
\(y\)		1.9283	2.8964	4.5919

\(x\)	0	1	2	3	4
\(\mathrm { P } ( X = x )\)	0.49	0.28	0.18	0.04	0.01