Questions - OCR MEI - A-Level Maths

OCR MEI C2 2010 June Q1

1 You are given that $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = \frac { u _ { n } } { 1 + u _ { n } } \end{aligned}$$ Find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Give your answers as fractions.

OCR MEI C2 2010 June Q2

2

Evaluate $\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }$.
Express the series $2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7$ in the form $\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )$ where $\mathrm { f } ( r )$ and $a$ are to be determined.

OCR MEI C2 2010 June Q3

3

Differentiate $x ^ { 3 } - 6 x ^ { 2 } - 15 x + 50$.
Hence find the $x$-coordinates of the stationary points on the curve $y = x ^ { 3 } - 6 x ^ { 2 } - 15 x + 50$.

OCR MEI C2 2010 June Q4

4 In this question, $\mathrm { f } ( x ) = x ^ { 2 } - 5 x$. Fig. 4 shows a sketch of the graph of $y = \mathrm { f } ( x )$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e5ac28f3-d61a-4b40-8b47-28c930761a28-2_789_887_1427_628} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} On separate diagrams, sketch the curves $y = \mathrm { f } ( 2 x )$ and $y = 3 \mathrm { f } ( x )$, labelling the coordinates of their intersections with the axes and their turning points.

OCR MEI C2 2010 June Q5

5 Find $\int _ { 2 } ^ { 5 } \left( 1 - \frac { 6 } { x ^ { 3 } } \right) \mathrm { d } x$.

OCR MEI C2 2010 June Q6

6 The gradient of a curve is $6 x ^ { 2 } + 12 x ^ { \frac { 1 } { 2 } }$. The curve passes through the point $( 4,10 )$. Find the equation of the curve.

OCR MEI C2 2010 June Q7

7 Express $\log _ { a } x ^ { 3 } + \log _ { a } \sqrt { x }$ in the form $k \log _ { a } x$.

OCR MEI C2 2010 June Q8

8 Showing your method clearly, solve the equation $4 \sin ^ { 2 } \theta = 3 + \cos ^ { 2 } \theta$, for values of $\theta$ between $0 ^ { \circ }$ and $360 ^ { \circ }$.

OCR MEI C2 2010 June Q9

9 The points $( 2,6 )$ and $( 3,18 )$ lie on the curve $y = a x ^ { n }$.
Use logarithms to find the values of $a$ and $n$, giving your answers correct to 2 decimal places.

OCR MEI C2 2010 June Q11

11

\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e5ac28f3-d61a-4b40-8b47-28c930761a28-4_775_768_260_733} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
\end{figure} A boat travels from P to Q and then to R . As shown in Fig. 11.1, Q is 10.6 km from P on a bearing of $045 ^ { \circ }$. R is 9.2 km from P on a bearing of $113 ^ { \circ }$, so that angle QPR is $68 ^ { \circ }$. Calculate the distance and bearing of R from Q .
Fig. 11.2 shows the cross-section, EBC, of the rudder of a boat. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e5ac28f3-d61a-4b40-8b47-28c930761a28-4_531_1490_1509_363} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
\end{figure} BC is an arc of a circle with centre A and radius 80 cm . Angle $\mathrm { CAB } = \frac { 2 \pi } { 3 }$ radians.
EC is an arc of a circle with centre D and radius $r \mathrm {~cm}$. Angle CDE is a right angle.
1. Calculate the area of sector ABC .
2. Show that $r = 40 \sqrt { 3 }$ and calculate the area of triangle CDA.
3. Hence calculate the area of cross-section of the rudder. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{e5ac28f3-d61a-4b40-8b47-28c930761a28-5_881_1378_255_379} \captionsetup{labelformat=empty} \caption{Fig. 12}
  \end{figure} A branching plant has stems, nodes, leaves and buds.
  - There are 7 leaves at each node.
From each node, 2 new stems grow.
At the end of each final stem, there is a bud.

OCR MEI C4 2006 January Q1

1 Solve the equation $\frac { 2 x } { x - 2 } - \frac { 4 x } { x + 1 } = 3$.

OCR MEI C4 2006 January Q2

2 A curve is defined parametrically by the equations $$x = t - \ln t , \quad y = t + \ln t \quad ( t > 0 )$$ Find the gradient of the curve at the point where $t = 2$.

OCR MEI C4 2006 January Q3

3 A triangle ABC has vertices $\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )$ and $\mathrm { C } ( 4,8,3 )$. By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle.

OCR MEI C4 2006 January Q4

4 Solve the equation $2 \sin 2 \theta + \cos 2 \theta = 1$, for $0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }$.

OCR MEI C4 2006 January Q5

5

Find the cartesian equation of the plane through the point ( $2 , - 1,4$ ) with normal vector $$\mathbf { n } = \left( \begin{array} { r } 1
- 1
2 \end{array} \right)$$
Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7
12
9 \end{array} \right) + \lambda \left( \begin{array} { l } 1
3
2 \end{array} \right)$$

OCR MEI C4 2006 January Q6

6

Find the first three non-zero terms of the binomial expansion of $\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }$ for $| x | < 2$.
Use this result to find an approximation for $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$, rounding your answer to
4 significant figures.
Given that $\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c$, evaluate $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$, rounding your answer to 4 significant figures.

OCR MEI C4 2006 January Q7

7 In a game of rugby, a kick is to be taken from a point P (see Fig. 7). P is a perpendicular distance $y$ metres from the line TOA. Other distances and angles are as shown. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{897205bc-2f93-4628-8f21-2ec7fd3b3699-3_509_629_513_715} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure}

Show that $\theta = \beta - \alpha$, and hence that $\tan \theta = \frac { 6 y } { 160 + y ^ { 2 } }$. Calculate the angle $\theta$ when $y = 6$.
By differentiating implicitly, show that $\frac { \mathrm { d } \theta } { \mathrm { d } y } = \frac { 6 \left( 160 - y ^ { 2 } \right) } { \left( 160 + y ^ { 2 } \right) ^ { 2 } } \cos ^ { 2 } \theta$.
Use this result to find the value of $y$ that maximises the angle $\theta$. Calculate this maximum value of $\theta$. [You need not verify that this value is indeed a maximum.]

OCR MEI C4 2006 January Q8

1 marks

8 Some years ago an island was populated by red squirrels and there were no grey squirrels. Then grey squirrels were introduced. The population $x$, in thousands, of red squirrels is modelled by the equation $$x = \frac { a } { 1 + k t } ,$$ where $t$ is the time in years, and $a$ and $k$ are constants. When $t = 0 , x = 2.5$.

Show that $\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { k x ^ { 2 } } { a }$.
Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after one year, calculate $a$ and $k$.
What is the long-term population of red squirrels predicted by this model? The population $y$, in thousands, of grey squirrels is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = 2 y - y ^ { 2 } .$$ When $t = 0 , y = 1$.
Express $\frac { 1 } { 2 y - y ^ { 2 } }$ in partial fractions.
Hence show by integration that $\ln \left( \frac { y } { 2 - y } \right) = 2 t$. Show that $y = \frac { 2 } { 1 + \mathrm { e } ^ { - 2 t } }$.
What is the long-term population of grey squirrels predicted by this model? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4)
Paper B: Comprehension
Monday
23 JANUARY 2006
Afternooon U Additional materials:
Rough paper
MEI Examination Formulae and Tables (MF2)
4754(B) \section*{Up to 1 hour
$$\text { o to } 1 \text { hour }$$} \section*{TIME} \section*{Up to 1 hour}
- Write your name, centre number and candidate number in the spaces at the top of this page.
- Answer all the questions.
- Write your answers in the spaces provided on the question paper.
- You are permitted to use a graphical calculator in this paper.
- Final answers should be given to a degree of accuracy appropriate to the context.
- The number of marks is given in brackets [ ] at the end of each question or part question.
- The insert contains the text for use with the questions.
- You may find it helpful to make notes and do some calculations as you read the passage.
- You are not required to hand in these notes with your question paper.
- You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
- The total number of marks for this section is 18.
For Examiner's Use
Qu. Mark
1
2
3
4
5
Total
1 Line 59 says "Again Party G just misses out; if there had been 7 seats G would have got the last one." Where is the evidence for this in the article? 26 parties, P, Q, R, S, T and U take part in an election for 7 seats. Their results are shown in the table below.
Party Votes (\%)
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Use the Trial-and-Improvement method, starting with values of $10 \%$ and $14 \%$, to find an acceptance percentage for 7 seats, and state the allocation of the seats.
Acceptance percentage, $\boldsymbol { a }$ \% 10\% 14\%
Party Votes (\%) Seats Seats Seats Seats Seats
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Total seats
Seat Allocation $\quad \mathrm { P } \ldots$... $\mathrm { Q } \ldots$ R ... S ... T ... $\mathrm { U } \ldots$.
Now apply the d'Hondt Formula to the same figures to find the allocation of the seats.
Round
Party 1 2 3 4 5 6 7 Residual
P 30.2
Q 11.4
R 22.4
S 14.8
T 10.9
U 10.3
Seat allocated to
Seat Allocation $\mathrm { P } \ldots$. Q … $\mathrm { R } \ldots$. S … T … $\mathrm { U } \ldots$. 3 In this question, use the figures for the example used in Table 5 in the article, the notation described in the section "Equivalence of the two methods" and the value of 11 found for $a$ in Table 4. Treating Party E as Party 5, verify that $\frac { V _ { 5 } } { N _ { 5 } + 1 } < a \leqslant \frac { V _ { 5 } } { N _ { 5 } }$.
4 Some of the intervals illustrated by the lines in the graph in Fig. 8 are given in this table.
Seats Interval Seats Interval
1 $22.2 < a \leqslant 27.0$ 5
2 $16.6 < a \leqslant 22.2$ 6 $10.6 < a \leqslant 11.1$
3 7
4
Describe briefly, giving an example, the relationship between the end-points of these intervals and the values in Table 5, which is reproduced below.

Complete the table above. \begin{table}[h]

	Round
Party	1	2	3	4	5	6	Residual
A	22.2	22.2	11.1	11.1	11.1	11.1	7.4
B	6.1	6.1	6.1	6.1	6.1	6.1	6.1
C	27.0	13.5	13.5	13.5	9.0	9.0	9.0
D	16.6	16.6	16.6	8.3	8.3	8.3	8.3
E	11.2	11.2	11.2	11.2	11.2	5.6	5.6
F	3.7	3.7	3.7	3.7	3.7	3.7	3.7
G	10.6	10.6	10.6	10.6	10.6	10.6	10.6
H	2.6	2.6	2.6	2.6	2.6	2.6	2.6
Seat allocated to	C	A	D	C	E	A

\captionsetup{labelformat=empty} \caption{Table 5}

\end{table} 5 The ends of the vertical lines in Fig. 8 are marked with circles. Those at the tops of the lines are filled in, e.g. • whereas those at the bottom are not, e.g. o.

Relate this distinction to the use of inequality signs.
Show that the inequality on line 102 can be rearranged to give $0 \leqslant V _ { k } - N _ { k } a < a$. [1]
Hence justify the use of the inequality signs in line 102.

OCR MEI C4 2006 June Q1

1 Fig. 1 shows part of the graph of $y = \sin x - \sqrt { 3 } \cos x$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-2_467_629_468_717} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Express $\sin x - \sqrt { 3 } \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 \leqslant \alpha \leqslant \frac { 1 } { 2 } \pi$.
Hence write down the exact coordinates of the turning point P .

OCR MEI C4 2006 June Q2

2

Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x } ,$$ where $A , B$ and $C$ are constants, find $B$ and $C$, and show that $A = 0$.
Given that $x$ is sufficiently small, find the first three terms of the binomial expansions of $( 1 + x ) ^ { - 2 }$ and $( 1 - 4 x ) ^ { - 1 }$. Hence find the first three terms of the expansion of $\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }$.

OCR MEI C4 2006 June Q3

3 Given that $\sin ( \theta + \alpha ) = 2 \sin \theta$, show that $\tan \theta = \frac { \sin \alpha } { 2 - \cos \alpha }$. Hence solve the equation $\sin \left( \theta + 40 ^ { \circ } \right) = 2 \sin \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

OCR MEI C4 2006 June Q4

4

The number of bacteria in a colony is increasing at a rate that is proportional to the square root of the number of bacteria present. Form a differential equation relating $x$, the number of bacteria, to the time $t$.
In another colony, the number of bacteria, $y$, after time $t$ minutes is modelled by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = \frac { 10000 } { \sqrt { y } } .$$ Find $y$ in terms of $t$, given that $y = 900$ when $t = 0$. Hence find the number of bacteria after 10 minutes.

OCR MEI C4 2006 June Q5

5

Show that $\int x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = - \frac { 1 } { 4 } \mathrm { e } ^ { - 2 x } ( 1 + 2 x ) + c$. A vase is made in the shape of the volume of revolution of the curve $y = x ^ { 1 / 2 } \mathrm { e } ^ { - x }$ about the $x$-axis between $x = 0$ and $x = 2$ (see Fig. 5). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-3_716_741_1233_662} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
Show that this volume of revolution is $\frac { 1 } { 4 } \pi \left( 1 - \frac { 5 } { \mathrm { e } ^ { 4 } } \right)$. Fig. 6 shows the arch ABCD of a bridge. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-4_378_1630_461_214} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure} The section from B to C is part of the curve OBCE with parametric equations $$x = a ( \theta - \sin \theta ) , y = a ( 1 - \cos \theta ) \text { for } 0 \leqslant \theta \leqslant 2 \pi$$ where $a$ is a constant.
Find, in terms of $a$,
(A) the length of the straight line OE,
(B) the maximum height of the arch.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$. The straight line sections AB and CD are inclined at $30 ^ { \circ }$ to the horizontal, and are tangents to the curve at B and C respectively. BC is parallel to the $x$-axis. BF is parallel to the $y$-axis.
Show that at the point B the parameter $\theta$ satisfies the equation $$\sin \theta = \frac { 1 } { \sqrt { 3 } } ( 1 - \cos \theta )$$ Verify that $\theta = \frac { 2 } { 3 } \pi$ is a solution of this equation.
Hence show that $\mathrm { BF } = \frac { 3 } { 2 } a$, and find OF in terms of $a$, giving your answer exactly.
Find BC and AF in terms of $a$. Given that the straight line distance AD is 20 metres, calculate the value of $a$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-5_748_1306_319_367} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .
Find the length AE .
Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30$$ Write down a vector normal to this plane.
Show that the vector $\left( \begin{array} { l } 4
3
5 \end{array} \right)$ is normal to the plane ABDE . Hence find the equation of the plane ABDE .
Find the angle between the planes ABC and ABDE . RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education \section*{MEI STRUCTURED MATHEMATICS} Applications of Advanced Mathematics (C4) \section*{Paper B: Comprehension} Monday 12 JUNE 2006 Afternoon Up to 1 hour Additional materials:
Rough paper
MEI Examination Formulae and Tables (MF2) TIME Up to 1 hour
- Write your name, centre number and candidate number in the spaces at the top of this page.
- Answer all the questions.
- Write your answers in the spaces provided on the question paper.
- You are permitted to use a graphical calculator in this paper.
- The number of marks is given in brackets [ ] at the end of each question or part question.
- The insert contains the text for use with the questions.
- You may find it helpful to make notes and do some calculations as you read the passage.
- You are not required to hand in these notes with your question paper.
- You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
- The total number of marks for this paper is 18.
For Examiner's Use
Qu. Mark
1
2
3
4
5
6
Total
1 The marathon is 26 miles and 385 yards long ( 1 mile is 1760 yards). There are now several men who can run 2 miles in 8 minutes. Imagine that an athlete maintains this average speed for a whole marathon. How long does the athlete take?
2 According to the linear model, in which calendar year would the record for the men's mile first become negative?
3 Explain the statement in line 93 "According to this model the 2-hour marathon will never be run."
4 Explain how the equation in line 49, $$R = L + ( U - L ) \mathrm { e } ^ { - k t } ,$$ is consistent with Fig. 2
initially,
for large values of $t$.
$\_\_\_\_$
5 A model for an athletics record has the form $$R = A - ( A - B ) \mathrm { e } ^ { - k t } \text { where } A > B > 0 \text { and } k > 0 .$$
Sketch the graph of $R$ against $t$, showing $A$ and $B$ on your graph.
Name one event for which this might be an appropriate model.
\includegraphics[max width=\textwidth, alt={}, center]{c64062c4-4cbd-41b2-9b4d-60a43dceb700-9_803_808_721_575}
$\_\_\_\_$

OCR MEI C4 2006 June Q6

6 A number of cases of the general exponential model for the marathon are given in Table 6. One of these is $$R = 115 + ( 175 - 115 ) \mathrm { e } ^ { - 0.0467 t ^ { 0.797 } }$$

What is the value of $t$ for the year 2012?
What record time does this model predict for the year 2012?
$\_\_\_\_$
$\_\_\_\_$

OCR MEI C4 2008 June Q1

1 Express $\frac { x } { x ^ { 2 } - 4 } + \frac { 2 } { x + 2 }$ as a single fraction, simplifying your answer.

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