Questions - OCR MEI C4 - A-Level Maths

OCR MEI C4 2013 January Q5

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

OCR MEI C4 2013 January Q6

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

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

1 marks

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 2009 June Q1

1 Express $4 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.
Hence solve the equation $4 \cos \theta - \sin \theta = 3$, for $0 \leqslant \theta \leqslant 2 \pi$.

OCR MEI C4 2009 June Q2

7 marks

2 Using partial fractions, find $\int \frac { x } { ( x + 1 ) ( 2 x + 1 ) } \mathrm { d } x$.
[0pt] [7]

OCR MEI C4 2009 June Q4

4 The part of the curve $y = 4 - x ^ { 2 }$ that is above the $x$-axis is rotated about the $y$-axis. This is shown in Fig. 4. Find the volume of revolution produced, giving your answer in terms of $\pi$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b4861178-720d-4803-a608-abef350efb0e-2_531_587_1204_778} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

OCR MEI C4 2009 June Q5

5 A curve has parametric equations $$x = a t ^ { 3 } , \quad y = \frac { a } { 1 + t ^ { 2 } }$$ where $a$ is a constant.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 t \left( 1 + t ^ { 2 } \right) ^ { 2 } }$.
Hence find the gradient of the curve at the point $\left( a , \frac { 1 } { 2 } a \right)$.

OCR MEI C4 2009 June Q6

6 Given that $\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3$, show that $\cot ^ { 2 } \theta - \cot \theta - 2 = 0$.
Hence solve the equation $\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3$ for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$. Section B (36 marks)

OCR MEI C4 2009 June Q7

7 When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point $\mathrm { A } ( 1,2,2 )$, and enters a glass object at point $\mathrm { B } ( 0,0,2 )$. The surface of the glass object is a plane with normal vector $\mathbf { n }$. Fig. 7 shows a cross-section of the glass object in the plane of the light ray and $\mathbf { n }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b4861178-720d-4803-a608-abef350efb0e-3_684_812_516_664} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure}

Find the vector $\overrightarrow { \mathrm { AB } }$ and a vector equation of the line AB . The surface of the glass object is a plane with equation $x + z = 2$. AB makes an acute angle $\theta$ with the normal to this plane.
Write down the normal vector $\mathbf { n }$, and hence calculate $\theta$, giving your answer in degrees. The line BC has vector equation $\mathbf { r } = \left( \begin{array} { l } 0
0
2 \end{array} \right) + \mu \left( \begin{array} { l } - 2
- 2
- 1 \end{array} \right)$. This line makes an acute angle $\phi$ with the normal to the plane.
Show that $\phi = 45 ^ { \circ }$.
Snell's Law states that $\sin \theta = k \sin \phi$, where $k$ is a constant called the refractive index. Find $k$. The light ray leaves the glass object through a plane with equation $x + z = - 1$. Units are centimetres.
Find the point of intersection of the line BC with the plane $x + z = - 1$. Hence find the distance the light ray travels through the glass object. \section*{[Question 8 is printed overleaf.]} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1PB. OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

OCR MEI C4 2009 June Q8

8 Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for $\pi$.

Fig. 8.1 shows a regular 12 -sided polygon inscribed in a circle of radius 1 unit, centre $\mathrm { O } . \mathrm { AB }$ is one of the sides of the polygon. $C$ is the midpoint of $A B$. Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4861178-720d-4803-a608-abef350efb0e-4_455_428_523_900} \captionsetup{labelformat=empty} \caption{Fig. 8.1}
\end{figure} (A) Show that $\mathrm { AB } = 2 \sin 15 ^ { \circ }$.
(B) Use a double angle formula to express $\cos 30 ^ { \circ }$ in terms of $\sin 15 ^ { \circ }$. Using the exact value of $\cos 30 ^ { \circ }$, show that $\sin 15 ^ { \circ } = \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$.
(C) Use this result to find an exact expression for the perimeter of the polygon. Hence show that $\pi > 6 \sqrt { 2 - \sqrt { 3 } }$.
In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4861178-720d-4803-a608-abef350efb0e-4_456_428_1621_900} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
\end{figure} (A) Show that $\mathrm { DE } = 2 \tan 15 ^ { \circ }$.
(B) Let $t = \tan 15 ^ { \circ }$. Use a double angle formula to express $\tan 30 ^ { \circ }$ in terms of $t$. Hence show that $t ^ { 2 } + 2 \sqrt { 3 } t - 1 = 0$.
(C) Solve this equation, and hence show that $\pi < 12 ( 2 - \sqrt { 3 } )$.
Use the results in parts (i)( $C$ ) and (ii)( $C$ ) to establish upper and lower bounds for the value of $\pi$, giving your answers in decimal form. \section*{ADVANCED GCE
MATHEMATICS (MEI)} 4754B
Applications of Advanced Mathematics (C4) Paper B: Comprehension Candidates answer on the question paper
Monday 1 June 2009
OCR Supplied Materials:
Morning
- Insert (inserted)
- MEI Examination Formulae and Tables (MF2)
Duration: Up to 1 hour
Other Materials Required:
- Rough paper
  \includegraphics[max width=\textwidth, alt={}, center]{b4861178-720d-4803-a608-abef350efb0e-5_122_442_1023_1370}
1 On lines 90 and 91, the article says "The average score for each player works out to be 0.25 points per round". Derive this figure. 2 Line 47 gives the inequality $b > c > d > w$.
Interpret each of the following inequalities in the context of the example from the 1st World War.
$b > w$
$c > d$
$\_\_\_\_$
$\_\_\_\_$
3 Table 3 illustrates a possible game where you always co-operate. In lines 98 and 99 the article says "Clearly the longer the game goes on the closer your average score approaches - 2 points per round and that of your opponent approaches 3 ." How many rounds have you played when your average score is - 1.999 ?
4 A Prisoner's Dilemma game is proposed in which $$b = 6 , c = 1 , d = - 1 \text { and } w = - 3 .$$ Using the information in the article, state whether these values would allow long-term co-operation to evolve. Justify your answer.
5 In a Prisoner's Dilemma game both players keep strictly to a Tit-for-tat strategy. You start with C and your opponent starts with D . The scoring system of $b = 3 , c = 1 , d = - 1$ and $w = - 2$ is used.
This table shows the first 8 out of many rounds. Complete the table.
Round You Opponent Your score Opponent's score
1 C D
2
3
4
5
6
7
8
⋯ ⋯ ⋯ ⋯ ⋯
Find your average score per round in the long run.
6 In the article, the scoring system is $b = 3 , c = 1 , d = - 1$ and $w = - 2$. In Axelrod's experiment, negative numbers were avoided by taking $b = 5 , c = 3 , d = 1$ and $w = 0$. State the effect this change would have on
the players' scores,
who wins.
$\_\_\_\_$
$\_\_\_\_$
7 Two companies, X and Y , are the only sellers of ice cream on an island. They both have a market share of about $50 \%$. Although their ice cream is much the same, both companies spend a lot of money on advertising.
What agreement might the companies reach if they decide to co-operate?
What advantage would a company hope to gain by 'defecting' from this agreement?
RECOGNISING ACHIEVEMENT

OCR MEI C4 2011 June Q1

1 Express $\frac { 1 } { ( 2 x + 1 ) \left( x ^ { 2 } + 1 \right) }$ in partial fractions.

OCR MEI C4 2011 June Q2

2 Find the first three terms in the binomial expansion of $\sqrt [ 3 ] { 1 + 3 x }$ in ascending powers of $x$. State the set of values of $x$ for which the expansion is valid.

OCR MEI C4 2011 June Q3

3 Express $2 \sin \theta - 3 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R$ and $\alpha$ are constants to be determined, and $0 < \alpha < \frac { 1 } { 2 } \pi$. Hence write down the greatest and least possible values of $1 + 2 \sin \theta - 3 \cos \theta$.

OCR MEI C4 2011 June Q4

4 A curve has parametric equations $$x = 2 \sin \theta , \quad y = \cos 2 \theta$$

Find the exact coordinates and the gradient of the curve at the point with parameter $\theta = \frac { 1 } { 3 } \pi$.
Find $y$ in terms of $x$.

OCR MEI C4 2011 June Q6

6 Fig. 6 shows the region enclosed by part of the curve $y = 2 x ^ { 2 }$, the straight line $x + y = 3$, and the $y$-axis. The curve and the straight line meet at $\mathrm { P } ( 1,2 )$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0a6247c9-ba64-4a8f-9e10-83986136cf56-2_643_933_1667_607} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure} The shaded region is rotated through $360 ^ { \circ }$ about the $y$-axis. Find, in terms of $\pi$, the volume of the solid of revolution formed.
[0pt] [You may use the formula $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$ for the volume of a cone.] Section B (36 marks)

OCR MEI C4 2011 June Q7

7 A piece of cloth ABDC is attached to the tops of vertical poles $\mathrm { AE } , \mathrm { BF } , \mathrm { DG }$ and CH , where $\mathrm { E } , \mathrm { F } , \mathrm { G }$ and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is $k$ metres. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0a6247c9-ba64-4a8f-9e10-83986136cf56-3_933_1436_518_351} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure}

Write down the vectors $\overrightarrow { \mathrm { AB } }$ and $\overrightarrow { \mathrm { AC } }$. Hence calculate the angle BAC .
Verify that the equation of the plane ABC is $x + y - 2 z + d = 0$, where $d$ is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane.
Given that $\mathrm { A } , \mathrm { B } , \mathrm { D }$ and C are coplanar, show that $k = 3$. Hence show that ABDC is a trapezium, and find the ratio of CD to AB .

OCR MEI C4 2011 June Q8

8 Water is leaking from a container. After $t$ seconds, the depth of water in the container is $x \mathrm {~cm}$, and the volume of water is $V \mathrm {~cm} ^ { 3 }$, where $V = \frac { 1 } { 3 } x ^ { 3 }$. The rate at which water is lost is proportional to $x$, so that $\frac { \mathrm { d } V } { \mathrm {~d} t } = - k x$, where $k$ is a constant.

Show that $x \frac { \mathrm {~d} x } { \mathrm {~d} t } = - k$. Initially, the depth of water in the container is 10 cm .
Show by integration that $x = \sqrt { 100 - 2 k t }$.
Given that the container empties after 50 seconds, find $k$. Once the container is empty, water is poured into it at a constant rate of $1 \mathrm {~cm} ^ { 3 }$ per second. The container continues to lose water as before.
Show that, $t$ seconds after starting to pour the water in, $\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 - x } { x ^ { 2 } }$.
Show that $\frac { 1 } { 1 - x } - x - 1 = \frac { x ^ { 2 } } { 1 - x }$. Hence solve the differential equation in part (iv) to show that $$t = \ln \left( \frac { 1 } { 1 - x } \right) - \frac { 1 } { 2 } x ^ { 2 } - x .$$
Show that the depth cannot reach 1 cm .

OCR MEI C4 2012 June Q1

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

OCR MEI C4 2012 June Q2

2 Find the first four terms in the binomial expansion of $\sqrt { 1 + 2 x }$. State the set of values of $x$ for which the expansion is valid.

OCR MEI C4 2012 June Q3

3 The total value of the sales made by a new company in the first $t$ years of its existence is denoted by $\pounds V$. A model is proposed in which the rate of increase of $V$ is proportional to the square root of $V$. The constant of proportionality is $k$.

Express the model as a differential equation. Verify by differentiation that $V = \left( \frac { 1 } { 2 } k t + c \right) ^ { 2 }$, where $c$ is an arbitrary constant, satisfies this differential equation.
The value of the company’s sales in its first year is $\pounds 10000$, and the total value of the sales in the first two years is $\pounds 40000$. Find $V$ in terms of $t$.

OCR MEI C4 2012 June Q5

5 Given the equation $\sin \left( x + 45 ^ { \circ } \right) = 2 \cos x$, show that $\sin x + \cos x = 2 \sqrt { 2 } \cos x$.
Hence solve, correct to 2 decimal places, the equation for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.

OCR MEI C4 2012 June Q6

6 Solve the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x ( x + 1 ) }$, given that when $x = 1 , y = 1$. Your answer should express $y$ explicitly in terms of $x$.

OCR MEI C4 2012 June Q7

7 Fig. 7a shows the curve with the parametric equations $$x = 2 \cos \theta , \quad y = \sin 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 } .$$ The curve meets the $x$-axis at O and P . Q and R are turning points on the curve. The scales on the axes are the same. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-3_513_661_632_685} \captionsetup{labelformat=empty} \caption{Fig. 7a}

\end{figure}

State, with their coordinates, the points on the curve for which $\theta = - \frac { \pi } { 2 } , \theta = 0$ and $\theta = \frac { \pi } { 2 }$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$. Hence find the gradient of the curve when $\theta = \frac { \pi } { 2 }$, and verify that the two tangents to the curve at the origin meet at right angles.
Find the exact coordinates of the turning point Q . When the curve is rotated about the $x$-axis, it forms a paperweight shape, as shown in Fig. 7b. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-3_321_385_1758_831} \captionsetup{labelformat=empty} \caption{Fig. 7b}
\end{figure}
Express $\sin ^ { 2 } \theta$ in terms of $x$. Hence show that the cartesian equation of the curve is $y ^ { 2 } = x ^ { 2 } \left( 1 - \frac { 1 } { 4 } x ^ { 2 } \right)$.
Find the volume of the paperweight shape.

OCR MEI C4 2012 June Q8

8 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation $x + 2 y - 3 z = 0$, as shown in Fig. 8. $\mathrm { A } ^ { \prime }$ is the point (2, 4, 1), and M is the midpoint of $\mathrm { AA } ^ { \prime }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9001b0d0-8d06-43f4-8831-23c0d6aef59d-4_563_716_413_635} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Show that $\mathrm { AA } ^ { \prime }$ is perpendicular to the plane $x + 2 y - 3 z = 0$, and that M lies in the plane. The vector equation of the line AB is $\mathbf { r } = \left( \begin{array} { l } 1
2
4 \end{array} \right) + \lambda \left( \begin{array} { r } 1
- 1
2 \end{array} \right)$.
Find the coordinates of B , and a vector equation of the line $\mathrm { A } ^ { \prime } \mathrm { B }$.
Given that $\mathrm { A } ^ { \prime } \mathrm { BC }$ is a straight line, find the angle $\theta$.
Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the $x$ - and $z$-axes).

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