C4app_jan13

This question paper consists of 4 printed pages.

OXFORD CAMBRIDGE AND RSA EXAMINATIONS

Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education

MEI STRUCTURED MATHEMATICS 4754(A)Applications of Advanced Mathematics (C4) Paper A

Monday 23 JANUARY 2006 Afternoon 1 hour 30 minutes

Additional materials:8 page answer bookletGraph paperMEI Examination Formulae and Tables (MF2)

TIME 1 hour 30 minutes

INSTRUCTIONS TO CANDIDATES

• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.

• Answer all the questions.

• 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.

INFORMATION FOR CANDIDATES

• The number of marks is given in brackets [ ] at the end of each question or part question.

• You are advised that an answer may receive no marks unless you show sufficient detail of theworking to indicate that a correct method is being used.

• The total number of marks for this paper is 72.

NOTE

• This paper will be followed by Paper B: Comprehension.

HN/3© OCR 2006 [T/102/2653] Registered Charity 1066969 [Turn over

2

Section A (36 marks)

1 Solve the equation [5]

2 A curve is defined parametrically by the equations

Find the gradient of the curve at the point where [5]

3 A triangle ABC has vertices , and . By calculating a suitablescalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle. [6]

4 Solve the equation for [6]

5 (i) Find the cartesian equation of the plane through the point with normal vector

[3]

(ii) Find the coordinates of the point of intersection of this plane and the straight line withequation

[4]

6 (i) Find the first three non-zero terms of the binomial expansion of for [4]

(ii) Use this result to find an approximation for , rounding your answer to 4 significant figures. [2]

(iii) Given that evaluate rounding your answer to

4 significant figures. [1]

1

4 20

1

-ÛıÙ x

xd ,1

4 212-

=ÛıÙ

( ) +x

x x cd arcsin ,

1

4 20

1

-ÛıÙ x

xd

� x � � 2.1

4 2- x

r =Ê

ËÁÁ

ˆ

¯˜˜

+Ê

ËÁÁ

ˆ

¯˜˜

7129

132

l .

n = -Ê

ËÁÁ

ˆ

¯˜˜

112

.

(2, �1, 4)

0° � q � 360°.2 sin 2q � cos 2q � 1,

C(4, 8, 3)B(2, 3, 4)A(�2, 4, 1)

t � 2.

x � t � ln t, y � t � ln t (t � 0) .

2x

x � 2�

4x

x � 1� 3.

4754(A) January 2006


3

Section B (36 marks)

7 In a game of rugby, a kick is to be taken from a point P (see Fig. 7). P is a perpendicular distancey metres from the line TOA. Other distances and angles are as shown.

Fig. 7

(i) Show that and hence that

Calculate the angle when [8]

(ii) By differentiating implicitly, show that [5]

(iii) Use this result to find the value of y that maximises the angle . Calculate this maximumvalue of [You need not verify that this value is indeed a maximum.] [4]

[Question 8 is printed overleaf.]

q .q

dqdy

�6(160 � y 2)(160 � y 2)2 cos2 q .

y � 6.q

tan q �6y

160 � y 2 .q � b � a ,

ba

q

A O T

P

6m 10m

ym

4

8 Some years ago an island was populated by red squirrels and there were no grey squirrels. Thengrey squirrels were introduced.

The population x, in thousands, of red squirrels is modelled by the equation

where t is the time in years, and a and k are constants. When ,

(i) Show that [3]

(ii) Given that the initial population of 2.5 thousand red squirrels reduces to 1.6 thousand after oneyear, calculate a and k. [3]

(iii) What is the long-term population of red squirrels predicted by this model? [1]

The population y, in thousands, of grey squirrels is modelled by the differential equation

When ,

(iv) Express in partial fractions. [4]

(v) Hence show by integration that

Show that [7]

(vi) What is the long-term population of grey squirrels predicted by this model? [1]

y �2

1 � e�2t .

ln .y

yt

22

-ÊËÁ

ˆ¯̃

=

12y � y 2

y � 1.t � 0

dy

dt� 2y � y 2.

dx

dt� �

kx 2

a.

x � 2.5.t � 0

x �a

1 � kt,


This question paper consists of 5 printed pages and 3 blank pages.

OXFORD CAMBRIDGE AND RSA EXAMINATIONS

Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education

MEI STRUCTURED MATHEMATICS 4754(A)Applications of Advanced Mathematics (C4) Paper A

Monday 12 JUNE 2006 Afternoon 1 hour 30 minutes

Additional materials:8 page answer bookletGraph paperMEI Examination Formulae and Tables (MF2)

TIME 1 hour 30 minutes


• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.








NOTE


HN/3© OCR 2006 [T/102/2653] Registered Charity 1066969 [Turn over

2


1 Fig. 1 shows part of the graph of

Fig. 1

Express in the form where and

Hence write down the exact coordinates of the turning point P. [6]

2 (i) Given that

where A, B and C are constants, find B and C, and show that [4]

(ii) Given that x is sufficiently small, find the first three terms of the binomial expansions ofand

Hence find the first three terms of the expansion of [4]

3 Given that show that

Hence solve the equation for [7]0° � q � 360°.sin (q � 40°) � 2sin q ,

tan q �sin a

2�cos a.sin (q � a) � 2sin q ,

3 � 2x 2

(1 � x) 2(1 � 4x) .

(1 � 4x) �1.(1 � x) �2

A � 0.

3 � 2x 2

(1 � x)2(1 � 4x) �A

1 � x�

B(1 � x)2 �

C

1 � 4x,

0 � a � 12 p .R � 0R sin (x � a),sin cosx x- 3

y

x

P

y x x= -sin cos .3

4754(A) June 2006

4754(A) June 2006 [Turn over

3

4 (a) The number of bacteria in a colony is increasing at a rate that is proportional to the square rootof the number of bacteria present. Form a differential equation relating x, the number ofbacteria, to the time t. [2]

(b) In another colony, the number of bacteria, y, after time t minutes is modelled by the differentialequation

Find y in terms of t, given that when Hence find the number of bacteria after10 minutes. [6]

5 (i) Show that [3]

A vase is made in the shape of the volume of revolution of the curve about the x-axisbetween and (see Fig. 5).

Fig. 5

(ii) Show that this volume of revolution is [4]14 4

15

p -ÊË

ˆ¯e

.

2O

y

x

x � 2x � 0y = x 1

2 e- x

�xe�2x dx � � 14 e�2x (1 � 2x) � c.

t � 0.y � 900

ddy

t y=

10000.

4


6 Fig. 6 shows the arch ABCD of a bridge.

Fig. 6

The section from B to C is part of the curve OBCE with parametric equations

for

where a is a constant.

(i) Find, in terms of a,

(A) the length of the straight line OE,

(B) the maximum height of the arch. [4]

(ii) Find in terms of [3]

The straight line sections AB and CD are inclined at 30° to the horizontal, and are tangents to thecurve at B and C respectively. BC is parallel to the x-axis. BF is parallel to the y-axis.

(iii) Show that at the point B the parameter satisfies the equation

Verify that is a solution of this equation.

Hence show that and find OF in terms of a, giving your answer exactly. [6]

(iv) Find BC and AF in terms of a.

Given that the straight line distance AD is 20 metres, calculate the value of a. [5]

BF � 32 a,

q � 23 p

sin ( cos ).q q= -1

31

q

q .dy

dx

0 � q � 2p ,x � a(q � sin q), y � a(1 � cos q)

30∞30∞

y

xOA F E D

B C

4754(A) June 2006

7

Fig. 7

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.

(i) Find the length AE. [2]

(ii) Find a vector equation of the line BD. Given that the length of BD is 15 metres, find thecoordinates of D. [4]

(iii) Verify that the equation of the plane ABC is

Write down a vector normal to this plane. [4]

(iv) Show that the vector is normal to the plane ABDE. Hence find the equation of the

plane ABDE. [4]

(v) Find the angle between the planes ABC and ABDE. [4]

435

Ê

ËÁÁ

ˆ

¯˜˜

�3x � 4y � 5z � 30.

D

B(–1, –7, 11)

C(–8, –6, 6)

O

y

x

z

A(0, 0, 6)

E(15, –20, 6)

5

4754(A) June 2006


• Write your name, centre number and candidate number in the spaces provided on the answer booklet.







ADVICE TO CANDIDATES

• Read each question carefully and make sure you know what you have to do before starting youranswer.


NOTE


This document consists of 6 printed pages and 2 blank pages.

HN/7 © OCR 2007 [T/102/2653] OCR is an exempt Charity [Turn over

ADVANCED GCE UNIT 4754(A)/01MATHEMATICS (MEI)Applications of Advanced Mathematics (C4)

Paper A

TUESDAY 23 JANUARY 2007 AfternoonTime: 1 hour 30 minutes

Additional materials:Answer booklet (8 pages)Graph paperMEI Examination Formulae and Tables (MF2)


1 Solve the equation [4]

2 Fig. 2 shows part of the curve

Fig. 2

(i) Use the trapezium rule with 4 strips to estimate giving your answer correct to

3 significant figures. [3]

(ii) Chris and Dave each estimate the value of this integral using the trapezium rule with 8 strips.Chris gets a result of 3.25, and Dave gets 3.30. One of these results is correct. Withoutperforming the calculation, state with a reason which is correct. [2]

1 3

0

2

+ÛıÙ

x xd ,

2 x

y

0

y x= +1 3 .

1x

�x

x � 2� 1.

2

© OCR 2007 4754A/01 Jan 07

3 (i) Use the formula for with and to show that

[4]

(ii) In triangle ABC, angle BAC = 45°, angle ACB = 30° and AB = 1 unit (see Fig. 3).

Fig. 3

Using the sine rule, together with the result in part (i), show that [3]

4 Show that

Hence, or otherwise, solve the equation for [7]

5 Find the first four terms in the binomial expansion of

State the range of values of x for which the expansion is valid. [5]

6 (i) Express in partial fractions. [3]

(ii) A curve passes through the point and satisfies the differential equation

Show by integration that [5]y �4x � 2x � 1

.

dy

dx�

y(2x � 1) (x � 1) .

(0, 2)

1(2x � 1) (x � 1)

1 313+( )x .

0° � q � 180°.1 � tan2q1�tan2q

� 2,

1 � tan2q1�tan2q

� sec 2q .

AC =+3 1

2.

A B

C

30∞

45∞

1

sin .1053 1

2 2∞ =

+f � 60°,q � 45°sin (q � f),

3

© OCR 2007 4754A/01 Jan 07 [Turn over


7 Fig. 7 shows the curve with parametric equations

The curve crosses the x-axis at points and , and the positive y-axis at C. D is themaximum point of the curve, and E is the minimum point.

The solid of revolution formed when this curve is rotated through 360° about the x-axis is used tomodel the shape of an egg.

Fig. 7

(i) Show that, at the point A, Write down the value of q at the point B, and find thecoordinates of C. [4]

(ii) Find in terms of q.

Hence show that, at the point D,

[5]

(iii) Solve this equation, and hence find the y-coordinate of D, giving your answer correct to 2decimal places. [5]

The cartesian equation of the curve (for ) is

(iv) Show that the volume of the solid of revolution of this curve about the x-axis is given by

Evaluate this integral. [6]

116

2 3 4

1

1

16 8 15 8p - - + -( )ÛıÙ-

x x x x xd .

y x x= -( ) -14

24 1 .

0 � q � p

2 cos2 q � 4 cos q �1 � 0.

dy

dx

q � 0.

D

E

A(1, 0) xB(–1, 0)

y

C

B(�1, 0)A(1, 0)

x � cos q , y � sin q � 18 sin 2q , 0 � q � 2p .

4

© OCR 2007 4754A/01 Jan 07

8 A pipeline is to be drilled under a river (see Fig. 8). With respect to axes Oxyz, with the x-axispointing East, the y-axis North and the z-axis vertical, the pipeline is to consist of a straight sectionAB from the point to the point directly under the river, and anotherstraight section BC. All lengths are in metres.

Fig. 8

(i) Calculate the distance AB. [2]

The section BC is to be drilled in the direction of the vector .

(ii) Find the angle ABC between the sections AB and BC. [4]

The section BC reaches ground level at the point

(iii) Write down a vector equation of the line BC. Hence find a and b. [5]

(iv) Show that the vector is perpendicular to the plane ABC. Hence find the cartesianequation of this plane. [5]

6i � 5j � 2k

C(a, b, 0).

3i � 4j � k

(E)x

z

y (N)

O

A(0, –40, 0)

B(40, 0, –20)

C(a, b, 0)

B(40, 0, �20)A(0, �40, 0)

5

© OCR 2007 4754A/01 Jan 07


• Write your name, centre number and candidate number in the spaces provided on the answer booklet.







ADVICE TO CANDIDATES

• Read each question carefully and make sure you know what you have to do before starting youranswer.


NOTE


This document consists of 6 printed pages and 2 blank pages.

HN/5 © OCR 2007 [T/102/2653] OCR is an exempt Charity [Turn over

ADVANCED GCE UNIT 4754(A)/01MATHEMATICS (MEI)Applications of Advanced Mathematics (C4)

Paper A

THURSDAY 14 JUNE 2007 AfternoonTime: 1 hour 30 minutes

Additional materials:Answer booklet (8 pages)Graph paperMEI Examination Formulae and Tables (MF2)


1 Express in the form where R and a are constants to be determined,and

Hence solve the equation for [7]

2 Write down normal vectors to the planes and

Hence show that these planes are perpendicular to each other. [4]

3 Fig. 3 shows the curve and part of the line

Fig. 3

The shaded region is rotated through 360° about the y-axis.

(i) Show that the volume of the solid of revolution formed is given by [3]

(ii) Evaluate this, leaving your answer in an exact form. [3]

4 A curve is defined by parametric equations

Show that the cartesian equation of the curve is [4]

5 Verify that the point lies on both the lines

Find the acute angle between the lines. [7]

r r=-

Ê

ËÁÁ

ˆ

¯˜˜

+-Ê

ËÁÁ

ˆ

¯˜˜

=Ê

ËÁÁ

ˆ

¯˜˜

+-

Ê

ËÁÁ

ˆ

¯˜˜

121

123

063

102

l mand .

(�1, 6, 5)

y �3 � 2x

2 � x.

x �1t � 1, y �

2 � t

1 � t.

p e d2

0

2y yÛ

ıÙ.

O

2

y

x

y � 2.y � ln x

x � 2y � z � 5.2x � 3y � 4z � 10

0° � q � 360°.sin q � 3 cos q � 1

0° � a � 90°.R sin (q � a),sin q � 3 cos q

2

© OCR 2007 4754A/01 June 07

6 Two students are trying to evaluate the integral

Sarah uses the trapezium rule with 2 strips, and starts by constructing the following table.

(i) Complete the calculation, giving your answer to 3 significant figures. [2]

Anish uses a binomial approximation for and then integrates this.

(ii) Show that, provided is suitably small, [3]

(iii) Use this result to evaluate approximately, giving your answer to 3 significant

figures. [3]

11

2

+ÛıÙ

-e dx x

1 112 1

218

2+( ) ª + -- - -e e ex x x .e�x

1 + -e x

x 1 1.5 2

1.1696 1.1060 1.06551 + e- x

11

2

+ÛıÙ

-e dx x.

3

© OCR 2007 4754A/01 June 07 [Turn over


7 Data suggest that the number of cases of infection from a particular disease tends to oscillatebetween two values over a period of approximately 6 months.

(a) Suppose that the number of cases, P thousand, after time t months is modelled by the equation

Thus, when

(i) By considering the greatest and least values of , write down the greatest and leastvalues of P predicted by this model. [2]

(ii) Verify that P satisfies the differential equation [5]

(b) An alternative model is proposed, with differential equation

As before,

(i) Express in partial fractions. [4]

(ii) Solve the differential equation (*) to show that

[5]

This equation can be rearranged to give

(iii) Find the greatest and least values of P predicted by this model. [4]

Pt

=-

1

212e sin

.

ln sin .2 1 1

2PP

t-Ê

Ëˆ¯ =

1P(2P�1)

P � 1 when t � 0.

dP

dt� 1

2 (2P 2 � P) cos t. (*)

dP

dt� 1

2 P 2 cos t.

sin t

t � 0, P � 1.P �2

2 � sin t.

4

© OCR 2007 4754A/01 June 07

8

Fig. 8

In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axesshown, the path of C is modelled by the parametric equations

where x and y are in metres.

(i) Show that

Verify that when Hence find the exact coordinates of the highest point A on

the path of C. [6]

(ii) Express in terms of Hence show that

[4]

(iii) Using this result, or otherwise, find the greatest and least distances of C from O. [2]

You are given that, at the point B on the path vertically above O,

(iv) Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. [4]

2 cos 2q � 2 cos q � 1 � 0.

x 2 � y 2 � 125 � 100 cos q.

q.x 2 � y 2

q � 13 p.

dy

dx� 0

dy

dx� �

cos q � cos 2qsin q � sin 2q

.

x � 10 cos q � 5 cos 2q, y � 10 sin q � 5 sin 2q , (0 � q � 2p),

x

y

C(x, y)

O

B A

5

© OCR 2007 4754A/01 June 07

ADVANCED GCE 4754/01AMATHEMATICS (MEI)

Applications of Advanced Mathematics (C4) Paper A

TUESDAY 22 JANUARY 2008 Afternoon

Time: 1 hour 30 minutesAdditional materials: Answer Booklet (8 pages)

Graph paperMEI Examination Formulae and Tables (MF2)


• Write your name in capital letters, your Centre Number and Candidate Number in the spacesprovided on the Answer Booklet.

• Read each question carefully and make sure you know what you have to do before startingyour answer.








NOTE


This document consists of 4 printed pages.

© OCR 2008 [T/102/2653] OCR is an exempt Charity [Turn over

2


1 Express 3 cos θ + 4 sin θ in the form R cos(θ − α), where R > 0 and 0 < α < 12π.

Hence solve the equation 3 cos θ + 4 sin θ = 2 for −π ≤ θ ≤ π. [7]

2 (i) Find the first three terms in the binomial expansion of1√

1 − 2x. State the set of values of x for

which the expansion is valid. [5]

(ii) Hence find the first three terms in the series expansion of1 + 2x√1 − 2x

. [3]

3 Fig. 3 shows part of the curve y = 1 + x2, together with the line y = 2.

Ox

y

1

2

Fig. 3

The region enclosed by the curve, the y-axis and the line y = 2 is rotated through 360◦ about the y-axis.Find the volume of the solid generated, giving your answer in terms of π. [5]

4 The angle θ satisfies the equation sin(θ + 45◦) = cos θ .

(i) Using the exact values of sin 45◦ and cos 45◦, show that tan θ = √2 − 1. [5]

(ii) Find the values of θ for 0◦ < θ < 360◦. [2]

5 Express4

x(x2 + 4) in partial fractions. [6]

6 Solve the equation cosec θ = 3, for 0◦ < θ < 360◦. [3]

© OCR 2008 4754/01A Jan08

3


7 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). Alldimensions are in centimetres.

O

x

y

z

B (15, 20, 0)C (15, 0, 0)

D (9, 6, 24)

G (3, 6, 24) F (3, 14, 24)

E (9, 14, 24)

A (0, 20, 0)

Fig. 7

(i) Write down the vectors−−→CD and

−−→CB. [2]

(ii) Find the length of the edge CD. [2]

(iii) Show that the vector 4i + k is perpendicular to the vectors−−→CD and

−−→CB. Hence find the cartesian

equation of the plane BCDE. [5]

(iv) Write down vector equations for the lines OG and AF.

Show that they meet at the point P with coordinates (5, 10, 40). [5]

You may assume that the lines CD and BE also meet at the point P.

The volume of a pyramid is 13× area of base × height.

(v) Find the volumes of the pyramids POABC and PDEFG.

Hence find the volume of the ornament. [4]

© OCR 2008 4754/01A Jan08 [Turn over

4

8 A curve has equation

x2 + 4y2 = k2,

where k is a positive constant.

(i) Verify that

x = k cos θ , y = 12k sin θ ,

are parametric equations for the curve. [3]

(ii) Hence or otherwise show thatdydx

= − x4y

. [3]

(iii) Fig. 8 illustrates the curve for a particular value of k. Write down this value of k. [1]

x

y

1

2

Fig. 8

O

(iv) Copy Fig. 8 and on the same axes sketch the curves for k = 1, k = 3 and k = 4. [3]

On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Itspath on the map is always at right angles to the contour it is crossing.

(v) Explain why the path of the stream is modelled by the differential equation

dydx

= 4yx

. [2]

(vi) Solve this differential equation.

Given that the path of the stream passes through the point (2, 1), show that its equation is y = x4

16.

[6]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.

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 2008 4754/01A Jan08

ADVANCED GCE 4754/01AMATHEMATICS (MEI)

Applications of Advanced Mathematics (C4) Paper A

WEDNESDAY 21 MAY 2008 Afternoon

Time: 1 hour 30 minutesAdditional materials: Answer Booklet (8 pages)

Graph paperMEI Examination Formulae and Tables (MF2)


• Write your name in capital letters, your Centre Number and Candidate Number in the spacesprovided on the Answer Booklet.

• Read each question carefully and make sure you know what you have to do before startingyour answer.








NOTE


This document consists of 4 printed pages.

© OCR 2008 [T/102/2653] OCR is an exempt Charity [Turn over

2


1 Expressx

x2 − 4+ 2

x + 2as a single fraction, simplifying your answer. [3]

2 Fig. 2 shows the curve y = √1 + e2x.

Ox

y

1

Fig. 2

The region bounded by the curve, the x-axis, the y-axis and the line x = 1 is rotated through 360◦ aboutthe x-axis.

Show that the volume of the solid of revolution produced is 12π(1 + e2). [4]

3 Solve the equation cos 2θ = sin θ for 0 ≤ θ ≤ 2π, giving your answers in terms of π. [7]

4 Given that x = 2 sec θ and y = 3 tan θ , show thatx2

4− y2

9= 1. [3]

5 A curve has parametric equations x = 1 + u2, y = 2u3.

(i) Finddydx

in terms of u. [3]

(ii) Hence find the gradient of the curve at the point with coordinates (5, 16). [2]

6 (i) Find the first three non-zero terms of the binomial series expansion of1

√1 + 4x2

, and state the set

of values of x for which the expansion is valid. [5]

(ii) Hence find the first three non-zero terms of the series expansion of1 − x2

√1 + 4x2

. [3]

7 Express√

3 sin x − cos x in the form R sin(x − α), where R > 0 and 0 < α < 12π. Express α in the form

kπ.

Find the exact coordinates of the maximum point of the curve y = √3 sin x − cos x for which 0 < x < 2π.

[6]

© OCR 2008 4754/01A Jun08

3


8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8.All dimensions are metres.

O

x

yz

C

F

B

E

D

A

upper surfaceof seam

lower surfaceof seam

Fig. 8

Relative to axes Ox (due east), Oy (due north) and O� (vertically upwards), the coordinates of thepoints are as follows.

A: (0, 0, −15) B: (100, 0, −30) C: (0, 100, −25)D: (0, 0, −40) E: (100, 0, −50) F: (0, 100, −35)

(i) Verify that the cartesian equation of the plane ABC is 3x + 2y + 20� + 300 = 0. [3]

(ii) Find the vectors−−→DE and

−−→DF. Show that the vector 2i − j + 20k is perpendicular to each of these

vectors. Hence find the cartesian equation of the plane DEF. [6]

(iii) By calculating the angle between their normal vectors, find the angle between the planes ABCand DEF. [4]

It is decided to drill down to the seam from a point R (15, 34, 0) in a line perpendicular to the uppersurface of the seam. This line meets the plane ABC at the point S.

(iv) Write down a vector equation of the line RS.

Calculate the coordinates of S. [5]

© OCR 2008 4754/01A Jun08 [Turn over

4

9 A skydiver drops from a helicopter. Before she opens her parachute, her speed v m s−1 after timet seconds is modelled by the differential equation

dvdt

= 10e−1

2t.

When t = 0, v = 0.

(i) Find v in terms of t. [4]

(ii) According to this model, what is the speed of the skydiver in the long term? [2]

She opens her parachute when her speed is 10 m s−1. Her speed t seconds after this is w m s−1, and ismodelled by the differential equation

dwdt

= −12(w − 4)(w + 5).

(iii) Express1(w − 4)(w + 5) in partial fractions. [4]

(iv) Using this result, show thatw − 4w + 5

= 0.4e−4.5t. [6]

(v) According to this model, what is the speed of the skydiver in the long term? [2]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.

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 2008 4754/01A Jun08

ADVANCED GCE

MATHEMATICS (MEI) 4754AApplications of Advanced Mathematics (C4) Paper A

Candidates answer on the Answer Booklet

OCR Supplied Materials:• 8 page Answer Booklet• Graph paper

• MEI Examination Formulae and Tables (MF2)

Other Materials Required:None

Tuesday 13 January 2009Morning

Duration: 1 hour 30 minutes

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• Write your name clearly in capital letters, your Centre Number and Candidate Number in the spaces providedon the Answer Booklet.

• Use black ink. Pencil may be used for graphs and diagrams only.

• Read each question carefully and make sure that you know what you have to do before starting your answer.• Answer all the questions.• Do not write in the bar codes.• 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.• 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 72.• This document consists of 4 pages. Any blank pages are indicated.

NOTE


© OCR 2009 [T/102/2653] OCR is an exempt Charity

2R–8H16 Turn over

2


1 Express3x + 2

x(x2 + 1) in partial fractions. [6]

2 Show that (1 + 2x)1

3 = 1 + 2

3x − 4

9x2 + . . . , and find the next term in the expansion.

State the set of values of x for which the expansion is valid. [6]

3 Vectors a and b are given by a = 2i + j − k and b = 4i − 2j + k.

Find constants λ and µ such that λ a + µb = 4j − 3k. [5]

4 Prove that cot β − cot α = sin(α − β)sin α sin β

. [3]

5 (i) Write down normal vectors to the planes 2x − y + ß = 2 and x − ß = 1.

Hence find the acute angle between the planes. [4]

(ii) Write down a vector equation of the line through (2, 0, 1) perpendicular to the plane 2x− y+ ß = 2.

Find the point of intersection of this line with the plane. [4]

6 (i) Express cos θ + √3 sin θ in the form R cos(θ − α), where R > 0 and α is acute, expressing α in

terms of π. [4]

(ii) Write down the derivative of tan θ .

Hence show that ä1

3π

0

1

(cos θ + √3 sin θ)2

dθ =√

3

4. [4]

© OCR 2009 4754A Jan09

3


7 Scientists can estimate the time elapsed since an animal died by measuring its body temperature.

(i) Assuming the temperature goes down at a constant rate of 1.5 degrees Fahrenheit per hour,

estimate how long it will take for the temperature to drop

(A) from 98 ◦F to 89 ◦F,

(B) from 98 ◦F to 80 ◦F. [2]

In practice, rate of temperature loss is not likely to be constant. A better model is provided by Newton’s

law of cooling, which states that the temperature θ in degrees Fahrenheit t hours after death is given

by the differential equation

dθ

dt= −k(θ − θ

0),

where θ0

◦F is the air temperature and k is a constant.

(ii) Show by integration that the solution of this equation is θ = θ0+ Ae−kt, where A is a constant.

[5]

The value of θ0

is 50, and the initial value of θ is 98. The initial rate of temperature loss is 1.5 ◦Fper hour.

(iii) Find A, and show that k = 0.031 25. [4]

(iv) Use this model to calculate how long it will take for the temperature to drop

(A) from 98 ◦F to 89 ◦F,

(B) from 98 ◦F to 80 ◦F. [5]

(v) Comment on the results obtained in parts (i) and (iv). [1]


© OCR 2009 4754A Jan09 Turn over

4

8 Fig. 8 illustrates a hot air balloon on its side. The balloon is modelled by the volume of revolution

about the x-axis of the curve with parametric equations

x = 2 + 2 sin θ , y = 2 cos θ + sin 2θ , (0 ≤ θ ≤ 2π).The curve crosses the x-axis at the point A (4, 0). B and C are maximum and minimum points on the

curve. Units on the axes are metres.

O

y

x

C

B

A (4, 0)

Fig. 8

(i) Finddy

dxin terms of θ . [4]

(ii) Verify thatdy

dx= 0 when θ = 1

6π, and find the exact coordinates of B.

Hence find the maximum width BC of the balloon. [5]

(iii) (A) Show that y = x cos θ .

(B) Find sin θ in terms of x and show that cos2 θ = x − 1

4x2.

(C) Hence show that the cartesian equation of the curve is y2 = x3 − 1

4x4. [7]

(iv) Find the volume of the balloon. [3]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable

effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be

pleased to make amends at the earliest possible opportunity.

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 2009 4754A Jan09

ADVANCED GCE



OCR Supplied Materials:• 8 page Answer Booklet• Graph paper



Monday 1 June 2009Morning


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NOTE



2R–9B09 Turn over

2


1 Express 4 cos θ − sin θ in the form R cos(θ + α), where R > 0 and 0 < α < 1

2π.

Hence solve the equation 4 cos θ − sin θ = 3, for 0 ≤ θ ≤ 2π. [7]

2 Using partial fractions, find ä x(x + 1)(2x + 1) dx. [7]

3 A curve satisfies the differential equationdy

dx= 3x2y, and passes through the point (1, 1). Find y in

terms of x. [4]

4 The part of the curve y = 4 − x2 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 π. [5]

y

x

Fig. 4

5 A curve has parametric equations

x = at3, y = a

1 + t2,

where a is a constant.

Show thatdy

dx= −2

3t(1 + t2)2.

Hence find the gradient of the curve at the point (a, 1

2a). [7]

6 Given that cosec2 θ − cot θ = 3, show that cot2 θ − cot θ − 2 = 0.

Hence solve the equation cosec2 θ − cot θ = 3 for 0◦ ≤ θ ≤ 180◦. [6]

© OCR 2009 4754A Jun09

3


7 When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts

at point A (1, 2, 2), and enters a glass object at point B (0, 0, 2). The surface of the glass object is a

plane with normal vector n. Fig. 7 shows a cross-section of the glass object in the plane of the light

ray and n.

n

q

f A

B

C

Fig. 7

(i) Find the vector−−→AB and a vector equation of the line AB. [2]

The surface of the glass object is a plane with equation x + ß = 2. AB makes an acute angle θ with the

normal to this plane.

(ii) Write down the normal vector n, and hence calculate θ , giving your answer in degrees. [5]

The line BC has vector equation r = ( 0

0

2

) + µ (−2−2−1

). This line makes an acute angle φ with the

normal to the plane.

(iii) Show that φ = 45◦. [3]

(iv) Snell’s Law states that sin θ = k sin φ , where k is a constant called the refractive index. Find k.

[2]

The light ray leaves the glass object through a plane with equation x + ß = −1. Units are centimetres.

(v) Find the point of intersection of the line BC with the plane x + ß = −1. Hence find the distance

the light ray travels through the glass object. [5]


Copyright Information

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 (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 2009 4754A Jun09

4

8 Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain

approximations for π.

(i) Fig. 8.1 shows a regular 12-sided polygon inscribed in a circle of radius 1 unit, centre O. AB

is one of the sides of the polygon. C is the midpoint of AB. Archimedes used the fact that the

circumference of the circle is greater than the perimeter of this polygon.

Fig. 8.1

A C B

O

1 1

(A) Show that AB = 2 sin 15◦. [2]

(B) Use a double angle formula to express cos 30◦ in terms of sin 15◦. Using the exact value of

cos 30◦, show that sin 15◦ = 1

2

√2 − √

3. [4]

(C) Use this result to find an exact expression for the perimeter of the polygon.

Hence show that π > 6√

2 − √3. [2]

(ii) 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.

Fig. 8.2

D F E

O

1

(A) Show that DE = 2 tan 15◦. [2]

(B) Let t = tan 15◦. Use a double angle formula to express tan 30◦ in terms of t.

Hence show that t2 + 2√

3t − 1 = 0. [3]

(C) Solve this equation, and hence show that π < 12(2 − √3). [4]

(iii) Use the results in parts (i)(C) and (ii)(C) to establish upper and lower bounds for the value of π,

giving your answers in decimal form. [2]

© OCR 2009 4754A Jun09

ADVANCED GCE



OCR Supplied Materials:• 8 page Answer Booklet• MEI Examination Formulae and Tables (MF2)


Friday 15 January 2010Afternoon


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77

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• Use black ink. Pencil may be used for graphs and diagrams only.• Read each question carefully and make sure that you know what you have to do before starting your answer.

• Answer all the questions.• Do not write in the bar codes.• 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.



indicate that a correct method is being used.• The total number of marks for this paper is 72.

• This document consists of 4 pages. Any blank pages are indicated.

NOTE



3R–9G08 Turn over

2


1 Find the first three terms in the binomial expansion of1 + 2x

(1 − 2x)2in ascending powers of x. State the

set of values of x for which the expansion is valid. [7]

2 Show that cot 2θ = 1 − tan2 θ

2 tan θ.

Hence solve the equation

cot 2θ = 1 + tan θ for 0◦ < θ < 360◦. [7]


x = e2t, y = 2t

1 + t.

(i) Find the gradient of the curve at the point where t = 0. [6]

(ii) Find y in terms of x. [2]

4 The points A, B and C have coordinates (1, 3, −2), (−1, 2, −3) and (0, −8, 1) respectively.

(i) Find the vectors−−→AB and

−−→AC. [2]

(ii) Show that the vector 2i − j − 3k is perpendicular to the plane ABC. Hence find the equation of

the plane ABC. [5]

5 (i) Verify that the lines r = (−5

3

4

) + λ(3

0−1

) and r = (−1

4

2

) + µ(2−1

0

) meet at the point (1, 3, 2).[3]

(ii) Find the acute angle between the lines. [4]

© OCR 2010 4754A Jan10

3


6 In Fig. 6, OAB is a thin bent rod, with OA = a metres, AB = b metres and angle OAB = 120◦. The

bent rod lies in a vertical plane. OA makes an angle θ above the horizontal. The vertical height BD

of B above O is h metres. The horizontal through A meets BD at C and the vertical through A meets

OD at E.

120°

a

b

O

A

B

C

DE

q

h

Fig. 6

(i) Find angle BAC in terms of θ. Hence show that

h = a sin θ + b sin(θ − 60◦). [3]

(ii) Hence show that h = (a + 12b) sin θ −

√3

2b cos θ. [3]

The rod now rotates about O, so that θ varies. You may assume that the formulae for h in parts (i)

and (ii) remain valid.

(iii) Show that OB is horizontal when tan θ =√

3b

2a + b. [3]

In the case when a = 1 and b = 2, h = 2 sin θ − √3 cos θ.

(iv) Express 2 sin θ − √3 cos θ in the form R sin(θ − α). Hence, for this case, write down the

maximum value of h and the corresponding value of θ. [7]



4

7 Fig. 7 illustrates the growth of a population with time. The proportion of the ultimate (long term)

population is denoted by x, and the time in years by t. When t = 0, x = 0.5, and as t increases,

x approaches 1.

t

x

0.5

1

Fig. 7

One model for this situation is given by the differential equation

dx

dt= x(1 − x).

(i) Verify that x = 1

1 + e−tsatisfies this differential equation, including the initial condition. [6]

(ii) Find how long it will take, according to this model, for the population to reach three-quarters of

its ultimate value. [3]

An alternative model for this situation is given by the differential equation

dx

dt= x2(1 − x),

with x = 0.5 when t = 0 as before.

(iii) Find constants A, B and C such that1

x2(1 − x) = A

x2+ B

x+ C

1 − x. [4]

(iv) Hence show that t = 2 + ln(x

1 − x) − 1

x. [5]

(v) Find how long it will take, according to this model, for the population to reach three-quarters of

its ultimate value. [2]







For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.



© OCR 2010 4754A Jan10

ADVANCED GCE



OCR Supplied Materials:• 8 page Answer Booklet• MEI Examination Formulae and Tables (MF2)

Other Materials Required:• Scientific or graphical calculator

Wednesday 9 June 2010

Afternoon


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**










NOTE



2R–9L24 Turn over

2


1 Expressx

x2 − 1+ 2

x + 1as a single fraction, simplifying your answer. [3]

2 Fig. 2 shows the curve y = √1 + x2.

x

y

0 1

1

Fig. 2

(i) The following table gives some values of x and y.

x 0 0.25 0.5 0.75 1

y 1 1.0308 1.25 1.4142

Find the missing value of y, giving your answer correct to 4 decimal places.

Hence show that, using the trapezium rule with four strips, the shaded area is approximately

1.151 square units. [3]

(ii) Jenny uses a trapezium rule with 8 strips, and obtains a value of 1.158 square units. Explain why

she must have made a mistake. [2]

(iii) The shaded area is rotated through 360◦ about the x-axis. Find the exact volume of the solid of

revolution formed. [3]

3 The parametric equations of a curve are

x = cos 2θ, y = sin θ cos θ for 0 ≤ θ < π.

Show that the cartesian equation of the curve is x2 + 4y2 = 1.

Sketch the curve. [5]

4 Find the first three terms in the binomial expansion of√

4 + x in ascending powers of x.

State the set of values of x for which the expansion is valid. [5]

© OCR 2010 4754A Jun10

3

5 (i) Express3(y − 2)(y + 1) in partial fractions. [3]

(ii) Hence, given that x and y satisfy the differential equation

dy

dx= x2(y − 2)(y + 1),

show thaty − 2

y + 1= Aex3

, where A is a constant. [5]

6 Solve the equation tan(θ + 45◦) = 1 − 2 tan θ, for 0◦ ≤ θ ≤ 90◦. [7]


7 A straight pipeline AB passes through a mountain. With respect to axes Oxyß, with Ox due East,

Oy due North and Oß vertically upwards, A has coordinates (−200, 100, 0) and B has coordinates(100, 200, 100), where units are metres.

(i) Verify that−−→AB = (300

100

100

) and find the length of the pipeline. [3]

(ii) Write down a vector equation of the line AB, and calculate the angle it makes with the vertical.

[6]

A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this

layer is x + 2y + 3ß = 320.

(iii) Find the coordinates of the point where the pipeline meets the layer of rock. [4]

(iv) By calculating the angle between the line AB and the normal to the plane of the layer, find the

angle at which the pipeline cuts through the layer. [5]


© OCR 2010 4754A Jun10 Turn over

4

8 Part of the track of a roller-coaster is modelled by a curve with the parametric equations

x = 2θ − sin θ, y = 4 cos θ for 0 ≤ θ ≤ 2π.

This is shown in Fig. 8. B is a minimum point, and BC is vertical.

y

xO A C

B

Fig. 8

(i) Find the values of the parameter at A and B.

Hence show that the ratio of the lengths OA and AC is (π − 1) : (π + 1). [5]

(ii) Finddy

dxin terms of θ. Find the gradient of the track at A. [4]

(iii) Show that, when the gradient of the track is 1, θ satisfies the equation

cos θ − 4 sin θ = 2. [2]

(iv) Express cos θ − 4 sin θ in the form R cos(θ + α).Hence solve the equation cos θ − 4 sin θ = 2 for 0 ≤ θ ≤ 2π. [7]










© OCR 2010 4754A Jun10

ADVANCED GCE


Candidates answer on the answer booklet.

OCR supplied materials:

• 8 page answer booklet(sent with general stationery)


Other materials required:• Scientific or graphical calculator

Friday 14 January 2011

Afternoon


**

44

77

55

44

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**


• Write your name, centre number and candidate number in the spaces provided on theanswer booklet. Please write clearly and in capital letters.

• Use black ink. Pencil may be used for graphs and diagrams only.• Read each question carefully. Make sure you know what you have to do before starting

your answer.• Answer all the questions.• Do not write in the bar codes.• You are permitted to use a scientific or 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.• 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 72.• This document consists of 4 pages. Any blank pages are indicated.

NOTE



RP–0F02 Turn over

2


1 (i) Use the trapezium rule with four strips to estimate ã 2

−2

√1 + ex dx, showing your working. [4]

Fig. 1 shows a sketch of y = √1 + ex.

x

y

–2 –1 1 2

Fig. 1

(ii) Suppose that the trapezium rule is used with more strips than in part (i) to estimate ã 2

−2

√1 + ex dx.

State, with a reason but no further calculation, whether this would give a larger or smaller estimate.

[2]

2 A curve is defined parametrically by the equations

x = 1

1 + t, y = 1 − t

1 + 2t.

Find t in terms of x. Hence find the cartesian equation of the curve, giving your answer as simply as

possible. [5]

3 Find the first three terms in the binomial expansion of1

(3 − 2x)3in ascending powers of x. State the

set of values of x for which the expansion is valid. [7]

4 The points A, B and C have coordinates (2, 0, −1), (4, 3, −6) and (9, 3, −4) respectively.

(i) Show that AB is perpendicular to BC. [4]

(ii) Find the area of triangle ABC. [3]

5 Show thatsin 2θ

1 + cos 2θ= tan θ. [3]

© OCR 2011 4754A Jan11

3

6 (i) Find the point of intersection of the line r = (−8−2

6

) + λ (−3

0

1

) and the plane 2x − 3y + ß = 11.

[4]

(ii) Find the acute angle between the line and the normal to the plane. [4]


7 A particle is moving vertically downwards in a liquid. Initially its velocity is zero, and after t seconds

it is v m s−1. Its terminal (long-term) velocity is 5 m s−1.

A model of the particle’s motion is proposed. In this model, v = 5(1 − e−2t).

(i) Show that this equation is consistent with the initial and terminal velocities. Calculate the

velocity after 0.5 seconds as given by this model. [3]

(ii) Verify that v satisfies the differential equationdv

dt= 10 − 2v. [3]

In a second model, v satisfies the differential equation

dv

dt= 10 − 0.4v2.

As before, when t = 0, v = 0.

(iii) Show that this differential equation may be written as

10(5 − v)(5 + v) dv

dt= 4.

Using partial fractions, solve this differential equation to show that

t = 14

ln(5 + v

5 − v). [8]

This can be re-arranged to give v = 5(1 − e−4t)

1 + e−4t. [You are not required to show this result.]

(iv) Verify that this model also gives a terminal velocity of 5 m s−1.

Calculate the velocity after 0.5 seconds as given by this model. [3]

The velocity of the particle after 0.5 seconds is measured as 3 m s−1.

(v) Which of the two models fits the data better? [1]


4

8 Fig. 8 shows a searchlight, mounted at a point A, 5 metres above level ground. Its beam is in the

shape of a cone with axis AC, where C is on the ground. AC is angled at α to the vertical. The beam

produces an oval-shaped area of light on the ground, of length DE. The width of the oval at C is GF.

Angles DAC, EAC, FAC and GAC are all β .

A A

B BD

D

G

C

C

F

E

E

5 m 5 mbb

ba

b

side view

Fig. 8

In the following, all lengths are in metres.

(i) Find AC in terms of α, and hence show that GF = 10 sec α tan β . [3]

(ii) Show that CE = 5(tan(α + β) − tan α).

Hence show that CE = 5 tan β sec2 α

1 − tan α tan β. [5]

Similarly, it can be shown that CD = 5 tan β sec2 α

1 + tan α tan β. [You are not required to derive this result.]

You are now given that α = 45◦ and that tan β = t.

(iii) Find CE and CD in terms of t. Hence show that DE = 20t

1 − t2. [5]

(iv) Show that GF = 10√

2t. [2]

For a certain value of β , DE = 2GF.

(v) Show that t2 = 1 − 1√

2.

Hence find this value of β . [3]




Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.





© OCR 2011 4754A Jan11

ADVANCED GCE


QUESTION PAPER

Candidates answer on the printed answer book.

OCR supplied materials:

• Printed answer book 4754A• MEI Examination Formulae and Tables (MF2)

Other materials required:

• Scientific or graphical calculator

Monday 13 June 2011

Morning



These instructions are the same on the printed answer book and the question paper.

• The question paper will be found in the centre of the printed answer book.

• Write your name, centre number and candidate number in the spaces provided on the printedanswer book. Please write clearly and in capital letters.

• Write your answer to each question in the space provided in the printed answer book.Additional paper may be used if necessary but you must clearly show your candidate number,

centre number and question number(s).

• Use black ink. Pencil may be used for graphs and diagrams only.• Read each question carefully. Make sure you know what you have to do before starting your

answer.

• Answer all the questions.• Do not write in the bar codes.

• You are permitted to use a scientific or graphical calculator in this paper.• Final answers should be given to a degree of accuracy appropriate to the context.


This information is the same on the printed answer book and the question paper.

• The number of marks is given in brackets [ ] at the end of each question or part question on the

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 72.• The printed answer book consists of 16 pages. The question paper consists of 4 pages. Any blank

pages are indicated.• This paper will be followed by Paper B: Comprehension.

INSTRUCTION TO EXAMS OFFICER / INVIGILATOR

• Do not send this question paper for marking; it should be retained in the centre or destroyed.


3R–1B23 Turn over

2


1 Express1

(2x + 1)(x2 + 1) in partial fractions. [5]

2 Find the first three terms in the binomial expansion of3√

1 + 3x in ascending powers of x. State the set

of values of x for which the expansion is valid. [5]

3 Express 2 sin θ − 3 cos θ in the form R sin(θ − α), where R and α are constants to be determined, and

0 < α < 12π.

Hence write down the greatest and least possible values of 1 + 2 sin θ − 3 cos θ. [6]


x = 2 sin θ, y = cos 2θ.

(i) Find the exact coordinates and the gradient of the curve at the point with parameter θ = 13π. [5]

(ii) Find y in terms of x. [2]

5 Solve the equation cosec2 θ = 1 + 2 cot θ, for −180◦ ≤ θ ≤ 180◦. [6]

6 Fig. 6 shows the region enclosed by part of the curve y = 2x2, the straight line x + y = 3, and the y-axis.

The curve and the straight line meet at P (1, 2).

x

y

Fig. 6

P (1, 2)

The shaded region is rotated through 360◦ about the y-axis. Find, in terms of π, the volume of the

solid of revolution formed. [7]

[You may use the formula V = 13πr2h for the volume of a cone.]

© OCR 2011 4754A Jun11

3


7 A piece of cloth ABDC is attached to the tops of vertical poles AE, BF, DG and CH, where E, F, 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.

y

x

z

F

O

E

A

(2, 0, 3) H

C (0, 4, 4)

G

D (–2, 4, )k

B (–2, 0, 1)

Fig. 7

(i) Write down the vectors−−→AB and

−−→AC. Hence calculate the angle BAC. [6]

(ii) Verify that the equation of the plane ABC is x + y − 2ß + d = 0, where d is a constant to be

determined.

Calculate the acute angle the plane makes with the horizontal plane. [7]

(iii) Given that A, B, D and C are coplanar, show that k = 3.

Hence show that ABDC is a trapezium, and find the ratio of CD to AB. [5]


© OCR 2011 4754A Jun11 Turn over

4

8 Water is leaking from a container. After t seconds, the depth of water in the container is x cm, and

the volume of water is V cm3, where V = 13x3. The rate at which water is lost is proportional to x, so

thatdV

dt= −kx, where k is a constant.

(i) Show that xdx

dt= −k. [3]

Initially, the depth of water in the container is 10 cm.

(ii) Show by integration that x = √100 − 2kt. [4]

(iii) Given that the container empties after 50 seconds, find k. [2]

Once the container is empty, water is poured into it at a constant rate of 1 cm3 per second. The

container continues to lose water as before.

(iv) Show that, t seconds after starting to pour the water in,dx

dt= 1 − x

x2. [2]

(v) Show that1

1 − x− x − 1 = x2

1 − x.

Hence solve the differential equation in part (iv) to show that

t = ln(1

1 − x) − 1

2x2 − x. [6]

(vi) Show that the depth cannot reach 1 cm. [1]




Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.





© OCR 2011 4754A Jun11

Tuesday 17 January 2012 – MorningA2 GCE MATHEMATICS (MEI)

4754A Applications of Advanced Mathematics (C4) Paper A

QUESTION PAPER

*4733130112*


These instructions are the same on the Printed Answer Book and the Question Paper.

• The Question Paper will be found in the centre of the Printed Answer Book.• Write your name, centre number and candidate number in the spaces provided on the

Printed Answer Book. Please write clearly and in capital letters.• Write your answer to each question in the space provided in the Printed Answer

Book. Additional paper may be used if necessary but you must clearly show your candidate number, centre number and question number(s).

• Use black ink. HB pencil may be used for graphs and diagrams only.• Read each question carefully. Make sure you know what you have to do before starting



This information is the same on the Printed Answer Book and the Question Paper.

• The number of marks is given in brackets [ ] at the end of each question or part question on the 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 72.• The Printed Answer Book consists of 16 pages. The Question Paper consists of 4 pages.

Any blank pages are indicated.• This paper will be followed by Paper B: Comprehension.


• Do not send this Question Paper for marking; it should be retained in the centre or recycled. Please contact OCR Copyright should you wish to re-use this document.

OCR is an exempt CharityTurn over

© OCR 2012 [T/102/2653]DC (KN/SW) 43992/4

Candidates answer on the Printed Answer Book.

OCR supplied materials:• Printed Answer Book 4754A• MEI Examination Formulae and Tables (MF2)



2

4754A Jan12© OCR 2012


1 Express x + 1––––––––x2(2x – 1)

in partial fractions. [5]

2 Solve, correct to 2 decimal places, the equation cot 2θ = 3 for 0° θ 180°. [4]

3 Express 3 sin x + 2 cos x in the form R sin(x + α), where R > 0 and 0 < α < π2 .

Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve y = f(x), where

f(x) = 3 sin x + 2 cos x, 0 x π. [7]

4 (i) Complete the table of values for the curve y = cos x .

x 0 π8

π4

3π8

π2

y 0.9612 0.8409

Hence use the trapezium rule with strip width h = π8 to estimate the value of the integral ∫0

cos x dx , giving your answer to 3 decimal places. [3]

Fig. 4 shows the curve y = cos x for 0 x π2 .

xO 2

y

Fig. 4

(ii) State, with a reason, whether the trapezium rule with a strip width of π16 would give a larger or smaller estimate of the integral. [1]

5 Verify that the vector 2i − j + 4k is perpendicular to the plane through the points A(2, 0, 1), B(1, 2, 2) and C(0, −4, 1). Hence find the cartesian equation of the plane. [5]

6 Given the binomial expansion (1 + qx) p = 1 − x + 2x2 + …, find the values of p and q. Hence state the set of values of x for which the expansion is valid. [6]

7 Show that the straight lines with equations r = 424

+ λ 301

and r =

–149

+ μ

–113

meet.

Find their point of intersection. [5]

π2

3

4754A Jan12 Turn over© OCR 2012


8 Fig. 8 shows a cross-section of a car headlight whose inside reflective surface is modelled, in suitable units, by the curve

x = 2t2, y = 4t, – 2 t 2 .

P(2t2, 4t) is a point on the curve with parameter t. TS is the tangent to the curve at P, and PR is the line through P parallel to the x-axis. Q is the point (2, 0). The angles that PS and QP make with the positive x-direction are θ and ϕ respectively.

x

y

Q (2, 0)

R

S

T

O

P (2t2, 4t)

Fig. 8

(i) By considering the gradient of the tangent TS, show that tan θ = 1t . [3]

(ii) Find the gradient of the line QP in terms of t. Hence show that ϕ = 2θ, and that angle TPQ is equal to θ. [8]

[The above result shows that if a lamp bulb is placed at Q, then the light from the bulb is reflected to produce a parallel beam of light.]

The inside surface of the headlight has the shape produced by rotating the curve about the x-axis.

(iii) Show that the curve has cartesian equation y2 = 8x. Hence find the volume of revolution of the curve, giving your answer as a multiple of π. [7]

4

4754A Jan12© OCR 2012

9

x cm

10 cm

Fig. 9

Fig. 9 shows a hemispherical bowl, of radius 10 cm, filled with water to a depth of x cm. It can be shown that the volume of water, V cm3, is given by

V = π (10x2 – 13 x

3).

Water is poured into a leaking hemispherical bowl of radius 10 cm. Initially, the bowl is empty. After t seconds, the volume of water is changing at a rate, in cm3 s−1, given by the equation

dVdt = k (20 – x),

where k is a constant.

(i) Find dVdx

, and hence show that πx dxdt = k. [4]

(ii) Solve this differential equation, and hence show that the bowl fills completely after T seconds, where T = 50π

k . [5]

Once the bowl is full, the supply of water to the bowl is switched off, and water then leaks out at a rate of kx cm3 s–1.

(iii) Show that, t seconds later, π (20 – x) dxdt = –k. [3]

(iv) Solve this differential equation.

Hence show that the bowl empties in 3T seconds. [6]


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 and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.



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.

The examination is in two parts:

Paper A (1 hour 30 minutes) Paper B (up to 1 hour)

Supervisors are requested to ensure that Paper B is not issued until Paper A has been collected in from the candidates.

Centres may, if they wish, grant a supervised break between the two parts of this examination.

Paper B should not be attached to the corresponding paper A script. For Paper A only the candidates’ printed answer books, in the same order as the attendance register, should be sent for marking; the question paper should be retained in the centre or recycled. For Paper Bonly the question papers, on which the candidates have written their answers, should be sent for marking; the insert should be retained in the centre or recycled. Any additional sheets used must be carefully attached to the correct paper.

For Paper B (Comprehension) only.

A standard English dictionary is allowed for the comprehension.(Dictionaries to be used in the examination must be thoroughly checked before the examination.) Full regulations are in the JCQ Regulations and Guidance booklet.

This notice must be on the Invigilator’s desk at all times during the morning of Thursday 14 June 2012.

Thursday 14 June 2012 – MorningA2 GCE MATHEMATICS (MEI)

4754 Applications of Advanced Mathematics (C4)

INSTRUCTIONS

OCR is an exempt Charity© OCR 2012 [T/102/2653]DC (LEO) 55776/1

*4715680612*

2

4754 Jun12© OCR 2012








QUESTION PAPER

*4715690612*

















© OCR 2012 [T/102/2653]DC (KN/SW) 43993/5





2

4754A Jun12© OCR 2012


1 Solve the equation 4xx + 1

– 32x + 1

= 1. [5]

2 Find the first four terms in the binomial expansion of 1 + 2x . State the set of values of x for which the expansion is valid. [5]

3 The total value of the sales made by a new company in the first t years of its existence is denoted by £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.

(i) Express the model as a differential equation.

Verify by differentiation that V = ( 12 kt + c)2, where c is an arbitrary constant, satisfies this differential

equation. [4]

(ii) The value of the company’s sales in its first year is £10 000, and the total value of the sales in the first two years is £40 000. Find V in terms of t. [4]

4 Prove that sec2θ + cosec2θ = sec2θ cosec2θ. [4]

5 Given the equation sin(x + 45°) = 2 cos x, show that sin x + cos x = 2 2 cos x .

Hence solve, correct to 2 decimal places, the equation for 0° x 360°. [6]

6 Solve the differential equation dydx

= yx(x + 1)

, given that when x = 1, y = 1. Your answer should express

y explicitly in terms of x. [8]

3

4754A Jun12 Turn over© OCR 2012


7 Fig. 7a shows the curve with the parametric equations

x = 2 cos θ, y = sin 2θ, – π2 θ π2 .

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.

x

y

R

Q

PO

Fig. 7a

(i) State, with their coordinates, the points on the curve for which θ = – π2 , θ = 0 and θ = π2. [3]

(ii) Find dydx

in terms of θ. Hence find the gradient of the curve when θ = π2 , and verify that the two tangents

to the curve at the origin meet at right angles. [5]

(iii) Find the exact coordinates of the turning point Q. [3]

When the curve is rotated about the x-axis, it forms a paperweight shape, as shown in Fig. 7b.

x

y

O

Fig. 7b

(iv) Express sin2θ in terms of x. Hence show that the cartesian equation of the curve is y2 = x2(1 − 14 x

2). [4]

(v) Find the volume of the paperweight shape. [4]

4

4754A Jun12© OCR 2012

8 With respect to cartesian coordinates Ox y z, 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 + 2y − 3z = 0, as shown in Fig. 8. A′ is the point (2, 4, 1), and M is the midpoint of AA′.

A (1, 2, 4)

(2, 4, 1)

C

B

M

x + 2y – 3z = 0

Fig. 8

(i) Show that AA′ is perpendicular to the plane x + 2y − 3z = 0, and that M lies in the plane. [4]

The vector equation of the line AB is r = 124

+ λ

1–12

.

(ii) Find the coordinates of B, and a vector equation of the line A′ B. [6]

(iii) Given that A′ BC is a straight line, find the angle θ. [4]

(iv) Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the x- and z-axes). [3]













This notice must be on the Invigilator’s desk at all times during the morning of Thursday 14 June 2012.


4754 Applications of Advanced Mathematics (C4)

INSTRUCTIONS

OCR is an exempt Charity© OCR 2012 [T/102/2653]DC (LEO) 55776/1

*4715680612*

2

4754 Jun12© OCR 2012








QUESTION PAPER

*4715690612*

















© OCR 2012 [T/102/2653]DC (KN/SW) 43993/5





2

4754A Jun12© OCR 2012


1 Solve the equation 4xx + 1

– 32x + 1

= 1. [5]

2 Find the first four terms in the binomial expansion of 1 + 2x . State the set of values of x for which the expansion is valid. [5]

3 The total value of the sales made by a new company in the first t years of its existence is denoted by £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.

(i) Express the model as a differential equation.

Verify by differentiation that V = ( 12 kt + c)2, where c is an arbitrary constant, satisfies this differential

equation. [4]

(ii) The value of the company’s sales in its first year is £10 000, and the total value of the sales in the first two years is £40 000. Find V in terms of t. [4]

4 Prove that sec2θ + cosec2θ = sec2θ cosec2θ. [4]

5 Given the equation sin(x + 45°) = 2 cos x, show that sin x + cos x = 2 2 cos x .

Hence solve, correct to 2 decimal places, the equation for 0° x 360°. [6]

6 Solve the differential equation dydx

= yx(x + 1)

, given that when x = 1, y = 1. Your answer should express

y explicitly in terms of x. [8]

3

4754A Jun12 Turn over© OCR 2012


7 Fig. 7a shows the curve with the parametric equations

x = 2 cos θ, y = sin 2θ, – π2 θ π2 .

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.

x

y

R

Q

PO

Fig. 7a

(i) State, with their coordinates, the points on the curve for which θ = – π2 , θ = 0 and θ = π2. [3]

(ii) Find dydx

in terms of θ. Hence find the gradient of the curve when θ = π2 , and verify that the two tangents

to the curve at the origin meet at right angles. [5]

(iii) Find the exact coordinates of the turning point Q. [3]

When the curve is rotated about the x-axis, it forms a paperweight shape, as shown in Fig. 7b.

x

y

O

Fig. 7b

(iv) Express sin2θ in terms of x. Hence show that the cartesian equation of the curve is y2 = x2(1 − 14 x

2). [4]

(v) Find the volume of the paperweight shape. [4]

4

4754A Jun12© OCR 2012

8 With respect to cartesian coordinates Ox y z, 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 + 2y − 3z = 0, as shown in Fig. 8. A′ is the point (2, 4, 1), and M is the midpoint of AA′.

A (1, 2, 4)

(2, 4, 1)

C

B

M

x + 2y – 3z = 0

Fig. 8

(i) Show that AA′ is perpendicular to the plane x + 2y − 3z = 0, and that M lies in the plane. [4]

The vector equation of the line AB is r = 124

+ λ

1–12

.

(ii) Find the coordinates of B, and a vector equation of the line A′ B. [6]

(iii) Given that A′ BC is a straight line, find the angle θ. [4]

(iv) Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the x- and z-axes). [3]













This notice must be on the Invigilator’s desk at all times during the afternoon of Friday 18 January 2013.

Friday 18 January 2013 – AfternoonA2 GCE MATHEMATICS (MEI)

4754/01 Applications of Advanced Mathematics (C4)

INSTRUCTIONS

OCR is an exempt Charity© OCR 2013 [T/102/2653]DC (LEG) 67965/1

*4734010113*

2

4754/01 Jan13© OCR 2013






Friday 18 January 2013 – AfternoonA2 GCE MATHEMATICS (MEI)

4754/01A Applications of Advanced Mathematics (C4) Paper A

QUESTION PAPER

*4734020113*

















© OCR 2013 [T/102/2653]DC (SLM/SW) 63877/2


OCR supplied materials:• Printed Answer Book 4754/01A• MEI Examination Formulae and Tables (MF2)



2

4754/01A Jan13© OCR 2013


1 Solve the equation xx

x1

2

1

11

+--= . [4]

2 Find the first four terms of the binomial expansion of x1 23- . State the set of values of x for which the

expansion is valid. [6]

3 The parametric equations of a curve are

, 2 , 0 2sin sinx y for G Gi i i r= = .

(i) Find the exact value of the gradient of the curve at the point where 61i r= . [4]

(ii) Show that the cartesian equation of the curve is y x x4 42 2 4= - . [3]

4 Fig. 4 shows the curve 1y ex2= + , and the region between the curve, the x-axis, the y-axis and the

line x = 2.

y

x2O

Fig. 4

(a) Find the exact volume of revolution when the shaded region is rotated through 360° about the x-axis. [4]

(b) (i) Complete the table of values, and use the trapezium rule with 4 strips to estimate the area of the shaded region. [3]

x 0 0.5 1 1.5 2

y 1.9283 2.8964 4.5919

(ii) The trapezium rule for 1 e xdx20

2

+; 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. [1]

3

4754/01A Jan13 Turn over© OCR 2013

5 Solve the equation 2 5sec tan2i i= , for 0 G Gi r . [6]

6 In Fig. 6, ABC, ACD and AED are right-angled triangles and BC = 1 unit. Angles CAB and CAD are i and z respectively.

D

C

B

1

EAzi

Fig. 6

(i) Find AC and AD in terms of i and z. [2]

(ii) Hence show that DEtan

tan1

i

z= + . [3]


7 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The Oxy plane is horizontal.

x

y

zE(1, 0, 3)

D(6, 0, 2)

F(0, 4, 0)

C(8, a, 0)

B(8, –a, 0)

A(0, –4, 0)O

Fig. 7

(i) Find the length of the ridge of the tent DE, and the angle this makes with the horizontal. [4]

(ii) Show that the vector 4 5i j k- + is normal to the plane through A, D and E.

Hence find the equation of this plane. Given that B lies in this plane, find a. [7]

(iii) Verify that the equation of the plane BCD is x z 8+ = .

Hence find the acute angle between the planes ABDE and BCD. [6]

4

4754/01A Jan13© OCR 2013

8 The growth of a tree is modelled by the differential equation

th h10 20d

d= - ,

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= .

(i) Write down the value of h for which th

0d

d= , and interpret this in terms of the growth of the tree. [1]

(ii) Verify that 1h 20 e. t0 1= - -_ i satisfies this differential equation and its initial condition. [5]

The alternative differential equation

th h200 400d

d 2= -

is proposed to model the growth of the tree. As before, h 0= when t 0= .

(iii) Using partial fractions, show by integration that the solution to the alternative differential equation is

1

1h

20

e

e.

. t

t

0 2

0 2

=+

--

-_ i.

[9]

(iv) What does this solution indicate about the long-term height of the tree? [1]

(v) After a year, the tree has grown to a height of 2 m. Which model fits this information better? [3]






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