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DC (SM/CGW) 34245/3

UCLES 2011 [Turn over

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSGeneral Certificate of Education Ordinary Level

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.Write in dark blue or black pen.You may use a pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.DO NOT WRITE IN ANY BARCODES.

Answer all the questions.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimalplace in the case of angles in degrees, unless a different level of accuracy isspecified in the question.The use of an electronic calculator is expected, where appropriate.You are reminded of the need for clear presentation in your answers.

At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question or partquestion.The total number of marks for this paper is 80.

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ADDITIONAL MATHEMATICS 4037/12

Paper 1 May/June 2011

2 hours

Candidates answer on the Question Paper.No Additional Materials are required.

For Examiners Use

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Total

www.Xtrem

ePapers.com

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4037/12/M/J/11 UCLES 2011

Mathematical Formulae

1. ALGEBRA

QuadraticEquation

For the equation ax2 + bx + c = 0,

xb b ac

a=

2 4

2.

Binomial Theorem

(a + b)n = an + (n

1 )an1b + (

n

2 )an2b2 + + (

n

r)anrbr+ + bn,

where n is a positive integer and ( nr) = n!(n r)!r! .

2. TRIGONOMETRY

Identities

sin2A + cos2A = 1

sec2A = 1 + tan2A

cosec2A = 1 + cot2A

Formulae forABC

a

sinA=

b

sinB=

c

sin C

a2 = b2 + c2 2bc cosA

=1

2bc sinA

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1 Find the value ofkfor which thex-axis is a tangent to the curve

y =x2 + (2k+ 10)x + k2 + 5. [3]

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2 The coefficient ofx3 in the expansion of (2 + ax)5 is 10 times the coefficient ofx2 in the

expansion of 1 + ax3 4

. Find the value ofa. [4]

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3 (a)

5

4

3

2

1

x2

y

O

The figure shows the graph ofy = k+ msinpx for 0 x, where k, m andp are positiveconstants. Complete the following statements.

k= ...................... m = ...................... p = ...................... [3]

(b) The function g is such that g(x) = 1 + 5cos3x . Write down

(i) the amplitude ofg, [1]

(ii) the period ofg in terms of. [1]

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4 You must not use a calculator in Question 4.

In the triangleABC, angleB = 90,AB = 4 + 2 2 and BC= 1 + 2 .

(i) Find tan C, giving your answer in the form k 2 . [2]

(ii) Find the area of the triangleABC, giving your answer in the form p + q 2, wherep and q

are integers. [2]

(iii) Find the area of the square whose side is of length AC, giving your answer in the forms + t 2, where s and t are integers. [2]

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5 (i) Show that 2x 1 is a factor of 2x3 5x2 + 10x 4. [2]

(ii) Hence show that 2x3 5x2 + 10x 4 = 0 has only one real root and state the value of thisroot. [4]

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6 The figure shows the graph of a straight line with 1gy plotted againstx. The straight line passes

through the pointsA(5,3) andB (15,5).

A (5,3)

B (15,5)

x

lg y

O

(i) Express lgy in terms ofx. [3]

(ii) Show that y = a (10bx) where a and b are to be found. [3]

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7 A team of 6 members is to be selected from 6 women and 8 men.

(i) Find the number of different teams that can be selected. [1]

(ii) Find the number of different teams that consist of 2 women and 4 men. [3]

(iii) Find the number of different teams that contain no more than 1 woman. [3]

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8 (i) Sketch the curve y = (2x 5)(2x + 1) for 1 x 3, stating the coordinates of the pointswhere the curve meets the coordinate axes. [4]

(ii) State the coordinates of the stationary point on the curve. [1]

(iii) Using your answers to parts (i) and (ii), sketch the curve y = (2x 5)(2x + 1)for 1 x 3. [2]

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9 The figure shows a circle, centre O, radius rcm. The length of the arcAB of the circle is 9cm.

AngleAOB is radians and is 3 times angle OBA.

rad

A

B

Orcm

9cm

(i) Show that =35

. [2]

(ii) Find the value ofr. [2]

(iii) Find the area of the shaded region. [3]

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10 Relative to an origin O, pointsA andB have position vectors 56 and 2913 respectively.

(i) Find a unit vector parallel to AB . [3]

The pointsA,B and C lie on a straight line such that 2AC = 3AB .

(ii) Find the position vector of the point C. [4]

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11 Solve

(i) 2cot2x 5cosecx 1 = 0 for 0

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12 Answer only one of the following two alternatives.

EITHER

The tangent to the curve y = 3x3 + 2x2 5x + 1 at the point where x = 1 meets they-axis at the

pointA.

(i) Find the coordinates of the pointA. [3]

The curve meets they-axis at the point B. The normal to the curve atB meets thex-axis at thepoint C. The tangent to the curve at the point where x = 1 and the normal to the curve atB meet

at the pointD.

(ii) Find the area of the triangleACD. [7]

OR

R

Q

y y = x (x3)2

xO

P

The diagram shows the curve y = x (x 3)2 . The curve has a maximum at the point P and

touches thex-axis at the point Q. The tangent at P and the normal at Q meet at the pointR. Findthe area of the shaded region PQR. [10]

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Start your answer to Question 12 here.

Indicate which question you are answering.

EITHER

OR

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Continue your answer here if necessary.

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the earliest possible opportunity.

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