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Name:______________________ ( ) Class:________

Marks:_______/70

SPRINGFIELD SECONDARY SCHOOLBETTER SELF FOR BETTER TOMORROW

Mid-Year Examination 2010

ADDITIONAL MATHEMATICS 4039/01

Paper 1

Sec 4 Normal Academic

7 May 2010 1 hour 45 minutes

Additional Materials: Writing Paper

___________________________________________________________________________

READ THESE INSTRUCTIONS FIRST

Write your name, register number and class in the spaces at the top of this page, on all the

work you hand in.

Write in dark blue or black pen on both sides of the paper.

You may use a soft pencil for any diagrams, graphs, tables or rough working.

Do not use paper clips, highlighters, glue or correction fluid.

Answerall questions.

Write your answers on the Writing Paper provided.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the

case of angles in degrees, unless a different level of accuracy is specified in the question.

The use of 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 part question.

The total number of marks for this paper is 70.

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Do not turn over this question paper until you are told to do so.

___________________________________________________________________________

This question paper consists of 5 printed pages

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Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation 02 =++ cbxax ,

a

acbbx

2

42 =

Binomial Expansion

( ) nrrnnnnn bbar

nba

nba

naba ++

++

+

+=+ ......

21

221

where n is a positive integer and!

)1)...(1(

)!(!

!

r

rnnn

rnr

n

r

n +=

=

2. TRIGONOMETRY

Identities

1cossin 22 =+ AAAA 22 tan1sec +=

cosec 2 A = 1 + cot 2 ABABABA sincoscossin)sin( =BABABA sinsincoscos)cos( =

BA

BABA

tantan1

tantan)tan(

=

AAA cossin22sin =

AAAAA

2222

sin211cos2sincos2cos===

A

AA

2tan1

tan22tan

=

Formulae for ABC

C

c

B

b

A

a

sinsinsin==

Abccba cos2222+=

Abc sin2

1=

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Answer all questions.

1 (i) Write down the inverse of the matrix .67

34

[2]

(ii) Hence solve the simultaneous equations

.01667

,0743

=++=++

yx

xy[3]

2 Find the range of values ofx for which2

2

59x

x

+. [4]

3 Find the range of values ofkfor which the liney = k 3x does not intersect the [4]

curvexy = 3.

4 (a) The coefficient ofx4 in the expansion of (1 +px)(1 2x)8 is -224.

Find the value ofp. [4]

(b) Find the term independent ofx in the expansion6

5 2

x

x [4]

5. The acute angleA is such that2

3cos =A .

Express in the formb

a, where a and b are integers,

(i) Asin , [1]

(ii) cotA . [2]

6. It is given that .3cos4)(f xx =

(i) State the amplitude of f. [1]

(ii) State the period of f, for x0 . [1]

(iii) Sketch the graph of )(f xy = for x0 .[3]

7 (i) Solve the equation, 12sin2 = , for 3600 x [3]

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(ii) Show that sin)90cos( =+ [3]

8 Solutions to this question by accurate drawing will not be accepted.

The diagram above, not drawn to scale, shows a circle with centre C(2, 3) and

radius 5 units. The circle cuts thex-axis atA andB.

The perpendicular bisector ofA and Ccuts thex-axis atD.Find

(i) the equation of the circle, [1]

(ii) the coordinates ofA, [3]

(iii) the midpoint ofAC, [1]

(iv) the gradient ofAC, [1]

(v) the coordinates ofD. [3]

9 Differentiate the following with respect tox.

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(i) 23

410

xx

[2]

(ii) 2)43(7

x [2]

10. Find the equation of the normal to the curvex

xy4

+= , at the pointPwhere [4]

x = 4.

11. A curve has the equation 2634 23 += xxxy .

(i) Find the coordinates of the stationary points. . [5]

(ii) Determine the nature of each of these stationary points. [3]

12. Evaluate the following

(i)4

11

0

dxx [2]

(ii) dxx

x )1(3

1

2 + [3]

13. The gradient of the tangent at a point on a curve is given by 22 +xx . [5]

Find the equation of the curve which passes through the point (2, 1).

End of Paper

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Springfield Secondary School

Mathematics Department

Marking Scheme

Test : Mid Year Exam 2010 Paper 1

Level: Sec 4 NA A.Maths

Q Marking points Mark

Awarded

Total

1. (i)Inverse =

47

36

2124

1

=3

1

47

36

B1

(determinant)

B1 (order in

matrix)

[2]

(ii)

=

=

=+=+

5

2

16

7

47

36

3

1

1667734

y

x

y

x

yxyx M1

M1

A1

[3]

2.

521

)5)(12(0

5920

259

2

2

+

+

x

xx

xx

xx

M1

M1

A2

[4]

3.

366

0)6)(6(

036

0)3)(3(4)(

04

033

3)3(

2

2

2

2

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Awarded

Total

rr xxr

)2()(6 165

Power ofx: 30- 5r- r= 0 r= 5

Term independent ofx: (-2)5

= -32

M1

M1A1

5. (i)

2

1sin =A B1 [1]

(ii)

3

tan

1cot

=

=A

AM1

A1

[2]

6. (i) Amplitude = 1 B1 [1]

(ii)Period =

3

2 B1 [1]

(iii

)

B1 y=cosx

B1 correct -

scale

B1 correct-

frequency

[3]

8. (i)

=

=

==

255,195,75,15

510,390,150,302

30,2

12sin0

basicangle M1

M1

A1

[3]

(ii)

sin

sin0

sin90sincos90cos

==

M1M1

A1

[3]

1

0.

(i) 222 5)3()2( =+ yx B1 [1]

(ii)

0)A(-2,

2-or)(6

16)2(

25)30()2(2

22

rejectx

x

x

==

=+M1

M1

A1

[3]

(iii

)

(iv)

1.5),0(:Midpt B1

B1

[1]

[1]

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Awarded

Total

Grad of AC=

4

3

)2(2

03

=

(v)

Grad of line passing through D =

3

4

Equation of line: c =1.5, y = 5.13

4+ x

D: ( 1.125, 0)

M1

M1

A1

[3]

9 (i)

3

2

32

8

10

3

d

d

8

10

3

d

d

x

x

x

y

xx

x

y

+=

+= M1

A1

[2]

(ii)

3

4

2

)43(

42

d

d

)43(

)]3)(43(2[7)0()43(

d

d

=

=

xx

y

x

xx

x

y M1

A1

[2]

1

0

3

110

3

4-

3

110

)4(3

45

3

4-

3

4-normalofgrad

4

3)4(

41 2

+=

=

+=

+=

=

==

=

xy

c

c

cxy

atxdx

dy

xdx

dy

M1

M1

[4]

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Awarded

Total

1

1

M1

A1

(i)

3)-(1,and)4

12,

2

1(-

3-or4

12y

1or2

1

)1)(12(0

6612 2

=

=

+=

=

x

xx

xxdx

dy

M1

M1

A1

A1, A1

[5]

(ii)

)(min,1

)(max,2

1

624

2

2

2

2

2

2

ptpositivedx

ydxAt

ptnegativedx

ydxAt

xdx

yd

==

==

=M1

A1

A1

[3]

1

2

(i)

(ii)

3

24

12

1

3

1

2

3

1

31

2)

1(

1

3

8

1

0

1

84

1

0

1

2

2

2

2

=

=

=+

=

=

x

xdx

xx

xxdx

M1

A1

M1

M1

A1

[2]

[3]

1

3

3

12

23

3

1

)2(22

2

3

21),1,2(

223

)2(

23

23

23

2

++=

=

++=

++=

+=

xxx

y

c

cpt

cxxxy

dxxxy M1

M1

M1

M1

A1

[5]

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