Tkss Prelim 2009 Am P1

Class

Reg Number

Candidate Name _____________________________________

TANJONG KATONG SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR

ADDITIONAL MATHEMATICS

4038/01

Paper 1 Wednesday 16 September 2009

2 hours

Additional Materials: Writing Paper Graph Paper READ THESE INSTRUCTIONS FIRST Write your name, class and index number in the spaces at the top of this page and on all separate writing paper used. Write in dark blue or black pen. You may use a soft pencil for any diagram or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Write your answers on the writing paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal in the case of angles in degree, unless a different level of accuracy is specified in the question. The use of a scientific 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 80.

This question paper consists of 5 printed pages.

[Turn over

2 Mathematical Formulae

1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x=

 b  b 2  4ac . 2a

Binomial Theorem n (a + b)n = a n +   a n  1 b + 1

n n  2 2 n   a b + . . . +   an  r br + . . . + bn, 2 r

n n! n( n  1).......(n  r  1) where n is a positive integer and   = = r!  r  (n  r )!r!

2. TRIGONOMETRY Identities sin2 A + cos2 A = 1. sec2 A = 1 + tan2 A. cosec2 A = 1 + cot2 A. sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B ∓ sin A sin B tan A  tan B tan (A ± B) = 1  tan A tan B sin 2A = 2 sin A cos A. cos 2A = cos2 A  sin2 A = 2 cos2 A  1 = 1  2 sin2 A 2 tan A tan 2A = 1  tan 2 A sin A + sin B = 2 sin 12 (A + B) cos 12 (A  B) sin A  sin B = 2 cos

1 2

(A + B) sin

cos A + cos B = 2 cos

1 2

(A + B) cos

1 2

(A  B)

(A + B) sin

1 2

(A  B)

cos A  cos B = 2 sin

1 2

1 2

(A  B)

Formulae for ABC a b c   . sin A sin B sin C

a 2 = b 2 + c2  2bc cos A. =

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1 bc sin A. 2

Prelim Add Mathematics P1

3

1.

 4  3  , find A−1 and use it to solve the simultaneous equations Given that A =  6 1  3y = 4x − 8 6x + y = 1.

2.

3.

4.

5.

Given that f x   k  1x 2  7kx  9k , where k is a constant not equal to 1. Find the range of values of k if f x   9 for all real values of x.

corresponding range.

[5]

Solve the following simultaneous equations 5 log x y + log y x = 2 x y = 64.

[6]

(a)

 5 Prove that sin θ + sin 2θ + sin 3θ + sin 4θ ≡ 4 cos θ cos sin . 2 2 Hence find all angles  between 0° and 180° for which sin θ + sin 2θ + sin 3θ + sin 4θ = 0

Express

[3]

[4]

x2  2 x  12 dx .

[4]

5

A curve has the equation y = cos 2x + 5cos x − 2, where 0 < x < 2π. dy (a) Find expression for (i) , dx d 2y (ii) . dx 2 (b)

[4]

x2 in partial fractions. x  12

Hence evaluate

7.

[5]

Sketch the graph of f x   x 2  x  6  1 , for  3  x  4 and write down the

(b)

6.

[4]

Find the coordinates of the stationary point of the curve and determine the nature of this point.

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[1] [1]

[5]

Prelim Add Mathematics P1

4 8.

In the figure, ABCD is a parallelogram with AB = 7 cm. The perpendicular from D meets the line AB at H and DH = 2.5 cm. The perpendicular from B meets the line AD produced at K and BK = 4  2 cm. K D

A

C

H

B

Find, in exact form,

9.

(i)

the length of AD,

[4]

(ii)

the perimeter of the parallelogram ABCD.

[1]

Given that y  3  e  x . dy > 0 for all real values of x. dx

(a)

Show that

(b)

Sketch the graph of y  3  e  x , indicating clearly on your graph, the intercept(s), if any.

(c)

[2]

[3]

Determine the equation of the straight line which would need to be drawn on your sketch to obtain a graphical solution of the equation ln (1 + x) = − x. [2]

7

10.

(a)

(b)

k   In the expansion of  x  2  where k ≠ 0, given that the coefficient of x   x4 is equal to the coefficient of x−2 , find the exact values of k. [4]

Find the first 3 terms in the expansion, in ascending powers of x, of (i)

1  x n

[2]

(ii)

1 2x 5

[1]

(iii)

Given that the expansion of 1  x  1  2x  in ascending powers of x is 1 + 15x + bx2 + . . . . , find the values of n and b.

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n

5

[3]

Prelim Add Mathematics P1

5 11. A

B

>

 O

4 cm

>

C

The diagram shows a trapezium ABCO inscribed in a semi-circle, centre O and radius 4 cm. OA makes an angle of  ( < 90° ) with the diameter of the circle. AB is parallel to OC and BC is perpendicular to both AB and OC. (a)

(b)

(c)

12.

Show that the perimeter, S of the trapezium is given by S = 4 + 12 cos  + 4 sin  .

[3]

Express 4 + 12 cos  + 4 sin  in the form 4 + R cos (  −  ), where R > 0 and 0° <  < 90°.

[3]

Find the value of  for which the perimeter of the trapezium is equal to 14 cm.

[3]

The running records of some track and field events are given in the table below. Distance (D metres) Time (T seconds)

200

400

800

1500

5000

19.8

43.9

103.5

214.9

800.4

It is claimed that these records follow the law T = kDn, where D is the distance in metres, T is the time in seconds and k and n are constants. (a)

Using graph paper, draw the graph of lg T against lg D.

(b)

Use your graph to estimate,

[3]

(i)

the value of k and of n,

[3]

(ii)

the record time for the 100 metres event.

[1]

End of Paper

Tanjong Katong Secondary School

Prelim Add Mathematics P1