2010AMathsPrelimP2

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TANJONG KATONG SECONDARY SCHOOLMID-YEAR EXAMINATION 2010SECONDARY FOUR EXPRESS

ADDITIONAL MATHEMATICS 4038/02Paper 2

Tuesday 21 September 2 hours 30 minutes

Additional Materials:Writing PaperGraph 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 100.

This question paper consists of 7 printed pages.[Turn over

Tanjong Katong Secondary School

2

Mathematical Formulae

1. ALGEBRAQuadratic Equation

For the equation ax2 + bx + c = 0,

x =

Binomial Theorem

= + + + . . . + + . . . + ,

where n is a positive integer and = =

2. TRIGONOMETRYIdentities

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 ± B) =

sin 2A = 2 sin A cos A

cos 2A = cos2 A sin2 A = 2 cos2 A 1 = 1 2 sin2 A

tan 2A =

sin A + sin B = 2 sin (A + B) cos (A B)

sin A sin B = 2 cos (A + B) sin (A B)

cos A + cos B = 2 cos (A + B) cos (A B)

cos A cos B = 2 sin (A + B) sin (A B)

Formulae for ABC

a2 = b2 + c2 2bc cos A

= ab sin C

1 (a) Solve the equation


3

(i) [ 4 ]

(ii) [ 5 ]

(b) Sketch the graph . [ 3 ]

2 (a) Prove that . [ 3 ]

(b) Hence solve the equation for the interval [ 4 ]

3 A particle moves in a straight line such that, t seconds after leaving a fixed point O, its velocity, v m/s, is given by . Find

(a) the initial velocity of the particle. [ 1 ]

(b) the acceleration of the particle when it is first instantaneously at rest. [ 3 ]

(c) the total distance travelled by the particle in the first 5 seconds. [ 3 ]

(d) the average speed over the first 5 seconds. [ 1 ]

4

The diagram shows part of the curve and the straight line .

The straight line meets the curve at y = 4.

(a) Show that the straight line meets the curve at . [ 2 ]

(b) Find the area of the shaded region. [ 5 ]

5


x

4

6

y

r

h

4

Red wax

8 cm 4cm

White wax

Diagram A Diagram B (cross-sectional view)

Diagram A shows a cylindrical candle stick of height 8 cm and radius 4 cm. The candle stick is made of white wax and red wax as shown in diagram B. The red wax forms an inverted cone of radius r cm and height h cm.

(a) Given that the sum of the radius and height of the cone is to remain constant at 5 cm, express h in terms of r . [ 1 ]

(b) Show that the volume, V, of the white wax is given by

[ 3 ]

(c) Find the value of r for which V is stationary. [ 3 ]

(d) Determine whether the stationary value of V is a maximum or a minimum. [ 2 ]

6 A circular archery target board with a diameter of 2 metres is mounted on a vertical wall of length 20 metres and height 8 metres.

Taking the bottom left corner of the wall as the origin and 1 unit to represent 1 metre,

(a) find the position of the centre of the target board, in coordinates form with reference to the wall, if it is represented by the equation

[ 2 ]

An archer, suffering from serious astigmatism, sees an exact image of the target board in addition to the actual one. This image appears to touch the actual target board on its right and at the same height from the floor.

(b) Find the equation that describes the image that he sees. [ 2 ]

(c) If he accidentally took aim at the extreme right of the image, describe where his arrow will land with reference to the centre of the actual target board. [ 1 ]

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


y

x

E(1, 2)

A

B

CD(0, 1)

5

The diagram shows a pentagon ABCDE in which AB is perpendicular to BC. The equation of AB is . The coordinates of points D and E are and

respectively.

(a) Given that DB is parallel to the line , show that the coordinates

of B is . [ 3 ]

(b) Find the area of triangle EBD. [ 2 ]

(c) Find the equation of BC. [ 2 ]

(d) Show that DBC = . [ 3 ]

8 A curve has the equation .

(a) Express in the form , where k is a constant. [ 4 ]

(b) Hence evaluate . [ 4 ]

9


AB

C

D

6

cm

The diagram above shows a right-angled triangle ABC with AB = cm and angle ABC = . A point D lies on CB such that angle CAD = .

(a) Express the side AC in the form of where is a positive integer. [ 1 ]

(b) Find the area of (i) triangle ABC [ 2 ](ii) triangle CAD [ 1 ]giving your answers in exact form.

(c) Hence or otherwise, show that the shortest distance of D from AB can be expressed as

cm. [ 3 ]

10

In the diagram above, A, B, C and D are points on the circle. AT is a tangent to the circle at A.

BDT is a straight line such that BD : DT is 3 : 2.

AC meets BD at E such that BE : ED is 2 : 1.

(a) Show that ATD is similar to BTA. [ 2 ]

(b) Prove that . [ 2 ]

(c) Prove that . [ 4 ]

11


A

12 cm

A T

D

E

B

C

7

In the diagram, AOB and ABC are both right-angled triangles. The sides AB and BC are of lengths 12 cm and 5 cm respectively. FC is a line parallel to OB. D is a point on FC such that DB is perpendicular to FC. OAB = FCB = and .

(a) Show that AF = .

Express AF in the form where R > 0 and . [ 4 ]

(b) Given that FC is , show that area of triangle AFC is

[ 2 ]

(c) Hence find the maximum value of the area of triangle AFC and the corresponding value of when it occurs. [ 2 ]

12 The table below shows experimental values of two variables x and y.

x 3 6 9 12 15

y 2.09 2.31 2.46 2.58 2.68

It is known that x and y are related by the equation , where p and q are constants.

(i) Plot against [ 5 ]

(ii) Use your graph to estimate the value of p and of q . [ 3 ]

(iii) By drawing a suitable straight line, find the value of x which satisfies the equation . [ 3 ]

End-of-Paper


x

y

O

Q (x, y)

R P (4, 0)

53 xy

C

BO

DF

5 cm

y

x

A(3, 1)

X (1, 5)

B

C (9, k)

O

D (5, 3)

8

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

Answers

1 (a) (b) 17(c)

2 (a) Equation is

3 (a) p = 4, q = 3

(b) Remaining factors are

4 (a) (i) or

(ii)

(b) No of solutions = 1

5 R = = 83.1, 212.96 (b) a = 5, b = 2, c = 37 (a) 1.52 (b)

8 (a) (b) (0,0) is a minimum point. is a maximum point.

9 (a)

10(b) units/s

11 (a) k = 5 (b) B = (2, 6) (c) 24 units2 ; 71.6


DC

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2010AMathsPrelimP2