Jurongville Prelim 2009 Am P1

JVSS AM Paper 1 Prelim 2009 1.

Find the range of values of x for which x 2  12  4 x  0 .

2.

Given that (225 p w ) 2  (243 p 3 )

3.

A cuboid has a square base of sides (3  2 2) cm and a volume of

3



2 3



[4]

3x5 y , evaluate w, x and y. p

[4]

(18  11 2) cm3 . Find, without using a calculator and giving your answer in

the form a  b 2 , where a and b are integers.

4.

5.

6.

(i) (ii)

the area of the square base of the cuboid, the height of the cuboid.

[2] [3]

(i) (ii)

Solve the equation 2 2 x  2  2 x  3 . Hence or otherwise, solve the equation 2 2 x  2  2  x  3 .

[4] [1]

 3 2  , find M-1 and hence, solve the simultaneous Given that M =   4 1   equations 3 y x3 2 4 x  y  14

x4 in partial fractions. (3x  5)( x  3)

(i)

Express

(ii)

Hence or otherwise, find the gradient of the curve y  at the point where x = 2.

3

[6]

[3]

x4 (3x  5)( x  3) [3]

JVSS AM Paper 1 Prelim 2009 7.

The figure below shows a square of sides 1 cm each. A point P on AB and a point Q on BC are such that QC = x cm and PB = kx cm, where k is a positive constant. (i)

Express the area of triangle OPQ in terms of k and x and show that it is equal to

(ii)

1 (kx2  x + 1) cm2 2

[2]

Find the minimum area of triangle OPQ in terms of k. A

P

kx cm

[4]

B

Q 1 cm

x cm

O

C 1 cm

8.

(i)

Find the term which is independent of x in the expansion of 16

1   2 x  6  . 2x   (ii)

[2]

Write down the first three terms of the expansion in ascending powers of x of (a)

 3x  1   2  

(b)

2  x 5

5

Hence, obtain the coefficient of x2 in the expansion of 5

3 2   2  2x  x  . 2  

[4]

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JVSS AM Paper 1 Prelim 2009

9.

x2  1 with respect to x. 2x  3

(i)

Differentiate

(ii)

Hence evaluate 

(i)

Differentiate 3 cos[( x 2  1)(3x  1) 2 ] with respect to x.

(ii)

Find the gradient of the curve y  ln(1  cos 3 x) where x 

(i)

Prove the identity 1 – cos2 + cos4 – cos6 = 4sin cos2 sin3. [4]

(ii)

Hence find all the angles  between 0 and  inclusive which satisfy the equation cos2 – cos4 + cos6 = 1. [4]

2

10.

11.

12.

x 2  3x

0 3  2 x  3

2

[2]

dx .

[5]

[4]

 . 2

[3]

The figure shows a trapezium ABCD in which AD is parallel to BC and AB is parallel to the x-axis. The point A, C and D are (4, 2), (7, 16) and (0, 10) respectively. The point X lies on BC such that AXB = 90. (i) (ii) (iii)

Find the equation of BC and of AX. Find the coordinates of B and X. Calculate the area of the trapezium ABCD.

y

[3] [3] [2]

C(7,16)

D(0,10)

X

B

A(4,2)

x

13.

(i)

The table below shows experimental values of two variables x and y. 5

JVSS AM Paper 1 Prelim 2009

x y

1 0

2 0.245

3 0.349

4 0.423

5 0.482

6 0.521

The two variables x and y are known to be related by the equation y  lg ax  b , where a and b are constants. Express the equation y  lg ax  b in a form suitable for drawing a straight line graph and explain how the graph can be drawn. Hence, by using suitable scales, draw the straight line graph. [4] (ii) (iii)

Use the graph to estimate the value of a and b. [2] If the x-coordinate of a point on the straight line is 4.5, evaluate the corresponding value of y. [2]

END OF PAPER

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JVSS AM Paper 1 Prelim 2009 Answer Key 1. x  6 or 2  x  3 1 2 2. x  6 , y  3 , w  3 3 3(i) 17  12 2 (ii) 42  29 2 4. -0.42, 0.42 5. x  3.09 ,  1.64 7 17 6(i)  (ii) 11 4( x  3) 4(3x  5) 4k  1 7(ii) 8k 15 45 2 8(i) 113.75 (ii)(a) 1  x  x  ... (b) 32  80 x  80 x 2  ... , 200 2 2 2 2( x  3 x  1) 2 9(i) (ii) 2 3 (2 x  3) 10(i) 6(1  3x)(6 x 2  x  3) sin[( x 2  1)(3x  1) 2 ] (ii) 3   2 3 11(ii) 0, , , , , 4 3 3 4 1 12(i) BC: y  2 x  30 AX: y  x (ii) B(14,2), X(12,6) (iii) 110 2 13(ii) a= 2.0, b = -1.0 (iii) y = 0.452

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