Smso 2007 Paper 1

SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG

SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG SEKOLAH MENENGAH SAYYIDINA ‘OTHMAN TUTONG

QUALIFYING EXAMINATION AUGUST 2007 UPPER SIXTH FORM MATHS PAPER 1 1 HOUR 45 MINUTES Instructions to students: This paper contains 10 questions. Please answer all the 10 questions in the answer booklet provided. The marks allocated for each questions are shown in the [ ] brackets. This paper has a total mark of 75. Calculator may be used in this paper.

NAME: _____________________________ DATE: _____________________________

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

x

 b  b 2  4ac . 2a

Sn =

n 2a  n  1d  2

Arithmetic series Un = a + (n – 1) d Geometric series Un = a r n – 1,

Sn =

Binomial Theorem





a 1 rn (r ≠ 1), 1 r

S 

a , (|r| < 1) 1 r

 n  n n (a  b) n  a n   a n1b   a n 2b 2  ...   a n r b r  ...  b n ,where n is a positive integer r  2 1 n n ! and    r!(n  r )! r (1  x) n  1  nx 

n(n  1) 2 n( n  1)(n  2) 3 x  x  ..., where n is rational and |x| < 1 2! 3!

TRIGONOMETRY Arc length of circle = rθ

(θ in radians) 2

Area of sector of circle = ½ r θ

tan  

(θ in radians)

sin  cos 

sin2 θ + cos2 θ = 1,

sec2 θ = 1 + tan2 θ,

cosec2 θ = 1 + cot2 θ.

VECTORS If a = a1i + a2j a3k and b = b1i + b2j b3k, then a.b = a1 b1 + a2b2 + a3b3 = |a||b| cos θ DIFFERENTIATION f (x)

f  (x)

xn ln x

nxn-1

ex sin x cos x tan x

ex cos x -sin x sec2x du dv u v dx dx du dv v u dx dx v2

uv

u v

1 x

20 . The straight line y = – 2x + 14 x intersects the curve at two points A and B.

1. The figure shows part of the curve y 

Show that the coordinates of point A and B are (2, 10) and (5, 4) respectively. [4] (ii) Find the volume obtained when the shaded region is rotated through 360° about the x - axis. [5]

(i)

2. The volume of an ice cube decreases at a constant rate of 3 cm3/s. Find the rate of change of its side (i) when its length is 4 cm, [3] (ii) when the volume of the cube is 27 cm3. [2]

3. (a) Show that the equation 15cos2θ = 13 + sin θ may be written as a quadratic equation in sin θ. Hence solve the equation, for 0° ≤ θ ≤ 360°.

[2] [4]

(b) (i) Sketch the graph of y = 2 cos 2x – 1, for 0 ≤ x ≤ 2π. (ii) Find the greatest value of y and of |y|.

[3] [2]

4. Find the first 4 terms in the expansion (2 – 3x)6 in ascending power of x. Hence, find the coefficient of x3 in (1 + 2x)(2 – 3x)6. [6]

5. In the diagram, ABC is an arc of a circle with centre O and radius 5 cm. The lines AD and CD are tangent to the circle at A and C respectively. Angle AOC = 2 . 3

(i) Show that the exact length of AD is 5 3 cm. (ii) Find the area of the sector AOC, giving your answer in terms of π. (iii)Calculate the area of the region enclosed by AD, DC and the arc ABC.

[2] [2] [3]

6. (a) How many terms of the AP, 1 + 3 + 5 + … are required to make a sum of 1521? [3] (b) The 6th term of a GP is 16 and the 3rd term is 2. Find the first term and the common ratio. [3]

7.

(i) Solve the simultaneous equations y = x2 – 3x + 2, y = 3x – 7 (ii) Interpret your solution to part (i) geometrically.

8. The points A, B, C have coordinates (3, -5), (4, -6), (11, 1) respectively. (a) Show that AB is perpendicular to BC. (b) Find the length of AC. (c) Find the perpendicular bisector of AC.

[3] [1]

[2] [2] [4]

9. (a) Given that the function f(x) = x2 – 2x + 5, for x ≥ 1, obtain f -1 (x), giving its domain and range. [5] (b) Function f and g are defined by f(x) = Find (i) g2 (ii) fg (iii) the value of x for which fg(x) = 3.

x , x  2 , and g(x) = 4x + 1. x2

[2] [3] [2]

10. In the diagram, OABCDEFG is a cube in which the length of each edge is 2 units. Unit vectors i, j, k are parallel to OA, OC , OD respectively. The mid-points of AB and FG are M and N respectively.

[3] (i) Express each of the vectors ON and MG in terms of i, j and k. (ii) Find the angle between the directions of ON and MG , correct to the nearest 0.1°. [4]