Prelim Amath (p2) 4e 2009

COMMONWEALTH SECONDARY SCHOOL PRELIMINARY EXAMINATION 2009 SECONDARY FOUR EXPRESS SECONDARY FIVE NORMAL ACADEMIC ADDITIONAL MATHEMATICS

4038/2

Paper 2

Time : 2

31 AUGUST 2009 Additional Materials : Answer Paper

1 hours 2

1030 – 1300

READ THESE INSTRUCTIONS FIRST Write your name, class and index number on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all questions. Write your answers on the separate answer 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 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 the 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 6 printed pages including this page.

CSS/PrelimExam 2009/Sec 4E&5N/AMath P2/Mrs M Loh/Page 1

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

−b ± b 2 − 4ac 2a

.

Binomial expansion  n  n −1  n  n −2 2  n  n −r r ( a + b) n = a n +  b + b + +  b + + b n , 1  a 2  a r  a       n 

n!

where n is a positive integer and  r   = r!(n − r )! =  

n(n −1)...( n − r +1) . r!

2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cos ec 2 A = 1 + cot 2 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 ) =

tan A ± tan B 1 tan A tan B

sin 2 A = 2 sin A cos A cos 2 A = cos 2 A − sin 2 A = 2 cos 2 A − 1 = 1 − 2 sin 2A

tan 2 A =

2 tan A 1 − tan 2 A

sin A + sin B = 2 sin

1 1 ( A + B ) cos ( A − B) 2 2

sin A − sin B = 2 cos

1 1 ( A + B ) sin ( A − B ) 2 2

1 1 ( A + B ) cos ( A − B ) 2 2 1 1 cos A − cos B = −2 sin ( A + B ) sin ( A − B ) 2 2 cos A + cos B = 2 cos

Formulae for ∆ ABC a b c = = sin A sin B sin C

.

a 2 = b 2 + c 2 − 2bc cos A 1 2

∆ = a bs inC

1.

(a)

The curve y = a – bcos4x where b > 0, is defined for 0 0 ≤ x ≤ 180 0 , has a maximum value of 5 and a minimum value of -1. CSS/PrelimExam 2009/Sec 4E&5N/AMath P2/Mrs M Loh/Page 2

(b)

2.

(i)

Find the value of a and of b.

[2]

(ii)

State the amplitude and period of the curve.

[2]

The volume of a right circular cone is 2 π m 3 . The radius of its base is ( 1 + 3 ) m. Find, without using a calculator, the height of the cone in the form x + y 3 , where x and y are integers. [3]

The roots of the equation x 2 - ( q – 5 )x =

q are 2

α and α + 5. Find the

possible values of q.

3. g( x ) = 3 -

2 x −4

[5]

is defined for the domain 0 ≤

x ≤ 5.

(a)

Sketch the graph of g( x ) and state the range of g corresponding to this domain. [3]

(b)

Find the set of values of

x for which g( x ) > 2.

[2]

4.

Express 2

6x + 4 in partial fractions. Hence show that (1 − 2 x)(1 + 3 x 2 )

6x + 4

∫ (1 − 2 x)(1 + 3x 1

2

)

d x = ln (

13 ). 36

[8]

5.

A particle moves in a straight line such that t seconds after passing through a fixed point O, its velocity, V ms −1 , is given by

1 − t  2 V =8  1 − e  

 .  

(i) [3]

Find the velocity when its acceleration is 2 ms −2 .

(ii)

Find the distance travelled during the fourth second of motion.

[3]

(iii)

Sketch the velocity-time graph for the motion of the particle.

[2]

CSS/PrelimExam 2009/Sec 4E&5N/AMath P2/Mrs M Loh/Page 3

6.

1 . 2

Sketch the graph of y = ln

(b)

Determine the equation of the straight line which would need to be drawn on the graph of y = ln 2 x −1 in order to obtain the graphical solution of the equation e 2 x + e 4 = 2xe 4 .

2 x −1

for

x >

(a)

[2]

7.

[3]

A

(a)

β

α

h

4

6

C D In the above triangle ABC, AD is the height of h units, BD = 4 units and DC = 6 units. Given that ∠ BAC = 45 0 , by considering tan( α + β ) or otherwise, find the value of h. [4] B

8.

(b) [3]

Given that sin( x+ 45 0 ) = cos x, evaluate tan x.

(a)

Show that

d (cos3x – 3 cos x) = 3 sin3x . dx 1

Hence, evaluate

∫ (3 sin

3

x − 2 sin x ) d x .

0.5

[5] (b)

Find the coordinates of the turning points on the curve y = cos x sin 2 x for 0 < x < π and determine the nature of the turning points. [7]

CSS/PrelimExam 2009/Sec 4E&5N/AMath P2/Mrs M Loh/Page 4

9. B

6 cm

P

Q

x0

A

B

C

Q

P

A

x0

Figure 1.

R

C Figure 2.

In figure 1, ∠ BAC = x ° and AB = 6 cm. P is the mid-point of AB, AC and PQ are each perpendicular to BC. ∆ PBQ is reflected along the line BC to form ∆ RBQ and the resulting shape is as shown in figure 2. (a)

Show that the perimeter, P cm , of ABRQC (figure 2) is given by P = 9 + 9 cos x° + 3 sin x° , [2]

(b)

(i)

Express 9 cos x° + 3 sin x° in the form Rcos( x 0 − α ) where R > 0 and α is acute. Hence, find the value of x when P = 15, [4]

(ii) Find the maximum value of P and the corresponding value of [3] (c)

Show that the area of ∆ RBQ is

9 sin 2 x° cm2 . 4

Find the ratio of the area of ABRQC : area of ∆ RBQ.

10. (a) form.

[4]

Differentiate with respect to x , leaving your answers in the simplest

(i)

(b)

x.

x ln(2 x 2 - 3)

(ii)

x2

Find the area enclosed by the curve y = -1 + lines

x =

1 and 2

[5]

1 + 2x

4 , the x2

x - axis and the

x = 5.

CSS/PrelimExam 2009/Sec 4E&5N/AMath P2/Mrs M Loh/Page 5

[5]

11.

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

ABCD is a rhombus where A and B are the points (–1, 0) and (2, 1) respectively. Given that DB is parallel to the line y – x = 3 and the centre of the rhombus lies on the y-axis, find (a) (b) (c)

the equation of AC the coordinates of C the area of the rhombus ABCD

(d)

the coordinates of a point E where point E lies on AC extended and is such that area of ∆ ABC is

[2] [3] [4]

1 of the area of ∆ ABE. 3

[3]

12.

In the diagram, P is a point of intersection of the two circles. The diameter MN of the larger circle is tangent to the smaller circle at point Q. PM and PN cut the smaller circle at points R and T respectively. If QT bisects ∠PQN, show that

(i)

PT = QT,

(ii)

QN NP = , NT NQ

P T

(iii) ∠PRQ = 2 ∠NQT, (iv) QM . PQ = RQ . MP

R M

Q

N[8]

CSS/PrelimExam 2009/Sec 4E&5N/AMath P2/Mrs M Loh/Page 6

END OF PAPER ANSWER KEY 1(a)(i) a = 2, b = 3 1(a)(ii) Amp = 3, Period = 90 ° 1(b) Height = [6 −3 3 ] m 2) q = 0 or 8 3(a) − 3 ≤ g ≤ 3 3(b) 1 2 1

< x < 2 12

4

6x

4) (1 − 2 x) + (1 + 3 x 2 ) 5(i) Velocity = 4m/s 5(ii) Distance travelled = 6.60m 6(b) Equation: y = x − 2 7(a) h = 34 7(b) tanx =

2 −1

8(a) -0.181 8(b) (0.955 , 0.385) is a maximum point; (2.19 , -0.385) is a minimum point 9(b)(i) R = 90 ; α = 18 .43 ; x = 69.2 9(b)(ii) max P = 18.5; x = 18.4 9(c) 5 : 1 2 10(a)(i) ln( 2 x − 3) +

10(a)(ii)

x(2 + 3x) (1 + 2 x)

10(b) Area = 2

4 2x − 3 2

3 2

7 sq units 10

11(a) Equation: y = −x −1 11(b) C = (1 , -2) 11(c) Area = 8 sq units 11(d) E = (5 , -6)

CSS/PrelimExam 2009/Sec 4E&5N/AMath P2/Mrs M Loh/Page 7