TRIAL STPM 2014 CONFIDENTIAL*

2014-2-JOHOR-SMKTunHababKotaTinggi_MATHS Q

SECTION A [45 marks]

Answer all questions in this section.

1. a) State the condition for function, f to be continues at a point x = c. [2 mark]

b) The function f is defined by

1

1

1

,

,

,1

)(

x

x

x

xeb

a

e

xf

x

, where a and b are positive constants.

Determine the exact values of a and b if f is continuous at x =1. [5 marks]

2. A curve has a equation x3 + xy + 2y3 = k where k is a constant.

a) Find dx

dyin terms of x and y. [3 marks]

b) If the tangent at a point on the curve is parallel to the y –axis. Show that the y-coordinate of

the point of contact with the curve satisfy the equation 216y6 + 4y3 + k = 0. Hence, find the possible

values of k in the case where the line x = -6 is a tangent to the curve. [5 marks]

3. Evaluate 2x

x

1dx. [3 marks]

Hence, find the exact value of 1

02sin–1 x dx. [5 marks]

4. Find the general solution of the differential equation 22 yydx

dyx . [6 marks]

5. By using Trapezium rule with 5 ordinates, estimate

4

0

ln( 6)x dx correct to 3 decimal places.[4 marks]

Explain, with the aid of a sketch, whether the Trapezium rule gives an over or under estimate of the

definite integral [3 marks]

6. Use standard Maclaurin Series to show that 2(1 2 )

ln1 3

x

x

= 2 31 43

72 3

x x x + … [4 marks]

State the range of values of x for which the series converges [2 marks]

Hence, estimate the value of

0.1 2

0

(1 2 )5ln

1 3

xdx

x

correct to three decimal places. [3 marks]

954/2[TRIAL]

*This question paper is CONFIDENTIAL until the examination is over. CONFIDENTIAL*

2014-2-JOHOR-SMKTunHababKotaTinggi_MATHS Q TRIAL STPM 2014

SECTION B [15 marks]

Answer one question only in this section.

7. a) Sketch on the same coordinate axes, the graphs of xy e and 2

1y

x

. [2 marks]

i) Verify that the equation (1 + x)ex – 2 = 0 has one real root. Show this root lies in the interval

( 0 , 1 ) [3 marks]

(ii) Use Newton-Raphson method with the initial estimate 0 0.5x to estimate the root correct to

3 decimal places [3 marks]

b) The graph of y = xe-x is shown in the diagram below.

i) Determine the coordinate A. [4 marks]

ii) The region R is bounded by the curve xxey , the positive x and y-axes, and the line x=a.

Find the area of R [3 marks]

8. Evaluate 𝑒∫45

2000−5𝑡𝑑𝑡

. [3 marks]

In an oil refinery, a storage tank contains 2000 m3 of gasoline that initially has 100 kg of an

additive dissolved in it. Starting from t = 0, gasoline containing 2 kg of additive per meter cube is

pumped into the tank at a rate of 40 m3 min-1. The well-mixed solution is pumped out at a rate of

45 m3min-1.

a) Let Q be the amount of additive in the tank at time t minutes. Show that the rate of change of Q

is given by 𝑑𝑄

𝑑𝑡 = 80 -

45𝑄

2000−5𝑡 [3 marks]

Hence, express Q in terms of t. [6 marks]

a) Calculate the concentration of the additive in the tank 20 min after the pumping process begins.

[3 marks]

954/2[TRIAL]

*This question paper is CONFIDENTIAL until the examination is over. CONFIDENTIAL*

y

x

R

a 0

A(a,b)