Ujian Penilaian Pra Pentaksiran

Semester 2, 2014

SMK Dato’ Jaafar, Johor Bahru

MATHEMATICS (T)

PAPER 2, 954/2 One and a half hours

Instuctions to candidates:

DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO

SO.

1) Answer all questions in Section A and any one question in Section B.

2) All necessary workings should be shown clearly.

3) Scientific calculators may be used. Programmable and graphic calculators are

prohibited.

4) The use of graphing utilities in the handset are prohibited.

Section A ( 45 marks )

1. (a) Find 2

3

2

1

xdx

x−∫ . [ 2 marks ]

(b) The formula for the volume of a sphere of radius r and its surface area are 34

3V rπ=

and 24A rπ= respectively.

Given that when r = 8m, the volume V is increasing at a rate of 10m3s

-1, find the rate

of increase of the surface area A at this instant. [ 4 marks ]

(c) Find the area enclosed by the parabola y2 = 4x and the line y = 2x – 4 . [ 4 marks ]

2. Given y = x2 + 2 ln(xy) where x and y are positive variables.

Find the values of dy

dx and

2

2

d y

dx when both x and y are equal to 1. [ 7 marks ]

3. Find the values of constants A and B such that for all values of x

( )6 16 2 4x A x B+ = + +

Hence, find the exact value of

1

2

2

6 16

4 13

xdx

x x−

+

+ +∫ [ 7 marks ]

4. It is given that y = ( tan x + sec x )2. Prove that cos 2

dyx y

dx= . [ 2 marks ]

Find Maclaurin’s series for y up to and including the term in x3 [ 5 marks ]

5. Determine the vertical asymptote of the graph of the function ( )2 1

, 11

xf x x

x

+= ≠ −

+.

Find ( )limx

f x→±∞

and hence, determine the horizontal asymptote of the graph of the

function f.

Sketch the graph of the function f.

Find 1f

− . State the domain and the range of 1f

− . [ 9 marks ]

6. Find the equation of the tangent to the curve y = ex at the point where x = a. Hence, find

the equation of the tangent to the curve y = ex passes through the origin.

The straight line y = mx intersects the curve y = ex in two distinct points.

Write an inequality for m. [ 5 marks ]

Section B ( 15 marks )

7. The temperature of a body, measured in degrees, at t minutes, is x. Newton’s law of

cooling states that the rate of fall in the temperature of the body placed in a room with a

constant temperature of x0 where x0 < x, is proportional to the difference of the

temperature of the body over the room temperature. Write a differential equation that

connects the variables x and t. The room temperature x0 is 27°.

Given that x = 63 when t = 0, and x = 45 when t = 6 ln 2. Prove that

627 36

t

x e−

= + [ 10 marks ]

Find by giving your answers correct to 3 significant figures,

(a) the fall in the temperature of the body after being left in the room for 7 minutes,

[ 2 marks]

(b) the time that passes before the body cools down to a range of 2° from the room

temperature. [ 3 marks ]

8. The diagram shows part of a circle with its centre at the origin.

The curve has equation

225y x= − .

(a) Use the trapezium rule with 10 intervals to find an approximation to the area of the

shaded region. [ 5 marks ]

(b) Does the trapezium rule overestimate or underestimate the true area? [ 1 mark ]

(c) find the exact area of the shaded region (geometry formula is not allowed here )

[ 6 marks ]

(d) By comparing your answers to parts (a) and (b), obtain an estimate for π to

2 decimal places. [ 3 marks ]

6

5

4

3

2

1

-1

2 4 6

f x( ) = 25-x2