Section A [45 marks]

Answer all questions in this section.

| x2 −1| , x <1
1. The function f is defined by f ( x )=
{ ( x−1 )2 , x ≥1

(a) Find lim ¿ and lim ¿ [2 marks]

(b) Determine whether f is continuous at x = 1. [2 marks]

(c) Sketch the graph of y = f(x). [3 marks]

1 1
2. Given that x=t− , y=2t + , where t is a nonzero parameter.
t t
dy 3
Show that. =2− 2
dx t +1

dy
Deduce that −1< <2 . [5 marks]
dx

3. A region R is bounded by a curve xy = 12 and a line 3x + 4y = 30.

(a) Find the coordinates of the points of intersection of the curve xy = 12 and the line
3x + 4y = 30. [ 3 marks]
(b) Calculate the area of the region R. [ 3 marks]
(c) Calculate the volume of the solid of revolution formed when this region R is rotated
through 360o about the x-axis. [3 marks]

y dy 2 2
4. (a) Use the substitution u= to solve the differential equation 2 xy − y + x =0.
x dx
[4 marks]
dy
(b) Solve the differential equation + y tan x=sin2 x , given that y = 1 when x = 0.
dx
[5 marks]

5. By using the Maclaurin’s series, write the first three non-zero terms of the expansion of sin x 2
and cos 2 x. [2 marks]
Hence,
1
(a) Find an approximation for the value of the integral ∫ sin x 2 dx, giving your answer
0
correct to three decimal places. [4 marks]

sin x 2
(b) Find lim . [3 marks]
x →0 1−cos 2 x

1
 1
ex
6. Evaluate ∫ x
dx. [3 marks]
−1 e +1
ex
Diagram below shows part of the graph of y = ,−1< x <1.
ex+ 1
y

0.8

0.5

0.2
x
-1 0 1

1
ex
If trapezium rule with 5 ordinates is used to estimate ∫ x
dx, what would you expect the
−1 e +1
result to be?
Explain the above result. [3 marks]

Section B [15 marks]

Answer only one question in this section.

5−2 x
7. The equation of a curve is y= .
x 2−4
dy
Find the coordinates of the stationary points and use the sign of to determine their
dx
nature. [6
marks]
State all the equations of asymptotes and hence sketch the graph. [6 marks]
Determine the set of values of k for which the equation 5−2 x=k ( x2 −4 ) does not have any
real roots. [3 marks]

dy
8. (a) Given that y=3−x , show that =−3−x ln x
dx
[3 marks]

(b) On the same coordinates axes, sketch the curves y = 4 – x2 and y=3−x . [3 marks]

(c) Verify that the curves intersect at point A(-1, 3) and the curves also intersect at point B
in the first quadrant whose x-coordinates α is the positive root of the equation

2
 x 2+ 3− x −4=0 . [3 marks]

Verify that 1 < α < 2 by calculations. [2 marks]

(d) By taking 2 as the first approximation to α, use Newton-Raphson method to obtain α