2014-2-SGOR-LaSallePJ_MATHS Q ByPnTeohAC

Section A

[45 marks]

Answer all questions in this section.

1. The function f is defined by

(a) Find and (2)

(b) Determine whether f is continuous at x = 1. (2) (c) Sketch the graph of y = f(x). (3)

2. A spherical ballon is inflated by gas such that its volume is increasing at a constant rate. Show that the rate of increase of the surface area of the ballon is inversely proportional to its

radius. (Volume of sphere = , surface area of sphere = (5)

3. Find the point of intersection of the curve and the line Hence show that the

area bounded by the curve, and the y-axis is Calculate also the volume of the solid

formed when the bounded area is rotated about the x-axis. (9)

4. In an experiment, the number of fish in a pond at time t (measured in days) can be expressed

by the differential equation Initially there are 50 fish in the pond. The pond

can only accommodate a maximum of 400 fish. Find the number of fish in the pond after 5

days. (7)

5. It is given that Show that

(i) (1)

(ii) (3)

Find the Maclaurin series for y, up to and including the term in (5)

Hence, evaluate (2)

6. Use the trapezium rule with six ordinates to evaluate to 4 decimal places. With the

aid of a sketch graph, explain whether the answer obtained overestimates or underestimates

the exact value. (6)

Section B

[15 marks]

Answer any one question in this section.

7. A curve is defined as

(a) Find the equations of the asymptotes of the curve. (4)

(b) Find the turning point of the curve and determine its nature. (5)

(c) Show that the curve does not exist for (4)

(d) Hence, sketch the curve. (2)

8. By sketching suitable graphs on the same axes, or otherwise, show that the equation

has exactly one positive root, and prove that this root lies between 0 and

1.(5)

(a) If is to be found using an iterative formula show that the equation

can be arranged in the form and

(3)

(b) By using as an initial approximation, show that the iterative formula

is not suitable for finding .

(3)

(c) Using the iterative formula with as an initial approximation, find

correct to 3 decimal places. (4)