2014-2-MELAKA-MunshiAbdullah_MATHS QA BY En. Tan Hun Kok

Section A (45 marks)

Answer all questions in this section

1. Sketch the graph of the function , where

( ) =

Find and .

Hence, show that is not continuous at point = 3. [5]

Determine whether is continuous at point = 0. [3]

2. A curve is defined by the parametric equations,

and .

If the normal to the curve at the point where meets the -axis at the point

(0, 7), show that . [6]

3. By mean of substitution , show that

. [5]

Hence or otherwise, find the exact value of . [4]

4. The current in an electric circuit at time satisfies the differential equation

.

Find in term of , given that = 2 when =0. [5]

State what happens to the current in this circuit for large values of . [1]

5. Given that , find in term of , and show that

. [5]

By differentiating this result, or otherwise, show that

Find the Maclaurin’s series of in ascending powers of up to and

including the term in . [5]

6. Use the trapezium rule with 5 ordinates, find an approximation for . Give

your answer correct to three significant figures. With the aid of a graph, explain why

the value obtained is bigger than the actual value. [6]

Section B (15 marks)

Answer any one question in this section

7. A function is defined by .

(a) Find the asymptotes and turning points of the curve. [5]

(b) Sketch the graph of . [3]

(c) Sketch the graph of and find the number of roots if [3]

(d) Given one of the root of lies in the interval [0,1]. By using Newton-

Raphson method with = 0.5, find the root correct to three decimal places. [4]

8. Water flows into a reservoir at a constant rate, . At the same time, water flows out at

a rate which is proportional to the depth of water in the reservoir. The depth of water

is m at time minutes. If the depth of the water reaches 0.5 m, it will remain at this

constant value. Show that

[3]

When = 0, the depth of water is 0.75 m and is decreasing at a rate of 0.11 ms-1.

(a) Find the time when the depth of water is 0.6m. [9]

(b) Find the depth of water at time = 10 minutes. [3]

SCHEME :

1. Sketch the graph of the function , where

( ) =

Find and .

Hence, show that is not continuous at point = 3. [5] Determine whether is continuous at point = 0. [3] Ans :

Graph D1, D1

( ) =

= = 0 A1

= = 2 A1

Since , does not exist,

is not continuous at = 3. A1 (5)

= =

= = M1

= ,

= 0. M1 (0) = 0

Since exist and is continuous at point x = 0. A1 (3) Total = 8 marks 2. A curve is defined by the parametric equations,

and .

If the normal to the curve at the point where meets the -axis at the point (0, 7), show

that . [6] Answer:

, B1

M1

=

m(normal) = M1

Eqn of normal is M1

At pt P(0,7) M1

A1 TOTAL = 6 marks

3. By mean of substitution , show that .

[5]

Hence or otherwise, find the exact value of . [4]

Answer:

= 8 – 8 cos 2 B1

M1

B1

M1

=

=

= A1 (5)

=

= M1

= 64 A1

= 64 M1

= 64

= A1(4)

Total = 9 marks 4. The current in an electric circuit at time satisfies the differential equation

. Find in term of , given that = 2 when =0. [5]

State what happens to the current in this circuit for large values of . [1] Answer:

M1

M1

M1

M1

A1 (5)

B1 (1)

Total = 6 marks

5. Given that , find in term of , and show that

. [5]

By differentiating this result, or otherwise, show that

Find the Maclaurin’s series of in ascending powers of up to and including the term in

. [5] Answer:

A1

M1

=

= A1

= M1

= = 0 A1 (5)

M1

A1

When x = 0, , , either B1

M1

= A1 (5)

Total = 10 marks

6. Use the trapezium rule with 5 ordinates, find an approximation for . Give your

answer correct to three significant figures. With the aid of a graph, explain why the value obtained is bigger than the actual value. [6] Answer:

0

tan 0 0.2679 0.5774 1 1.7321

M1

B1

= M1

= 0.710 A1 (graph) B1

The value obtained is bigger than the actual value because the graph is concave upwards / the trapezium is higher than the curve. A1 Total = 6 marks

7. A function is defined by .

Find the asymptotes and turning points of the curve. [5] Sketch the graph of . [3] Sketch the graph of and find the number of roots if [3] Given one of the root of lies between 0 and 1. By using Newton-Raphson method with = 0.5, find the root correct to three decimal places. [4] Answer :

Asymptotes : B1 B1

M1

=

M1

The turning point is (-1, A1

D1 any part correct D1 any second part correct D1 all correct (3)

D1 for D1 for no of roots = 3 A1 (3)

B1

M1

A1

A1 (4) Total = 15 marks 8. Water flows into a reservoir at a constant rate, . At the same time, water flows out at a rate which is proportional to the depth of water in the reservoir. The depth of water is m at time minutes. If the depth of the water reaches 0.5 m, it will remain at this constant value. Show that

[3]

When = 0, the depth of water is 0.75 m and is decreasing at a rate of 0.11 ms-1. Find the time when the depth of water is 0.6m. [9] Find the depth of water at time = 10 minutes. [3] Answer:

and

B1

Given ,

M1

A1 (3)

(a)

B1

M1

= 0, = 0.75 ,

M1

A1

Also, , M1

22 A1

A1 M1 A1 (9)

M1

M1 A1 (3)

Total = 15 marks