2014 2 MELAKA MunshiAbdullah Maths QA
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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]
Transcript of 2014 2 MELAKA MunshiAbdullah Maths QA
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
