2014-2-SELANGOR-SMKMethodist(ACS) Klang_MATHS QA byRabiahIdris

SECTION A [45 marks]

Answer all questions in this section.

1. The function f is defined by {

(a) Determine whether exist. Hence, determine if f is continuous at 3 [5m]

(b) Sketch the graph of f. [3m]

2. Given that

prove that

[5m]

3. Using integration by parts, show that ∫

[6m]

4. A swimming pool with a rectangular base and having vertical sides of height, h is initially full of

water. The cleaner drain out the water through the drain plug outlet. The water drains out from

the outlet hole in the horizontal has of the pool at a rate which at any instant, is proportional to

the square root of the depth of the water at that instant.

If x is the depth of the water, at time t after the drain started is represented by the equation

√ and the swimming pool is exactly half empty after an hour, find the further time

that elapse before the pool is completely empty. [9m]

5. Using Maclaurin theorem, find the expansion of up to the term in x3. [5m]

Hence, evaluate

[2m]

6. (a) Given ∫

∫ , determine the expression of . [3m]

(b)Diagram below shows parts of the curve

.

Given A is the area of the region bounded by the curve

, the x-axis, the line x = 0.5 and

the line 1.5, (i) write an expression of A as an integral in terms of x, [1m]

(ii) estimate the value of A to three decimal places by using five ordinates in trapezium rule, [4m]

(iii) determine whether the estimation by trapezium rule in part (ii) is an over estimate or under estimate.

[2m]

3 x

y

1

0

𝑦 𝑥

𝑥

SECTION B

[15 MARKS]

Answer any one question in this section.

7. The function f is defined by

.

(a) State all asymptotes of f. [2m]

(b) Find the stationary point of f, and determine its nature. [6m]

(c) Obtain the intervals, where

(i) f is concave upwards, and

(ii) f is concave downwards.

Hence, determine the coordinates of the point of inflexion. [5m]

(d) Sketch the graph of y=f(x). [2m]

8. A particle moves from rest along a horizontal straight line. At time t s, the displacement and

velocity of the particle are x m and v m/s, is given by

√

Express v and x in terms of. t. [7m]

Find the velocities of the particle when its acceleration is zero for the first and second times. Find

also the distance travelled by the particle between the first and second times its acceleration is

zero. [8m]

ANSWER SCHEME Mathematics T(STPM 2014-TERM 2 TRIAL EXAMINATION)

1. The function f is defined by {

(c) Determine whether exist. Hence, determine if f is continuous at 3 [5m]

(d) Sketch the graph of f. [3m]

Answer: (a)

m1(choose the correct function)

Since M1

exists A1

f(3)=5 , f is not continuous at x=3. M1A1

(b)

2. Given that

prove that

[5m]

Answer:

M1A1(M1-using product rule)A1-all correct)

M1

B1(subs )

A1

3. Using integration by parts, show that ∫

[6m]

∫

∫

M1A1(M1-using int. by parts, A1All correct)

∫ M1

*

+

A1

⌊ ⌋ M1(Substitution)

A1

3 -1

8

5

x

y

D1(2 graphs)

D1(2 points)

D1(labeling) ●

○

4. A swimming pool with a rectangular base and having vertical sides of height, h is initially full of

water. The cleaner drain out the water through the drain plug outlet. The water drains out from

the outlet hole in the horizontal has of the pool at a rate which at any instant, is proportional to

the square root of the depth of the water at that instant.

If x is the depth of the water, at time t after the drain started is represented by the equation

√ and the swimming pool is exactly half empty after an hour, find the further time

that elapse before the pool is completely empty. [9m]

Answer:

√ , ∫

∫ M1(separate the variables)

A1

√ M1

√ √ A1

√

√ M1(substitution)

√

√ √ √ √ A1

√ ( √ )√ √

√

√ √ M1

Elapse time=3h25m-1h=2h25m M1A1

5. Using Maclaurin theorem, find the expansion of up to the term in x3. [5m]

Hence, evaluate

[2m]

Answer:

M1(1

st and 2

nd derivatives)

M1(subs x=0)

A1(for all f(0),f’(0), f’’(0),f’’’(0))

M1(subs f(0),f’(0), f’’(0),f’’’(0))

A1

=

=

M1

=

A1

6. (a) Given ∫

∫ , determine the expression of . [3m]

(b)Diagram below shows parts of the curve

.

Given A is the area of the region bounded by the curve

, the x-axis, the line x = 0.5 and

the line 1.5,

(i) write an expression of A as an integral in terms of x, [1m]

(ii) estimate the value of A to three decimal places by using five ordinates in trapezium rule,

[4m]

(iii) determine whether the estimation by trapezium rule in part (ii) is an over estimate or

under estimate. [2m]

Answer: ∫

∫ M1A1

∫

By comparing,

A1

(i) ∫

dx B1

x y y

0.5 0.95885

0.75 0.90885

1 0.84147

1.25 0.75919

1.5 0.6650

Total 1.62385 2.5095

d=(1.5-0.5)/4 =0.25 B1

(ii)

[ ] M1A1

(3 d. p.) A1

(iii) The estimation is under estimate because the curve is concaved downward. M1A1

3 x

y

1

0

𝑦 𝑥

𝑥

7. The function f is defined by

.

(e) State all asymptotes of f. [2m]

(f) Find the stationary point of f, and determine its nature. [6m]

(g) Obtain the intervals, where

(iii) f is concave upwards, and

(iv) f is concave downwards.

Hence, determine the coordinates of the point of inflexion. [5m]

(h) Sketch the graph of y=f(x). [2m]

Answer:

(a) Asymptotes are x=0. and y=0. B1B1

(b)

M1A1

M1

A1

(

)

M1

(

) is a maximum turning point. A1

(c) M1

A1

M1

f’’(x) - 0 +

f is concaved upwards in the interval (4.48, ∞).

f is concaved downwards in the interval (0, 4.48) A1(for both)

Inflexion point is (

) B1

8. A particle moves from rest along a horizontal straight line. At time t s, the displacement and

velocity of the particle are x m and v m/s, is given by

√

Express v and x in terms of. t. [7m]

Find the velocities of the particle when its acceleration is zero for the first and second times. Find

also the distance travelled by the particle between the first and second times its acceleration is

zero. [8m]

Answer:

∫ ∫( √ )

√

M1

M1

[ √ ] A1

∫

∫[ √ ] B1

*

√

+ M1

(

√

)

√

M1

*

√

+

√

A1 (Total=7m)

When acceleration=0,

√

√

M1A1

√

B1

√

B1

√

√

√

B1

√

√

√

B1

The distance travelled= √

√

M1A1 (Total=8m)

•(2.7183, 0.3679)

•(4.48, 0.3347)

•

0.5 0

D1

D1

(d)