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2014 2 NSEMBILAN KGVseremban Maths QA

2014 2 NSEMBILAN KGVseremban Maths QA

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STPM TRIAL 2014 MATHEMATICS T 2
STPM TRIAL 2014 MATHEMATICS T 2

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08/22/2014

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2
2014-2-NSEMBILAN-KGVseremban_MATHS QA Rose
Section A
[45 marks]
all
questions in this section.
1.

Evaluate (a)

im
√ 
+−
.
[3 marks]
x→∞

[7.2/ III/ L] (b)

lim
√ 
+6−6
, [3 marks]

x→0

[7.2/ III/ H] 2.

Given that
 = 
 ,
find the value of


when
= 1. [6 marks] [8.1/ I/ L] 3.

y
=
+

4y = x Diagram 1 R x 0

Diagram 1 shows a portion of the curve
=
+

and the line 4y = x. By using the substitution
 = tan
, show that

+
 = ∫ 

√

and evaluate this integral. Hence, find the volume of revolution generated when region R is rotated completely about the x-axis. Give your answer correct to three significant figures. [11 marks] [9.2/ III/ H] 4.

Find the solution of the differential equation

= 
satisfying the condition y = 2 when x = 3. Give your answer in the form y = f(x). [5 marks] [10/ II/ M] 5.

Given that
 = √ 18
, show that

= 4
[2 marks]
By further differentiation of this result, or otherwise, show that Maclaurin’s series for y in ascending
powers of x is
3

 
7



[6 marks]
[11/ III/ L,M] 6.

By sketching the graphs of
= 
−
and
=
−
on the same diagram, show that the equation (3
–
x)
−

–
1 = 0 has only one real root. [4 marks] [12.1/ II/ L] Use the Newton-Raphson method, with the initial approximation
= 0.5, to find the root correct to three decimal places. [5 marks] [12.2/ III/ H]

3
Section B
[15 marks]
one
question in this section.
7.

Show that the curve
=

,
> 0
has a stationary point at
,

and determine whether this is a local maximum or minimum. [6 marks] Sketch the curve
=

,
> 0
[2 marks] [8.2/II/M] Show that the area of the region bounded by the curve
=

,
the x-axis and the line
=

,
is equal to the area of the region bounded by the curve
=

,
the x- axis and the line x = e. [7 marks] [9.2/II/M] 8

a. A piece of wire 8 cm is cut into two pieces. One of length x cm, is made into a circle. The other piece is bent to form a square. Find an expression for the total area of the circle and the square, in terms of r, where r is the radius of the circle. As x varies, find the radius of the circle which makes the total area a minimum. [8 marks] [8.2/II/M]  b. Use the trapezium rule with 5 ordinates to obtain an approximation of




is an over estimate or an under estimate. [3 marks] [12.2/II/M]

4
Maths T (Marking Scheme)
1(a). lim
√ 
+−

x→∞
= lim



M1
x→∞

= lim
+

−

M1
x→∞
= 1 A1 [3] (b). lim
√ 
+6−6

x→0

= lim
√ 
+6−6
.
√ 
+6+6√ 
+6+6

M1
x→0

= lim

+6−6
√ 
+6+6

M1
x→0
= lim
√ 
+6+6

x→0
=
A1 [3] 2.
 = 
=
ln
=
3ln  ln
M1

=



M1 y = 1,
ln = 1
M1
=

A1

=

(


)
M1 2

= 3




= 3
A1 [6] 3.
 = tan

 = 
 
M1 x = 0,
tan = 0

= 0
x =
√ 3
,
tan = √ 3

=
both correct B1

+
 = ∫



 
√

M1
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