2014 2 PENANG SMJK Chung Ling BW MATHS QA
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2014-2-PENANG-SMJKChungLingBW_MATHS QA byOngLeeGhaik Section A [45 marks] Answer all questions in this section. 1. Given that . otherwise , 2 , 2 2 , 2 ) ( f x x x x Find ) ( f lim 2 x x , ) ( f lim 2 x x , ) ( f lim 2 x x and ) ( f lim 2 x x , determine whether f is continuous at x = – 2 and x = 2. [6] 2. A right pyramid has a square base of side x m and a total surface area 72 m 2 . Show that the volume V cm 3 is given by 4 2 2 4 144 x x V . [4] If x varies, find the value of x which V is a maximum and obtain the maximum value of V. [Volume of pyramid = 3 1 (base area x height) [6] 3. Show that the equation 0 5 2 3 x x has a root that lies between 2 and 3. [3] Show that the equation can be rearranged in the form 3 1 ) 5 2 ( x x . [1] Use an iterative method to find the root correct to three decimal places. [4] 4. Find the particular solution y in terms of x for the differential equation 3 2 1 2 3 ) 1 3 ( 2 d d ) 1 3 )( 1 2 ( 3 x y x x y x x given that y = 1 when x = 0. [9] 5. Using the Maclaurin series, evaluate ) 1 ( 2 sin 2 lim 5 . 0 2 0 x x e x x x . [6] 6. Using Trapezium rule with seven ordinates, find the value of 0 3 d ) 2 3 ( ln x x correct to three decimal places. [4] Determine whether the value obtained from the Trapezium rule is an underestimate or overestimate, give a reason. [2]
Transcript of 2014 2 PENANG SMJK Chung Ling BW MATHS QA
2014-2-PENANG-SMJKChungLingBW_MATHS QA byOngLeeGhaik
Section A [45 marks]
Answer all questions in this section.
1. Given that
.otherwise,2
,22,2)(f
x
xxx
Find )(flim2
xx
, )(flim2
xx
, )(flim2
xx
and )(flim2
xx
, determine whether f is continuous at
x = – 2 and x = 2. [6]
2. A right pyramid has a square base of side x m and a total surface area 72 m2. Show that the
volume V cm3 is given by 422 4144 xxV . [4]
If x varies, find the value of x which V is a maximum and obtain the maximum value of V.
[Volume of pyramid = 3
1(base area x height) [6]
3. Show that the equation 0523 xx has a root that lies between 2 and 3. [3]
Show that the equation can be rearranged in the form 3
1
)52( xx . [1]
Use an iterative method to find the root correct to three decimal places. [4]
4. Find the particular solution y in terms of x for the differential equation
3
2
123)13(2d
d)13)(12(3 xyx
x
yxx
given that y = 1 when x = 0. [9]
5. Using the Maclaurin series, evaluate )1(
2sin2lim
5.020
xx ex
xx. [6]
6. Using Trapezium rule with seven ordinates, find the value of 0
3d)23(ln xx correct to
three decimal places. [4]
Determine whether the value obtained from the Trapezium rule is an underestimate or
overestimate, give a reason. [2]
Section B [15 marks]
Answer any one question in this section.
7. Given that xy 1tanln
(a) Show that 0d
d12
dx
d1
2
22
x
yx
yx . [4]
(b) Using Maclaurin’s Theorem, show that the series expansion for y
is ...6
1
2
11 32 xxx . State the range of values of x for which the expansion of y is valid.
[8]
(c) Using the series expansion in (b), where x = 1, estimate the value of correct to three
decimal places. [3]
8. A research has been set up on an island to study a particular species of turtle. Initially,
there are 25 turtles on the island. After t years the number of turtles x satisfies the differential
equation )(20
1
d
dxkx
kt
x , where k is a constant.
(a) Show that k = 100 if it is known that the rate of growth is 0.45 turtle per year when
x =10. [1]
(b) What is the maximum rate of growth? [4]
(c) Obtain the solution of the differential equation and sketch the curve. [6]
(d) Find
(i) the number of turtles after 30 years. [2]
(ii) the time ( to nearest year) when the number of turtles is 50. [2]
