3 954/2/2014 SMK PBP1 kuyuhwah 2014-2-SGOR-BANDARPuchong_MATHS QA kuyuwah Section A [45 marks] Answer all questions in this section 1. The function is defined by () = { 2||β , β 0 1 , =0 Determine whether ) ( lim 0 x f xexists [5 marks] 2. The function is defined by () = 1β4 2 1+4 2 , where β (a) Find β² () and determine whether is a decreasing or an increasing function. [5 marks] (b) Determine the ) ( lim x f x . [2 marks] 3. The diagram shows the curve = 2 ln and its minimum point . (a) Find the exact values of the coordinates of . [5 marks] (b) Find the exact value of the area of the shaded region bounded by the curve, the x-axis and the line =. [5 marks] 4. Show that β« tan = sec . [3 marks] Hence, find the particular solution of the differential equation cot += 2 sin , which satisfy the condition =2 when = 0. Give your answer in the form = () [5 marks]
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3 954/2/2014 SMK PBP1
kuyuhwah
2014-2-SGOR-BANDARPuchong_MATHS QA kuyuwah Section A [45 marks]
Answer all questions in this section
1. The function π is defined by
π(π₯) = {2|π₯|βπ₯
π₯, π₯ β 0
1 , π₯ = 0
Determine whether )(lim0
xfx
exists [5 marks]
2. The function π is defined by
π(π₯) =1β4π2π₯
1+4π2π₯ , where π₯ β π
(a) Find πβ²(π₯) and determine whether π is a decreasing or an increasing function. [5 marks]
(b) Determine the )(lim xfx
. [2 marks]
3.
The diagram shows the curve π¦ = π₯2 ln π₯ and its minimum point π.
(a) Find the exact values of the coordinates of π. [5 marks] (b) Find the exact value of the area of the shaded region bounded by the curve, the x-axis
and the line π₯ = π. [5 marks]
4. Show that πβ« tan π₯ ππ₯ = sec π₯. [3 marks]
Hence, find the particular solution of the differential equation
cot π₯ππ¦
ππ₯+ π¦ =
πππ 2π₯
sin π₯ , which satisfy the condition π¦ = 2 when π₯ = 0.
Give your answer in the form π¦ = π(π₯) [5 marks]
4 954/2/2014 SMK PBP1
kuyuhwah
5. If π¦ = π‘ππβ1π₯, show that
π2π¦
ππ₯2 + 2π₯ (ππ¦
ππ₯)
2
= 0 and π3π¦
ππ₯3 + 4π₯ (ππ¦
ππ₯) (
π2π¦
ππ₯2) + 2 (ππ¦
ππ₯)
2
= 0 [5 marks]
Using Maclaurinβs Theorem, express π‘ππβ1π₯ as a series of ascending powers of π₯ up to the
term in π₯3. [4 marks]
6. Show that the equation π₯3 + 7π₯ β 1 = 0 has a real root in the interval [0,1].
Show also that this equation can be rearranged in the form =1
π₯2+7 . [3 marks]
Hence, use the iterative method to find this root correct to three decimal places, given that
π₯0 = 1 [3 marks]
Section B Answer any one question in this section
7. In a rabbit farm there are 500 rabbits and one rabbit is infected with Myxomatosis, a
devastating viral infection, in the month of April. The farm owner has decided to cull the rabbits if 20% of the population is infected. The rate of increase of the number of infected rabbits, π₯, at
π‘ days is given by the differential equation ππ₯
ππ‘= ππ₯(500 β π₯) where π is a constant.
Assuming that no rabbits leave the farm during the outbreak, (a) show that
x=500
1+499πβ500ππ‘ [8 marks]
(b) If it is found that, after two days, there are five infected rabbits, show that
π =1
1000 ππ
499
99 [3 marks]
(c) determine the number of days before culling will be launched. [4 marks]
8. Given that π¦ = 3π₯, find ππ¦
ππ₯ in term of π₯. [3 marks]
(a) (i) Find the exact value of β« 3π₯2
0 ππ₯ [2 marks]
(ii) Use the trapezium rule with 5 ordinates, to find, in surd form, an approximate value of
β« 3π₯2
0 ππ₯.
State a reason why the approximated value is greater than the true value of the definite integral. [5 marks]
(b) Given that the equation π₯(3π₯) = 2 has one real root and it lies in the interval [0,1].
Use the Newton-Raphson method with first approximation 0.8, find the root of the equation correct to three decimal places. [5 marks]
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