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2014-2-SGOR-BANDARPuchong_MATHS QA kuyuwah Section A [45 marks]
Answer all questions in this section
1. The function 𝑓 is defined by
𝑓(𝑥) = {2|𝑥|−𝑥
𝑥, 𝑥 ≠ 0
1 , 𝑥 = 0
Determine whether )(lim0
xfx
exists [5 marks]
2. The function 𝑓 is defined by
𝑓(𝑥) =1−4𝑒2𝑥
1+4𝑒2𝑥 , where 𝑥 ∈ 𝑅
(a) Find 𝑓′(𝑥) and determine whether 𝑓 is a decreasing or an increasing function. [5 marks]
(b) Determine the )(lim xfx
. [2 marks]
3.
The diagram shows the curve 𝑦 = 𝑥2 ln 𝑥 and its minimum point 𝑀.
(a) Find the exact values of the coordinates of 𝑀. [5 marks] (b) Find the exact value of the area of the shaded region bounded by the curve, the x-axis
and the line 𝑥 = 𝑒. [5 marks]
4. Show that 𝑒∫ tan 𝑥 𝑑𝑥 = sec 𝑥. [3 marks]
Hence, find the particular solution of the differential equation
cot 𝑥𝑑𝑦
𝑑𝑥+ 𝑦 =
𝑐𝑜𝑠2𝑥
sin 𝑥 , which satisfy the condition 𝑦 = 2 when 𝑥 = 0.
Give your answer in the form 𝑦 = 𝑓(𝑥) [5 marks]
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5. If 𝑦 = 𝑡𝑎𝑛−1𝑥, show that
𝑑2𝑦
𝑑𝑥2 + 2𝑥 (𝑑𝑦
𝑑𝑥)
2
= 0 and 𝑑3𝑦
𝑑𝑥3 + 4𝑥 (𝑑𝑦
𝑑𝑥) (
𝑑2𝑦
𝑑𝑥2) + 2 (𝑑𝑦
𝑑𝑥)
2
= 0 [5 marks]
Using Maclaurin’s Theorem, express 𝑡𝑎𝑛−1𝑥 as a series of ascending powers of 𝑥 up to the
term in 𝑥3. [4 marks]
6. Show that the equation 𝑥3 + 7𝑥 − 1 = 0 has a real root in the interval [0,1].
Show also that this equation can be rearranged in the form =1
𝑥2+7 . [3 marks]
Hence, use the iterative method to find this root correct to three decimal places, given that
𝑥0 = 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, 𝑥, at
𝑡 days is given by the differential equation 𝑑𝑥
𝑑𝑡= 𝑘𝑥(500 − 𝑥) where 𝑘 is a constant.
Assuming that no rabbits leave the farm during the outbreak, (a) show that
x=500
1+499𝑒−500𝑘𝑡 [8 marks]
(b) If it is found that, after two days, there are five infected rabbits, show that
𝑘 =1
1000 𝑙𝑛
499
99 [3 marks]
(c) determine the number of days before culling will be launched. [4 marks]
8. Given that 𝑦 = 3𝑥, find 𝑑𝑦
𝑑𝑥 in term of 𝑥. [3 marks]
(a) (i) Find the exact value of ∫ 3𝑥2
0 𝑑𝑥 [2 marks]
(ii) Use the trapezium rule with 5 ordinates, to find, in surd form, an approximate value of
∫ 3𝑥2
0 𝑑𝑥.
State a reason why the approximated value is greater than the true value of the definite integral. [5 marks]
(b) Given that the equation 𝑥(3𝑥) = 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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