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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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