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2014-2-PEARK-HuaLianTaiping_MATHS QA HanPohChoo

Section A [45 marks]

Answer all questions in this section

1. A function f is defined by

– 1

3 x – 1, x ≤ -3

f(x) = √𝑥 + 3 , -3 < x < 0

𝑥 + 2

𝑥−2 , x ≥ 0

i. Show that the f(x) is continuous at x = -3. [3marks]

ii Sketch the graph of y = f(x). [4marks] 2. Show that f(x) = x2 is differentiable at x = 0. [4marks]

3. Given that cos y = x, show that 𝑑𝑦

𝑑𝑥= −

1

√1−𝑥2.

Hence, show that ∫ 𝑐𝑜𝑠−1𝑥 𝑑𝑥 = 𝑥𝑐𝑜𝑠−1𝑥 − √1 − 𝑥2 + 𝐶. [6 marks]

4. A curve C is given parametrically by the equation tx 2 , 21 ty .

i. Show that the normal at the point with parameter t has equation 2tt2ty2x3

[4marks] ii. The normal at the point T, where t = 2, cuts C again at the point P, where t = p.

Show that 0184 2 pp and hence deduce the coordinates of P. [5marks]

5.(a) Solve each of the following differential equations yxdx

dyx

3

[4marks]

(b) During a cooling process, the rate of change of temperature of a hot object, satisfies the differential equation

where T is the temperature at time tin minutes, S is the surrounding temperature and k is a constant. A chicken is taken from the oven at 150oC and placed at room temperature of 30oC. In 2 minutes, the chicken’s temperature is 90oC. How long will it take the chicken to cool to 60oC? [6 marks]

6. Given that y = 1

1+sin 2𝑥, show that (1 + sin 2x)

2

2

x

y

d

d+ 4cos 2x

x

y

d

d- 4y sin 2x = 0 Hence,

Find find the first 3 terms of in Maclaurin’s series of y. [5 marks]

(a) Use the series expansion above, estimate the value of dxy1.0

1.0 correct to 4 decimal places.

[2 marks]

(b) find the first 2 terms of in Maclaurin’s series of 𝑑𝑦

𝑑𝑥. [2 marks]

)ST(kdt

dT

2

Section B [15 marks] Answer any one question in this section.

7. A curve C has equation 2

1

)4( 2 xy for 11 x . The regions R is enclosed by C, the x-

axis and the lines 1x and 1x (see diagram).

i. Find the exact value of the area of R. [5 marks] ii. Find the exact value of the volume generated when R is rotated through four right angles about

the x-axis. [5 marks] iii. Show that the volume generated when R is rotated through two right angles about the y-axis is

)324( . [5 marks]

8. (a) Show that the equation 𝑥3 − 5𝑥 + 1 = 0 has one real root which lies between 𝑥 = 0 and 𝑥 = 1 [2 marks]

The iterative formula , derived from the above equation can be written in the form

𝑥𝑛+1 = √5𝑥𝑛 − 13 or 𝑥𝑛+1 =

1

5(𝑥𝑛

3 + 1).

Using 𝑥0 = 0.5, show that only one of these formula will enable you to find this root

and determine this root correct to four decimal places. [6 marks]

(b) A quarter circle of radius 1 unit with equation y = √1 − 𝑥2 . Show that

𝑦 = ∫ √(1 − 𝑥21

0)𝑑𝑥 =

𝜋

4

Estimate the value of π correct to 1 decimal place using the trapezium rule with 4 intervals [7 marks]

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SMJK HUA LIAN TAIPING PERAK

MATHEMATICS T PAPER 2 TRIAL 2014

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