Mid Semester 2 STPM 2021

Name : ………………………………………………… Class : ………..…….…..

I.C Number : ……………………………........…………

MATHEMATICS T
PAPER 2
954/2
One and half hours

KOLEJ TINGKATAN ENAM SHAH ALAM

JALAN TIMUN 24/1
SHAH ALAM SELANGOR DARUL EHSAN

PEPERIKSAAN PERTENGAHAN SEMESTER 2 STPM 2021

MATHEMATICS T
954/2

PAPER 2

DURATION : 1 1 Hour
2

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE Section A
ALLOWED TO DO SO
1
Write your name, I.C number and class in the spaces
provided. 2
Answer all questions in Section A. 3
Answer one question only in Section B. 4
Write the answers on your examination test pad. 5
Begin each answer on a new page of the answer 6
sheet.
All working should be shown. Section B
Scientific calculators may be used. Programmable
7
and graphic display calculators are not allowed.
8
Answers may be written in English or Malay. TOTAL

Prepared by: Checked by: Validated by:

………………………………...… …………………………………… ………………………….…….….
Pn Chin Li Mei Puan Maziah Binti Muhamad Zain Puan Shirin Ahmad Sapiuddin
Head of Matematics T Unit Science and Mathematics Head of Senior Assistant
Department KTESA

1

Section A [45 marks]
Answer all questions

1. Given that f(x) = 3 2+ . Find lim ( ) and lim ( ) [6 marks]
3 2−8 −3 →−31 →∞
[5 marks]
2. A function f is defined as follow : [2 marks]

f(x) 2 +4 < −2 [3 marks]
2 2+9 +10 = −2 [5 marks]
[6 marks]
=2 > −2 [4 marks]

1−√ 2−3 [3 marks]

{ +2 [4 marks]

(a) Show that lim ( ) exists

→−2

(b) Determine whether f is continuous at x = −2

3. A curve is defined implicitly as y√ −√ – = 1

(a) Determine in terms of x and y


(b) Find the values of y and when x = 1


4. Given that y = cos − , show that 2 + 2y =0
+ 2


5. Show that = 1 [3√2 − 4]
10
∫04 2 3

6. (a) Expess 1 in partial fractions
2 2 − 7 + 6

Determine the value of p, where p >3 such that ∫3 1 dx = ln 9
2 2−7 +6 7

( ) Given y = 2x + 2 ln cosx . Show that = 2 3x .



Hence evaluate ∫04 3 [7 marks]

2

Section B [15 marks]
Answer one question only
You may answer all the questions but, only the first answer will be marked.

7. (a) The diagram below shows a semi ellipse, .

A container is formed by rotating the above semi ellipse through 2π radians about the y-axis. Initially, the
container is empty. Water is then poured into the container at a constant rate in 3 −1 Time taken to
fill up the container is 16 minutes.

(i) Find the volume of the water inside the container when the height of the water level from the base of the
container is h m, h ≤ 3.

(ii) Calculate the increasing rate of the volume of the water inside the container.

(iii) Find the rate of change of the water level inside the container at the instant when h = 2 m. [7 marks]

(b) A rectangle is inscribed in an ellipse with equation 2 + 2 = 1. The sides of the rectangle are
25 4

parallel to the axes. Show that the area of the rectangle is A = 8 x √25 − 2
5

Prove that the area is a maximum when x = 5√2 . Deduce the maximum value of the area of the
2

rectangle. [8 marks]

8. The equation of a curve is given by f (x) = −3
( −2)( +1)

(a) Find lim ( ) and state all the asymptotes. [3 marks]

→∞

(b) Find the coordinates of the stationary points on the curve and determine their nature. [9 marks]
Sketch the curve. [3 marks]

(c) Determine the set of values of k such that the equation x-3 = k(x-2)(x+1)
does not have any real roots.

3

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