STPM 2021 Trial Exam Paper

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

I.C Number : …………………………………
MATHEMATICS (T)
954/1, FEBRUARY 2021

KOLEJ TINGKATAN ENAM SHAH ALAM
JALAN TIMUN 24/1

SHAH ALAM SELANGOR DARUL EHSAN

SEMESTER 1 TRIAL EXAMINATION STPM 2021

MATHEMATICS (T)

PAPER 1
DURATION: 1 HOUR 30 MINUTES

Instruction to candidates:

DO NOT OPEN THIS QUESTION PAPER UNTIL YOU ARE TOLD TO DO SO.

Answer all questions in Section A.

Answer one question only in Section B. You may answer
all questions but, only the first answer will be marked.

Write the answers on your examination test pad. Begin
each answer on a new page. All workings should be shown.

Non-exact numerical answers may be given correct to
three significant figures, or one decimal place in the case of
angles in degrees, unless a different level of accuracy is
specified in the question, Scientific calculators may be used.
Programmable and graphic display calculators are
prohibited.

Prepared by: Checked by: Validated by:
…………………………… …………………………… …………………………..
Baizura Mohamed Idris
Chin Li Mei Maziah Muhammad Zain
Head of Mathematics T Unit Science & Mathematics Head

1

SECTION A (45 MARKS)
Answer all questions.

1. Sketch the graphs of y= ⃒1 − 2x ⃒ and y = 1 on the same axes. Hence, solve the


following inequality.

⃒1 − 2 ⃒ > 1 [8 marks]




2. If is small enough for powers of higher than the third to be neglected, show that

( + 1 − ( − 1 = 1 ( + 3
8 3)
)2 )2 2

Use this result to deduce a rational approximation to √6 − √2 . Deduce a rational

approximation to √6 + √2 . [7 marks]

2 13 [5 marks]
3. A matrix M is given by (−1 0 4) .

3 10

Find the inverse of M by using elementary row operations method.

4. Plot on an Argand diagram the points A and B representing the complex numbers

1 = 1 + 2 and 2 = 2 + 3 respectively. Draw the triangle OAB.
If 3 = 1 + 3 , find the complex numbers (a) 1 3 and (b) 2 3 and show your

results as the points A’ and B’ on the Argand diagram. Draw the triangle OA’B’.
[9 marks]

2

5. The equation of an ellipse is given as
4 2 + 9 2 + 8 − 36 + 4 = 0

a. Obtain the standard form for the equation of the ellipse. [3 marks]

b. Find the coordinates of the centre C, the foci 1 and 2 of the ellipse.
[4 marks]

c. Sketch the ellipse, and indicate the points C, 1 and 2 on the ellipse. [2 marks]

6. Three vectors a = 2 + , = 5 + and = 3 + 8 are such that a and b are
perpendicular and the scalar product of a and c is −34 . Find

a. the values of m and n , [4 marks]
b. the angle between a and c . [3 marks]

3

SECTION B (15 MARKS)
Answer any one question in this section.

7. Let = ∑ =1 −1 , where ≠ 0 . Prove that

(1 − )
= 1 −

State the condition in order that lim exists and find this limit in terms of a and r.

→∞

[5 marks]

a. Find the smallest integer n so that

1 + 3 + (3)2 + ⋯ + (3) > 3.4 [5 marks]
44 4

b. Find the sum to infinity of

22(1 − )2 + 23(1 − )3 + 24(1 − )4 + ⋯ + 2 (1 − ) + ⋯

and determine the set of values of for this sum to be valid. [5 marks]

8. The position vectors of the points P, Q and R with respect to an origin O are respectively
given by,

p = 3i , q = 2j and r = 9k .

a. Find the exact value of the cosine of angle PRQ. [4 marks]

b. Calculate the area of the triangle PRQ. [3 marks]

c. Show that the volume of the tetrahedron OPQR is 1 ∣→ , (→ × →) ∣ , and
6


calculate its value. [6 marks]

d. Hence find the perpendicular distance from O to the plane PQR. [2 marks]

1

[The volume of a tetrahedron is 3 (base area)(perpendicular height).

4