Second Assessment STPM 2021

2020- Semester 1 Assessment
Section A [45 marks]

Answer all questions in this section.

1 Given that 34−2x = 53y+x and 3x− y  252 y−1 = 1, show that x2 − 3y2 − 6xy + 4x + 16y − 8 = 0 .
[4 marks]

2 Express 2 as partial fraction. [3 marks]
r2 −1

Hence, find a simple expression forS = n 2 and determine whether S converges when
r=2 r2 −1

n is large. [5 marks]

3. A system of linear equations is given by

2 + + = ,
2 − + 2 = 0,
2 + + = 3,

where and are real numbers. Write the augmented matrix for the system and show
that it may be reduced to

2 1 1 [4 marks]
(0 −2 1 | − )

0 0 − 1 3 −

Hence, determine the values of and so that the system of linear equation has

(a) a unique solution, [2 marks]
(b) infinitely many solutions, [2 marks]
(c) no solution. [2 marks]

4 The complex number u is defined by u = 5 , where the constant a is real.
a + 2i

(a) Express u in the form x + iy where x and y are real. [2 marks]

(b) Find the value of a for which arg (u*) = 3 , where * denotes the complex conjugate
4

of u. [5 marks]

5 The equation of a conic is 9x2 + 4y2 + 54 x + 32y + 109 = 0.

(a) Obtain its standard form and identify the conic. [4 marks]

(b) Find the coordinates of its centre, vertices and foci. Hence, sketch its graph.

[5 marks]

6 It is given that a = pi – j + 4k , b = qj + 3k and a x b = i – 6j – 2k, where p and q are

constants.

(a) Determines the values of p and q. [3 marks]

(b) Find (i) the area of the parallelogram with sides a and b. [2 marks]

(ii) the angle between a and b. [2 marks]

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

7 A polynomial p(x) = 4x4 + ax3 + bx2 + 8x − 3 has a factor x + 3 and gives a remainder

−10 when it is divided by 2x +1. Determine the values of the constants a and b.

[5 marks]

Show that 1 is the zero of p(x) and factorise completely p(x) . [5 marks]
2 [5 marks]

Hence, find the set of values of x for p(x)  (2x −1)(x + 3)(x − 2) .

 1 2 r −3  1  2r −1
 3   3 
8. (a) Given that ur = + .

(i) Express n in terms of A 1 − B  and determine the values of A and B.
9n 
 ur

r =1

[7 marks]

(ii) Determine the sum to infinity of this series. [2 marks]

(b) Evaluate 1 .

102 r
r =1

Hence, express 1.878787… (1. 8̇ 7̇ ) as a fraction in its lowest form. [6 marks]