1. SEKOLAH MENENGAH KEBANGSAAN DATO’ JAAFAR
JOHOR BAHRU
MONTHLY TEST 1 (JULY 2012) DATE: 1/8/2012
SUBJECT: MATHEMATICS T (954) DAY: WEDNESDAY
CLASS: PRA-U-S1-BIO TIME: 10.30 – 12.00 pm
Answer all questions in Section A and any one question in Section B. Answer may be written in either
English or Bahasa Malaysia. All necessary working should be shown clearly.
Section A [45 marks]
Answer all questions in this section.
x 2 − 3x
1 Express in partial fractions. [5 marks]
x 2 − 2x +1
2 Express cos x + 3 sin x in the form R cos( x −α) where R is a positive and α is acute. Hence, solve
the equation cos x + 3 sin x = 2 for 0 ≤ x < 360  . [6 marks]
2 1
3 Given that log a ( 3x − 4a ) + log a 3 x = + log a (1 − 2a ) where 0 < a < , find x. [7 marks]
log 2 a 2
2 1
4 Find the set of values of x which satisfies +1 ≥ 4 − . [7 marks]
x 2−x
2t 1−t 2 θ
5 By using the substitution sin θ = and cosθ = , where that t = tan , show that
1+ t 2
1+ t 2
2
1 − sin θ 1 1 
= tan 2  π − θ  . [3 marks]
1 + sin θ 4 2 
1 1  0

Hence, or otherwise, by using the substitution θ = π , show that tan 22 2  = 2 −1 .
  [4 marks]
4  
6 The polynomial p( x ) = ax 3 − 8 x 2 + bx + 6 , where a and b are real constants, is divisible by x 2 − 2 x − 3 .
(a) Find a and b. [5 marks]
(b) For these values a and b, factorise p(x) completely. [2 marks]
(c) Show that 2 is a zero of the polynomial 3x − 14 x + 11x + 16 x − 12 . Hence, solve the equation
4 3 2
3 x 4 − 14 x 3 + 11x 2 + 16 x − 12 = 0 by using the polynomial p(x). [6 marks]
Section B [15 marks]
Answer any one question in this section.
1

2. 7 Function f and g are defined by
1
f ( x) = and g ( x ) = 2 x −1 .
x
(a) State the domain and range of f and g. [2 marks]
(b) Sketch the graph of f and g. [2 marks]
(c) Find the composite function f  g and its domain and range. [5 marks]
(d) Find the f − and g − . State also its domain and range for each function.
1 1
[6 marks]
8 The polynomial p( x ) = x 4 + ax 3 − 7 x 2 − 4ax + b has a factor x + 3 and, when divided by x − 3 , has
remainder 60. Find the values of a and b, and factorise p(x) completely. Sketch the graph of p(x).
[12 marks]
1
Using the substitution y = , solve the equation 12 y 4 − 8 y 3 − 7 y 2 + 2 y + 1 = 0 . [3 marks]
x
END OF QUESTIONS
2