MTH111 Multiple choice Questions

1) In the process of dividing the polynomials (c) 𝑥12 − 4096 = 2

𝑃(𝑥)𝑎𝑛𝑑 𝑄(𝑥), we usually arrived at a (d) 𝑥 + 2 = 2
𝑟(𝑥)
results form 𝑞(𝑥) + 𝑄(𝑥). where 𝑞(𝑥) is 8) The remainder when 𝑃(𝑥) = 2𝑥 4 + 𝑘𝑥 3 −
11𝑥 2 + 4𝑥 + 12 is divided by (𝑥 − 3) is 60.
called …
Find the value of k.
(a) Quotient
(a) 3
(b) Remainder
(b) 2
(c) Divisor
(c) 1
(d) Dividend.
(d) -1
2) If 𝑃(𝑐) = 0, 𝑡ℎ𝑒𝑛 𝐶 is called …. Or ….. of
9) The differences between remainder theorem
the polynomial 𝑃(𝑥).
and the factor theorem are … and ….
(a) Zero or the root
Respectively.
(b) Factor or multiple
(a) 𝑷(𝒙) = 𝒓 𝒂𝒏𝒅 𝑷(𝒄) = 𝟎
(c) Leading or Monic
(b) 𝑃(𝑐) = 0 𝑎𝑛𝑑 𝑃(𝑐) = 𝑟
(d) Factor or leading
(c) 𝑃(𝑥) = 𝑥 − 𝑐 𝑎𝑛𝑑 𝑄(𝑥) = 𝑥 − 𝑐
3) When 𝑃(𝑥) is divided by 𝑄(𝑥), the degree
(d) all of the above.
of the remainder must be …
10) A polynomial 𝑃(𝑥) has a factor (𝑥 − 𝑐) if
(a) Less than the 𝑸(𝒙)
and only if 𝑃(𝑐) equals to …
(b) Greater than the degree of Q(𝑥)
(a) c
(c) Equal to the degree of 𝑃(𝑥)
(b) r
(d) All of the above.
(c) 0 (zero)
4) The coefficient of the term with the highest
(d) 4.
power of the unknown is called ...
11) In the process of long division, we write
(a) Monomial
zero (0) coefficient when any ………… is
(b) Leading coefficient
missing.
(c) Linear
(a) Monomial
(d) Cubic
(b) Leading coefficient
5) If 𝑃𝑛 (𝑥)has degree n and 𝑄𝑛 (𝑥) has degree
(c) Term
m, then their product has degree?
(d) Divisor.
(a) 𝒎 + 𝒏
12) When you divide the polynomial:
(b) 𝑚 − 𝑛
𝑃(𝑥) = 3𝑥 3 − 2𝑥 2 + 5𝑥 − 7by:
(c) 𝑚𝑛
𝑄(𝑥) = 𝑥 − 2,the remainder is …
(d) 𝑚2 𝑛
(a) 19
6) The process of carrying out the division for
(b) 9
polynomials is called ….
(c) 18
(a) Remainder theorem
(d) 𝑥 + 2
(b) Indices
13) When the polynomial 𝑃(𝑥) is divided by
(c) Division algorithm
(𝑥 − 𝑐), the remainder is ….
(d) Factor theorem
(a) 𝑷(𝒄)
7) When is (𝑥 + 2) a factor of 𝑥12 − 4096?
(b) 𝑥 − 𝑐
(a) 𝒙 = −𝟐 𝑝(𝑥)
(b) 𝑥12 − 4096 = 0 (c) 𝑥−𝑐
 (d) 𝑃(𝑥). (b) Monic polynomial
14) In the process of synthetic division, the (c) Cubic polynomial
divisor must be ….. (d) None of the above.
(a) Linear 21) A polynomial of degree five is called ….
(b) Cubic (a) Quadratic
(c) Quartic (b) Cubic
(d) Quadratic (c) Quartic
15) When is (𝑥 + 4) a factor of the polynomial (d) Linear.
𝑃(𝑥) = 𝑥 3 + 𝑥 2 + 𝑎𝑥 + 8? 22) The expression (3𝑥 2 + 5𝑥 − 12) − 2(𝑥 2 +
(a) 𝒂 = −𝟏𝟎 4𝑥 + 9) is equivalent to which of the
(b) 𝑥 = 10 following:
(c) 𝑥 = −10 (a) 𝒙𝟐 − 𝟑𝒙 − 𝟑𝟎
(d) 𝑎 = 0 (b) 𝑥 2 + 13𝑥 + 6
16) Given that 𝑃(𝑥) = 10𝑥 4 + 5𝑥 3 + 3𝑥 2 + (c) 5𝑥 2 + 𝑥 − 18
7𝑥& 𝑄(𝑥) = 2𝑥 5 − 4𝑥 4 − 2𝑥 3 + 5𝑥 2 + 10𝑥 − (d) 𝑥 2 + 3𝑥 − 21𝑠
5. Find 𝑃(𝑥) + 𝑄(𝑥). 23) Rational function is a quotient of two ……..
(a) 𝟐𝒙𝟓 + 𝟔𝒙𝟒 + 𝟑𝒙𝟑 + 𝟖𝒙𝟐 + 𝟏𝟕𝒙 − 𝟓 functions
(b) 2𝑥 5 − 6𝑥 4 − 2𝑥 3 + 5𝑥 2 + 10𝑥 − 5 (a) Linear
(c) 2𝑥 5 − 4𝑥 4 + 5𝑥 2 + 10𝑥 − 5 (b) Polynomial
(d) 2𝑥 5 − 4𝑥 4 − 2𝑥 3 + 5𝑥 2 − 5 (c) Quadratic
17) Given that 𝑃(𝑥) = 𝑥 3 + 4𝑥 2 + (d) Cubic
5𝑥and 𝑄(𝑥) = 2𝑥 3 − 1. Find 𝑃(𝑥) × 𝑄(𝑥). 24) Under which condition could we say
𝑔(𝑥)
(a) 𝑥 − 1 𝑓(𝑥) = ℎ(𝑥) is rational function?
(b) 2𝑥 6 − 3𝑥 4 + 2𝑥 3 (a) If and only ifℎ(𝑥) = 0
(c) 𝟐𝒙𝟔 + 𝟖𝒙𝟓 + 𝟏𝟎𝒙𝟒 − 𝒙𝟑 − 𝟒𝒙𝟐 − 𝟓𝒙 (b) If and only if𝑔(𝑥) ≠ 0
(d) 2𝑥 5 + 8𝑥 (c) If and only if 𝑓(𝑥) ≠ 0
18) The remainder when 𝑥 3 + 𝑎𝑥 2 + 𝑏𝑥 + 1 is (d) If and only if𝒉(𝒙) ≠ 𝟎
divided by (𝑥 − 1) is 7.When divided by 25) For what valueof 𝑥 could the rational
(𝑥 − 2), the remainder is 39. Find 𝑎 𝑎𝑛𝑑 𝑏. 𝑥 2 −𝑥
function, 𝑓(𝑥) = 16−𝑥 2 be said to be
(a) (−5, 10)
(b) (𝟏𝟎, −𝟓) undefined?
(c) (15, 5) (a) If |𝒙| = 𝟒
(d) (−5, 15) (b) If 𝑥 = 4
19) Find the values of 𝑘 for which 𝑃(𝑥) = 𝑥 5 + (c) If |𝑥| > 4
(d) If 𝑥 < 4
𝑘𝑥 4 − 2𝑥 + 1 leaves a remainder 5 when
26) What is the domain of the rational function
divided by (𝑥 + 2). 𝑡+2
(a) 2 𝑓(𝑡) = 𝑡−7?
(b) 5 (a) 𝒕 ≠ 𝟕 (i.e. all real values except 𝟕)
(c) -27 (b) 𝑡 = 7
(d) -2 (c) 𝑡 > 7
20) A polynomial whose leading coefficient is (d) 𝑡 < 7
one is called… 27) What is the quotient, when the polynomial
(a) Leading polynomial 3𝑥 2 + 17𝑥 + 20 is divided by 𝑥 + 4?
 𝑥+6
(a) 3𝑥 + 12𝑥 a) 𝑥+2
(b) 5𝑥 − 20 𝑥+2
b)
(c) 3𝑥 2 + 12𝑥 𝑥+5
𝒙+𝟔
(d) 𝟑𝒙 + 𝟓 c) 𝒙+𝟓
𝒙𝟑 +𝒙𝟐 +𝟒𝒙 𝑥+5
28) If is expressed as a quotient together d)
𝒙𝟐 +𝒙−𝟐 𝑥+6
with a fraction whose numeration is of lower 1 2 4
32) What is:𝑥−1 − 𝑥+5 + 𝑥−1 , in a single rational
degree than denominator, it gives?
6𝑥
function?
a) 𝑥 − 𝑥 2 −𝑥+2 2𝑥+26
a)
6𝑥 (𝑥−1)(𝑥+5)
b) 𝑥 + 𝑥 2 −𝑥+2 𝟑𝒙+𝟐𝟕
b)
𝟔𝒙 (𝒙−𝟏)(𝒙+𝟓)
c) 𝒙 + 𝒙𝟐 +𝒙−𝟐 2𝑥+27
6𝑥
c)
(𝑥−1)(𝑥+5)
d) 𝑥 − 𝑥 2 +𝑥−2 3𝑥+26
d)(𝑥−1)(𝑥+5)
 Fig. 1

 Info 1:
TheSysthetic division approach of:
2𝑥 4 − 10𝑥 2 − 23𝑥 + 6
gives:
𝑥−3
3
√2 0 −10 −23 6
6 18 24 3
---------------------------------
2 6 8 1 9
Use the information in the last row of the info1
to answer Questions 33 to 36
33) What is the Quotient of the division process in
29) What is the domain of the rational function Info 1?
represented by the graph in Fig 1? a) 2𝑥 4 + 6𝑥 3 + 8𝑥 + 1
(a) 𝑥 = 2 b) 2𝑥 4 + 6𝑥 3 + 8𝑥 2 + 𝑥 + 9
(b) 𝒙 ≠ 𝟐 c) 𝟐𝒙𝟑 + 𝟔𝒙𝟐 + 𝟖𝒙 + 𝟏
(c) 𝑥 = 1.5 d) 𝑥 − 3
(d) 𝑥 = 2.5 34) What is the remainder of the division process
(e) 𝑥 ≠ 2.5 in Info 1?
30) The fraction so formed (if the reminder is a) 6
placed over the divisor) in the division of b) 8
10𝑥 2 + 7𝑥 − 19by2𝑥 + 3 is: c) 1
𝟕 d) 9
a) 𝟐𝒙+𝟑
5𝑥−4
b) 35) What fraction is formed, if the remainder is
2𝑥+3
7
c) placed upon the divisor, in the process in
5𝑥−4
5𝑥−4 Info 1?
d) 2
7
a)
𝑥 2 +8𝑥+12 𝑥−3
31) The simplified form of 𝑥 2 +7𝑥+10 is: 𝟗
b) 𝒙−𝟑
 6 b) 2
c) 𝑥−3
1 c) -1
d) 𝑥−3 d) -4
36) What is the result of the division process? 1
41) How can be partitioned into
8 𝑥 3 + 𝑥2 + 𝑥 + 1
a) 2𝑥 4 + 6𝑥 3 + 8𝑥 + 1 + 𝑥−3
partial fraction decomposition?
𝟑 𝟐 𝟗 𝐴 𝐵 𝐶
b) 𝟐𝒙 + 𝟔𝒙 + 𝟖𝒙 + 𝟏 + 𝒙−𝟑 a) − 𝑥 2 + 𝑥 2 −2
𝑥
8
c) 2𝑥 4 + 6𝑥 3 + 8𝑥 + 1 − 𝑥−3 b)
𝐴 𝐵
+ 𝑥 2 + 𝑥 2 −2
𝐶
𝑥
9
d) 2𝑥 3 + 6𝑥 2 + 8𝑥 + 1 − 𝑥−3 c)
𝐴 𝐵
+ 𝑥 2 + 𝑥 2 +2
𝐶
𝑥
37) The following are the methods often used in 𝑨 𝑩𝒙+𝑪
d) +
Partial fraction decomposition except? 𝒙+𝟏 𝒙𝟐 +𝟏
42) The partial fraction decomposition of
a) Cover-up rule
𝑥 3 + 𝑥 2 + 4𝑥
b) Least cost rule is:
𝑥2 + 𝑥 − 2
c) Elimination method 4𝑥 + 2
a) 𝑥 + 𝑥 2 + 𝑥 − 2
d) Method of undetermined coefficients 4𝑥 + 2
e) 𝑎 𝑜𝑟 𝑏 𝑜𝑟 𝑐 or their combination b) 𝑥2 + 𝑥 − 2
4 2
 Info. 2: c) + (𝑥− 1)
6𝑥−9 (𝑥 + 2)
Rational function: 𝑥 2 −1 𝟒 𝟐
d) 𝒙 + (𝒙 + 𝟐) + (𝒙− 𝟏)
Use the function in Info.2 to answer
 Info. 3: By synthetic division:the process:
Question 38 and 39.
3𝑥 4 − 8𝑥 3 + 9𝑥 + 54
38) How can function, in Info.2, be decompose is given by:
𝑥−2
to partial fraction? 2
√3 −8 0 9 54
𝑨 𝑩
a) + 𝒙+𝟏 6 −4 𝐵 2
𝒙−𝟏
b)
𝐴
+ (𝑥−1)2
𝐵 ---------------------------------
𝑥−1
3 𝐴 −4 1 𝐶
𝐴 𝐵
c) + 𝑥 2 −1 Use the details in Info.3 to answer Q43 to Q45
𝑥−1
𝐴 𝐵𝑥+𝐶 43) What value of 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 makes the synthetic
d) + (𝑥+1)2
𝑥−1
division, inInfo.3, complete?
39) What is the value of 𝐴, 𝐵 𝑎𝑛𝑑/𝑜𝑟 𝐶
a) 𝐴 = 6, 𝐵 = 1 𝑎𝑛𝑑 𝐶 = 2
(according to your choice in Q38)
b) 𝐴 = −8, 𝐵 = 1 𝑎𝑛𝑑 𝐶 = 54
decomposes the function in Info.2 to partial
c) 𝑨 = −𝟐, 𝑩 = −𝟖 𝒂𝒏𝒅 𝑪 = 𝟓𝟔
fractions
−𝟑 𝟏𝟓
e) 𝐴 = −8, 𝐵 = −9 𝑎𝑛𝑑 𝐶 = 54
a) 𝑨 = 𝒂𝒏𝒅 𝑩 = 44) If 3𝑥 4 − 8𝑥 3 + 9𝑥 + 54 = 𝑓(𝑥)inInfo.3,
𝟐 𝟐
3 −15
b) 𝐴 = 2 𝑎𝑛𝑑 𝐵 = which of the letters in the synthetic division
2
−3 15 represents 𝑓(2) and what is the value of your
c) 𝐴 = . 𝐵= 𝑎𝑛𝑑 𝐶 = 6
2 2 chosen letter?
3 −15
d) 𝐴 = 2 𝑎𝑛𝑑 𝐵 = 𝑎𝑛𝑑 𝐶 = −9 a) 𝑪 = 𝟓𝟔
2
𝑥 2 +10𝑥−36 b) 𝐴 = −2
40) The decomposition of 𝑥(𝑥−3)2
into partial
c) 𝐶 = 2
𝑃 𝑄 𝑅
fractions is given by: + 𝑥−3 + (𝑥−3)2 . What is d) 𝐴 = 6
𝑥
the value of 𝑅 in the partial fraction?
a) 1
  Fig. 2
(b) 4
(c) 5
(d) 7
51) If 𝑎 𝑥 = 𝑎 𝑦 then
(a) 𝑎 = 𝑥
(b) 𝑦 = 𝑎
(c) 𝒙 = 𝒚
(d) 𝑎 𝑥 = 𝑦
52) Evaluate log7 49
(a) 2
(b) 5
1
Use Fig. 2 to answer Questions 45 to 47 (c)
2
(d) 3
45) What is the value of 𝑓(−1) of the function 53) Simplify log 64 + 2 log 5 − 2 log 40
represented by Fig. 2 (a) 3
a) 0 (b) 7
b) 0.5 (c) 0
c) 1 (d) 1
d) 1.5 54) Evaluate log 4 𝑥 = −1
46) How can you describe the graph in Fig.2 ? 1
(a) 2
a) A pole 𝟏
b) A pillar (b) 𝟒
1
c) An Asymptote (c) 3
d) A Polar (d) 2
47) What type of function does Fig.2 represent? 55) Determine the value of x in the equation
a) Linear function log 2 16 = 𝑥
b) Quadratic function (a) 3
c) Rational function (b) 7
d) Polynomial function
(c) 4
48) IF 𝑦 = log 𝑎 𝑥, then (d) 2
(a) 𝑎 𝑥 = 𝑦 56) Simplify log 3 (4𝑥 + 1) − log 3 (3𝑥 − 5) = 2
(b) 𝒂𝒚 = 𝒙 (a) 3
(c) 𝑥 𝑎 = 𝑦 (b) 2
(d) 𝑦 𝑎 = 𝑥 (c) 4
49) The x- intercept of graph of logarithms (d) 8
function is equal to 1 since 57) IF log10 𝐴 = 4What is A
(a) 𝑎 = 1 (a) 2
(b) 𝑎2 = 0 (b) 30
(c) 𝒂𝟎 = 𝟏 (c) 45
(d) 𝑎 𝑥 = 1 (d) 10000
50) Determine the value of x in the equation 58) Solve log 2 (𝑥 + 3) = 2
log 4 3𝑥 − 4 log 4 3 = 0 (a) 5
(a) 3 (b) 7
 (c) 4 1
(c) 3 log 2
(d) 10 1
(d) 3 log√8
59) Find P if log 𝑝 49 = 2
(a) 7 67) Determine the value of x if log10 5 +
(b) 5 2 log10 (𝑥 + 2) − log10 (2𝑥 + 4) = 1
(c) 6 (a) 2
(d) 10 (b) 4
log 24 (c) 3
60) Evaluate
log 23 (d) 5
𝟒
(a) 𝟑 68) Solve for x in the equation log 6 (𝑥 + 9) =
(b) 2 1 + log 6 (𝑥 + 3) − log 6 (𝑥 + 2)
(c) 5 (a) 𝑥 = 1 𝑜𝑟 𝑥 = 2
(d) 3 (b) 𝒙 = 𝟎 𝒐𝒓 𝒙 = −𝟓
61) Simplify log 2 32 = 𝑥 (c) 𝑥 = 0 = 𝑜𝑟 𝑥 = 5
(a) 5 (d) 𝑥 = −1 𝑜𝑟 𝑥 = −2
(b) 1 69) Evaluate log 5 25 = 𝑥
(c) 3 (a) 2
(d) 4 (b) 1
62) Solve 3𝑥+1 = 15 (c) 3
(a) 1.465 (d) 6
(b) 2.127 70) Simplify log 3 (𝑥 − 7) = 2 − log 3 (𝑥 + 1)
(c) 3.246 (a) -2 or 8
(d) 5 (b) 2 or 8
63) Evaluate log 3 27 = 𝑥 (c) 3 or 4
(a) 3 (d) -3 or -4
(b) 2 71) Solve for x and y in the equation
𝑦
(c) 7 log 4 𝑥 − 1 + log 4 2 = 1,
(d) 9 log 2 (𝑥 + 1) + log 2 (𝑦) = 2
64) Simplify log10 240 − log10 36 + log10 15 (a) 2 or 5
(a) 5 (b) 3 or -4
(b) 2 (c) -3 or 4
(c) 7 (d) 3 or 5
(d) 3 72) Evaluate log 3 𝑥 + log 3 3 = 1
30 5 400
65) Evaluate log10 10 − 2 log10 9 + log10 243 (a) 1
(a) 2− log10 5 (b) 5
(c) 3
(b) 1log10 15
(d) 2
(c) 1
73) ………. Is the measure of an angle whose
(d) 1.5
vertex is at the center of a circle, and that it
log√8
66) Simplify 1
log 8 intersect an arc equal in length to (360) of the
1
(a) 3 circumference.
𝟏 (a) Circumference
(b) 𝟐
 (b) Radian (b) 750
(c) Circle (c) 1200
(d) Degree (d) 1350
74) ………... is the measure of an angle whose 81) All the followings are special angles except?
vertex is at the center of a circle, and that it 𝟒𝝅
(a) 𝒓𝒂𝒅
𝟕
intersect an arc equal in length to the radius 𝜋
(b) 6 𝑟𝑎𝑑
of the circle.
(a) Circumference (c) 450
3𝜋
(b) Radian (d) 𝑟𝑎𝑑
4
(c) Circle 82) All the followings identities are true except?
(d) Degree (a) sin2 𝜃 + cos 2 𝜃 = 1
75) One degree (10) is equal to…… in rad (b) 1 + tan2 𝜃 = sec 2 𝜃
(a) 360 rad (c) 𝐬𝐢𝐧𝟐 𝜽 = 𝟏 + 𝐜𝐨𝐬𝟐 𝜽
(b) 180 rad (d) cos2 𝜃 = 1 − sin2 𝜃
𝝅
(c) 𝟏𝟖𝟎 rad 5
83) Given that given that sin 𝜃 = 13 and that 𝜃
𝑟𝑎𝑑
(d) is acute, evaluate cos 𝜃
180
76) One radian (1 rad) is equal to …. In degree 5
(a)
𝟏𝟖𝟎𝒐 13
(a) 13
𝝅 (b)
𝜋 12
(b) rad 𝟏𝟐
180 (c) 𝟏𝟑
𝜋
(c) 4
180 (d) 13
(d) 1800
84) Tan 950 is same as ………….
77) Express 3150 in radian
π (a) – tan750
(a) 12 rad
(b) tan450
𝟕𝛑
(b) 𝟒
𝐫𝐚𝐝 (c) tan250
(c)
π
rad (d) − tan850
180
3𝜋
85) Cos2200 is same as …………
(d) 𝑟𝑎𝑑 (a) –cos400
2
78) Express 500 in radian (b) –cos500
𝜋
(a) 6 𝑟𝑎𝑑 (c) sin400
3𝜋 (d) cos750
(b) 𝑟𝑎𝑑
2 86) Express sin1250 in terms of the
𝟓𝝅
(c) 𝒓𝒂𝒅 trigonometric ration of an acute angle
𝟏𝟖
(d)
3𝜋
𝑟𝑎𝑑 (a) Cos450
5
3𝜋 (b) sin450
79) Express 𝑟𝑎𝑑 in degree (c) sin550
2
(a) 2700 (d) cos550
(b) 700 87) if cos A =
−4
and A is an obtuse angle,
15
(c) 1000
determine tanA
(d) 2300
√𝟐𝟎𝟗
80) Express
3𝜋
𝑟𝑎𝑑 in degree (a)
𝟒
4
0 √209
(a) 145 (b) − 4
 (c)
√209 95) Sin2A is the same as …….
15
(a) sin 𝐴 cos 𝐴
√209
(d) − 15 (b) 1 − 2 Cos 2 𝐴
1+𝑠𝑖𝑛𝑥
88) The expansion of (1+𝑐𝑜𝑠𝑥)(1+𝑐𝑜𝑠𝑒𝑐𝑥) is?
1+𝑠𝑒𝑐𝑥 (c) 𝟐 𝐬𝐢𝐧 𝑨 𝐜𝐨𝐬 𝑨
(d) 1 + 2 sin2 𝐴
(a) Cosecx 3
(b) secx 96) Given that cos2A = 5, find the value of tanA
2
(c) tanx (a) ± 3
(d) cotx 3
1 1 (b) ± 5
89) The value of 1 + in a 𝟏
−𝐶𝑜𝑠𝑥 1 + 𝐶𝑜𝑠𝑥
(c) ± 𝟐
simplified form is …….
1
(a) 2sec 2 x (d) ± 3
(b) sec 2 x 97) What is the value of sin(A + B) + sin(A – B)
(c) cosec 2 x (a) 𝟐 𝐬𝐢𝐧 𝑨 𝐜𝐨𝐬 𝑩
(d) 2𝐜𝐨𝐬𝐞𝐜 𝟐 𝐱 (b) 2 cos 𝐴 sin 𝐵
90) The simplified form of (1 +𝑡𝑎𝑛2 𝑥) (1 (c) −2 sin 𝐴 cos 𝐵
−sin2 𝑥) is ……….. (d) −2 cos 𝐴 sin 𝐵
(a) cos2 𝑥 98) What is the value of sin(A + B) − sin(A – B)
(b) 1 (a) 2 sin 𝐴 cos 𝐵
(c) 0 (b) 2 cos 𝐴 sin 𝐵
(d) Tanx (c) −2 sin 𝐴 cos 𝐵
91) The simplified form of cos 𝑥 tanx is (d) −2 cos 𝐴 sin 𝐵
(a) sinx 99) What is the value of cos(A + B) + cos(A – B)
(b) tanx (a) 2 sin 𝐴 cos 𝐵
(c) cos 𝑥 (b) −2 cos 𝐴 cos 𝐵
(d) cotx (c)−2 sin 𝐴 sin 𝐵
𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥
92) The expression 𝐶𝑜𝑠𝑥 − is same as (d) 𝟐 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩
𝐶𝑜𝑡𝑥
(a) sinx 100) What is the value of cos(A + B) − cos(A – B)
(b) 0 (a) 2 sin 𝐴 cos 𝐵
(c) cosx (b) −2 cos 𝐴 cos 𝐵
(d) 1 (c) −𝟐 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩
93) The value of cos150 in surd form is ……. (d) 2 cos 𝐴 cos 𝐵
𝐴
√𝟐(√𝟑+ 𝟏 101) From half angles formular, we have sin( )
(a) 2
𝟒
1 as …..
(b)
√2 𝟏− 𝐜𝐨𝐬𝐀
√3 (a) √
(c) 𝟐
2
1+ cosA
(d) 1 (b) √
2
94) Cos2A is the same as ………
1− cosA
(a) 1 − 2 Cos2 𝐴 (c) √1+ cosA
(b) 𝟏 − 𝟐 𝐬𝐢𝐧𝟐 𝑨
1+ cosA
(c) 1 + 2 sin2 𝐴 (d) √1− cosA
(d) 2 sin2 𝐴 − 1
 𝐴
102) From half angles formular, we have cos( 2 ) ii. b2 = a2 + c 2 − 2bc cos B
as ….. iii. c 2 = b2 + a2 − 2ba cos C
iv. 2bc cos A = b2 + c 2 − a2
1− cosA
(a) √ (a) i, ii and iii
2
(b) iii and iv
𝟏+ 𝐜𝐨𝐬𝐀
(b) √ (c) i, ii, iii and iv
𝟐
(d) i and ii
1− cosA
(c) √1+ cosA 108) Given the equation sinѲ = p, then
(a) For a given value of Ѳ, p is unique.
1+ cosA
(d) √1− cosA (b) If p is known, then Ѳ may have many
𝐴 solutions
103) From half angles formular, we have cos( 2 )
(c) All of the above
as ….. (d) None of the above
1− cosA 1
(a) √ 109) If sinѲ= 2, then two possible value of Ѳ
2
are?
1+ cosA
(b) √ (a) 300 and 1500
2

𝟏− 𝐜𝐨𝐬𝐀
(b) 600 and 1500
(c) √𝟏+ 𝐜𝐨𝐬𝐀 (c) 300 and 600
1+ cosA
(d) 300 and 1200
(d) √1− cosA 110) Given that cos450 = p, what is the value of p
1
104) Sin7tsin3t as a sum or difference can be (a) 2
written as……. 1
1
(b) − 2
(a) 2(sin10t – cos4t) 1
1
(c) −
√2
(b) 2(sin10t – sin4t) 𝟏
𝟏 (d)
√𝟐
(c) − 𝟐 (𝒄𝒐𝒔𝟏𝟎𝒕 – 𝒄𝒐𝒔𝟒𝒕)
111) The expansion of tan(arctan3x + arctan2x)
1
(d) 2(cos10t – sin4t) gives?
𝐴 tan3x−tan2x
2tan( )
2 (a) 1+tan3xtan2x
105) Given that tanA = 𝐴 and that
1−𝑡𝑎𝑛2 ( ) 𝟑𝐱+𝟐𝐱
2 (b)
4 𝐴 𝟏 − (𝟑𝐱)(𝟐𝐱)
tanA = 3, find tan( 2 ) 3x − 2x
(c) 1+
(a)− 2 or 0 (3x)(2x)
tan3x+tan2x
(b) 1 or −2 (d) 1 − tan3xtan2x
(c) −1 or 2 112) Find the possible values of 𝑆𝑖𝑛𝑥 given that
(d) 2 or 0 8Cos2 𝑥 + 6Sinx – 9 = 0
106) All the followings are true except? (a) 14.80 or 300
(a) asinB = bsinB (b) 14.80 or −300
(b) bsinC = csinB 1
(c) 4 or − 2
1
𝑐 𝑎
(c) 𝑠𝑖𝑛𝐶 = sin 𝐴 1 1
(d) or
(d) asinA = bsinB 4 2

107) Which of the followings is/are correct? 113) Given triangle ABC with a = 7.2, b = 8.9
i. a2 = b2 + c 2 − 2ac cos A and B = 550, determine A
 (a) 400 (a)
√3− 1
2√2
(b) 45.10
√3− 1
(c) 41.50 (b) 2√3
(d) 51.50 𝟏−√𝟑
(c)
114) Given triangle ABC with A = 1050, b = 3.6 𝟐√𝟐
and c = 7.5, determine a 1+√3
(d) 2√2
(a) 7.5 Sin3A Cos3A
(b) 7.2 121) Simplify −
SinA CosA
(c) 3.6 (a) Sin3A
(d) 4.5 (b) CosA
115) Given that 2Cos2 𝑥 = 2 – sinx the two (c) − 2
possible values of sinx are? (d) 2
1−Cos2A
(a) 0 or 0.5 122) Simplify 1+Cos2A
(b) 1 or 0.5
(a) 𝐓𝐚𝐧𝟐 𝐀
(c) 1.5 or 2
(b) Cos2 A
(d) 00 or 300
(c) 2
116) What is the value of sin 750 in surd form?
(d) 1
√3− 1
(a) 123) In any complex number, 𝑍 = 𝑥 + 𝑖𝑦, then
2√2
√3 if 𝑥 = 0 the number 𝑍 is;
(b) 2 (a) Zero complex number
1
(c) (b) Purely imaginary
√2
√𝟑+ 𝟏 (c) Purely real
(d)
𝟐√𝟐 (d) Negative complex number
117) Express 3cosѲ – 2sinѲ in the form 124) Simplify 𝑖 8 + 𝑖 19
Rsin(Ѳ + 𝛼) (a) 1
(a) √𝟏𝟑 𝐬𝐢𝐧(Ѳ + 560) (b) 1+𝑖
(b) √13 sin(Ѳ − 560) (c) 1-𝒊
(c) √3 sin(Ѳ + 560) (d) -𝑖
(d) √12 sin(Ѳ + 560) 125) Given that 𝑍1 = 2 + 4𝑖 and 𝑍2 = 3 + 5𝑖.
118) Express 2cosѲ – 3sinѲ in the form Find 𝑍1 − 𝑍2 .
(a) −𝟏 − 𝒊
Rsin(Ѳ −α)
(b) 1 − 𝑖
(a) √𝟏𝟑 𝐬𝐢𝐧(Ѳ + 560)
(c) 5 + 9𝑖
(b) √13 sin(Ѳ − 560)
(d) 5 − 𝑖
(c) √3 sin(Ѳ + 560) 126) Find the complex conjugate to 1 + 8𝑖
(d) √12 sin(Ѳ + 560) (a) 8𝑖
1
119) If sin500cosx + cos500sinx = 2 what are (b) −𝟏 − 𝟖𝒊
the possible value of x? 00 ≤ x ≤ 3600 (c) 1 − 8𝑖
(a) 1000 or 1500 (d) 8𝑖 − 1
1 2
(b) 1000 or 3400 127) Find 𝑥 − 𝑦 if + 𝑍 = 1 + 𝑖.
𝑍
(c) 1500 or 3400 (a) 6/10
(d) 1500 or 2400 (b) -6/10
120) The value of Cos1050 in surd form is? (c) 12/10
 (d) 3/10 (a) De moivre’s theorem
128) Given that 1 + 𝑖 = 𝑧, find 𝑧10 (b) De Alarmbert’s theorem
(a) 𝟐𝟓 (𝐜𝐨𝐬 𝟐𝝅 + 𝒊 𝐬𝐢𝐧 𝟐𝝅) (c) Cauchy’s Root Test
(b) 210 (cos 5𝜋 + 𝑖 sin 5𝜋) (d) De moivre and Abel formula
(c) 210 (cos 2𝜋 + 𝑖 sin 2𝜋) 136) The Geometric representation of complex
(d) 25 (cos 5𝜋 + 𝑖 sin 5𝜋) numbers is called
129) Find the values of 𝑥 and 𝑦 from the (a) Agand diagram
equation 2𝑦 + 𝑖𝑥 = 4 + 𝑥 − 𝑖 (b) Argard diagram
(a) (3/2, −1) (c) Argand diagram
(b) (−𝟏, 𝟑/𝟐) (d) Ardgand diagram
(c) (𝑖, 3/2𝑖) 137) The modulus of a complex number 𝑍 =
(d) (1, 2) 𝑥 + 𝑖𝑦 is defined as
130) Find the values of 𝑎 and 𝑏 from the (a) 𝒁 = 𝒙 − 𝒊𝒚
equation 𝑎 + 𝑖𝑏 = (2 + 𝑖)2 (b) 𝑍 = √𝑥 2 + 𝑦 2
(a) (3, 4) √𝑥 2 +𝑦 2
(b) (4 ,3) (c) 𝑧 = 𝑥+𝑦
(c) (2, 3) (d) 𝑍 = 3√𝑥 2 + 𝑦 2
(d) (1,4) 138) Polar form of complex number is;
131) Express as a complex number (2 + 3𝑖) + (a) 𝑟(𝑡𝑎𝑛𝜃 + 𝑖𝑐𝑜𝑡𝜃)
(−4 + 5𝑖) − (9 − 3𝑖/3) (b) 𝑟(𝑠𝑒𝑐𝜃 + 𝑖𝑐𝑜𝑠𝑒𝑐𝜃)
(a) (−𝟓 + 𝟗𝒊) (c) 𝒓(𝒄𝒐𝒔𝜽 + 𝒊𝒔𝒊𝒏𝜽)
(b) (9𝑖 − 5) (d) 𝑟(𝑠𝑖𝑛𝜃 + 𝑖𝑐𝑜𝑠𝜃)
(c) (5 + 9𝑖) 139) If 𝑍1 = 2 + 𝑖 and 𝑍2 = 3 + 5𝑖 then
(d) (5 − 8𝑖) 𝑖𝑅𝑒(𝑧1 − 𝑧2 )
132) Simplify (−5 + 3𝑖)(−4 + 8𝑖) (a) 1
(a) (4 − 52𝑖) (b) I
(b) (−4 + 52𝑖) (c) 2i
(c) (−4 − 51𝑖) (d) 2
(d) (−𝟒 − 𝟓𝟐𝒊) 140) |𝑍1 + 𝑍2 | =
133) Simplify (−1 − 2𝑖)(−4 + 3𝑖) (a) > |𝑍1 | + |𝑍2 |
11 2
(a) 25 − 25 𝑖 (b) ≤ |𝒁𝟏 | + |𝒁𝟐 |
2 11 (c) ≤ 𝑍1 + 𝑍2
(b) 25 − 25 𝑖
9 11
(d) > 𝑍1 + 𝑍2
(c) 25 − 25 𝑖 141) |𝑍1 − 𝑍2 | =
−𝟐 𝟏𝟏
(d) 𝟐𝟓 + 𝟐𝟓 𝒊 (a) ≥ |𝒁𝟏 | − |𝒁𝟐 |
(b) ≤ |𝑍1 | − |𝑍2 |
134) Euler’s formula is as follows;
(c) ≤ 𝑍1 − 𝑍2
(a) 𝑒 𝑖𝜃 = 𝑠𝑖𝑛𝜃 + 𝑖𝑐𝑜𝑠𝜃
(d) > 𝑍1 − 𝑍2
(b) 𝒆𝒊𝜽 = 𝒄𝒐𝒔𝜽 + 𝒊𝒔𝒊𝒏𝜽
142) The general form of complex number Z is
(c) 𝑒 𝑖𝜃 = 𝑐𝑜𝑠𝜃 − 𝑖𝑠𝑖𝑛𝜃 given by ;
(d) 𝑒 𝑖𝜃 = 𝑠𝑖𝑛𝜃 − 𝑖𝑐𝑜𝑠𝜃 (a) 𝑍 = 𝑎 − 𝑖𝑏
135) Given that 𝑍 = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃) is a non- (b) 𝒁 = 𝒂 + 𝒊𝒃
zero complex number, then 𝑧 𝑛 = (c) 𝑍 = 𝑎 − 2𝑖𝑏
𝑟 𝑛 (𝑐𝑜𝑠𝑛𝜃 + 𝑖𝑠𝑖𝑛𝜃) I s called
(d) 𝑧 = 𝑖 2
143) Simplify 𝑖 3 + 𝑖 2 151) For 𝑧 = 𝑥 + 𝑖𝑦 then
(a) 𝑖 − 2 (a) 𝒙, 𝒚 𝑬 𝑹
(b) −𝒊 − 𝟏 (b) 𝑥, 𝑦 𝐸 ∁
(c) 1 − 𝑖 (c) 𝑥, 𝑦 𝐸 𝑁
(d) 𝑖 5 (d) 𝑥, 𝑦 𝐸 𝑄
144) Given that 𝑧 = 𝑐 + 𝑖𝑑, find 𝑐 + 𝑑 if 152) Given that 𝑧 = 𝑥 + 𝑖𝑦 and 𝑤 = 𝑎 + 𝑖𝑏
1 2 then 𝑧𝑤 =
+ 𝑍 = 1 + 𝑖.
𝑍
(a) 𝒙𝒂 − 𝒚𝒃 + 𝒊(𝒙𝒃 + 𝒚𝒂)
(a) 𝟏𝟐/𝟏𝟎
(b) 𝑥𝑎 + 𝑦𝑏 + 𝑖(𝑥𝑏 − 𝑦𝑎)
(b) 3/10
(c) 𝑥𝑎 − 𝑦𝑏 + 𝑖(𝑥𝑏 − 𝑦𝑎)
(c) 9/10
(d) 𝑥𝑎 − 𝑦𝑏 + 𝑖(𝑥𝑦 + 𝑦𝑏)
(d) 5/6
153) Given (𝑥, 𝑦) 𝜊 (𝑎, 𝑏) = [𝑥𝑎 − 𝑦𝑏, 𝑥𝑏 + 𝑦𝑎],
145) Find 𝑎/𝑏 from the equation 𝑎 + 𝑖𝑏 =
find (1,0) 𝜊 (𝑎, 𝑏).
(3 + 𝑖)2
(a) (−1,0)
(a) 3/4
(b) (0,1)
(b) 8/5
(c) (𝟏, 𝟎)
(c) 3/5
(d) (−𝑎, 𝑏)
(d) 𝟒/𝟑
154) Find 𝑍 + 𝑍̅ if 𝑍 = 𝑥 + 𝑖𝑦
146) Write (−7 + 𝑖)(3 − 𝑖) in the form (𝑥, 𝑦)
(a) 2𝑦
(a) (−𝟐𝟎, 𝟏𝟎)
(b) 𝟐𝒙
(b) (−19,10)
(c) 2𝑥𝑦
(c) (−20, 9)
(d) 0
(d) (10, −20)
155) Find 𝑍 − 𝑍̅ if 𝑍 = 𝑥 + 𝑖𝑦
147) If 𝑍1 𝑎𝑛𝑑 𝑍2 are complex numbers, then;
(a) 𝟐𝒊𝒚
(a) 𝑍 − 𝑍̅ = 2𝑎
(b) 2𝑦
(b) 𝑍 − 𝑍̅ = 2𝑎𝑖
̅ = 𝟐𝒊𝒃 (c) 2𝑥
(c) 𝑍 − 𝒁
(d) 2𝑥𝑦
(d) 𝑍 − 𝑍̅ = 𝑎2 + 𝑏 2 1
148) If 𝑍1 𝑎𝑛𝑑 𝑍2 are complex numbers, then 156) If 𝑍 ≠ 0, then =
𝑧
𝑍
(a) 𝑍 + 𝑍̅ = 2𝑖𝑏 (a) 𝑍𝑍̅
(b) 𝒁 + 𝒁̅ = 𝟐𝒂 𝑍̅
(b) 𝑍 2
(c) 𝑍 + 𝑍̅ = 2𝑎𝑖
̅
𝒁
(d) 𝑍 + 𝑍̅ = 𝑎2 + 𝑏 2 (c) 𝒁𝒁̅
149) one of the following expressions is correct; 1
(d) 𝑍 − Z
(a) 𝑍𝑍̅ = 2𝑖𝑏
(b) 𝑍𝑍̅ = 2𝑎 157) 𝑍̿ = Z and 𝑧̅ = 0 iff
(a) 𝑍 = 1
(c) 𝑍𝑍̅ = (𝑎2 + 𝑏 2 )𝑖
̅ = 𝒂𝟐 + 𝒃𝟐 (b) 𝑍 = 𝑖
(d) 𝒁𝒁
𝑍 |𝑍 | (c) 𝑍 = 2
150) |𝑍1 | = |𝑍1 |, provided;
2 2 (d) 𝒁 = 𝟎
(a) 𝑍2 = 𝑍1 158) Find the conjugate of 3𝑖
(b) 𝑍2 ≠ 1 (a) 3
(c) 𝑍1 ≠ 1 (b) −𝟑𝒊
(d) 𝒁𝟏 ≠ 𝟎 (c) −3
 (d) −3𝑖 − 1
159) For 𝑧 = 𝑥 + 𝑖𝑦 ,𝑦 = 0 .Then 𝑍 is
(a) Purely Imaginary
(b) Zero Complex Number
(c) Negative Complex Number
(d) Purely Real
160) Simplify 𝑖 19
(a) i
(b) 𝑖 2
(c) – 𝒊
(d) 1
161) Given that 𝑍1 = 2 + 4𝑖 and 𝑍2 = 3 +
5𝑖.Find 𝑍1 + 𝑍2
(a) −1 − 𝑖
(b) 𝟓 + 𝟗𝒊
(c) 1 − 𝑖
(d) 5 − 𝑖
162) Find the conjugate of 4𝑖 − 1
(a) −𝟒𝒊 − 𝟏
(b) 4𝑖 + 1
(c) −4𝑖 + 1
(d) −4𝑖