Complex Numbers &Quadratic Equations MCQs Part V

1) 1+i2+i4+i6+……+i2n is
a) +ve
b) -ve
c) 0
d) cannot be found

2) i2+i4+i6+…..(2 n+1) terms =
a) i
b) -i
c) 1
d) -1

3) If x + iy =
a)
b)
c)
d)

4) If (a + ib) (c + id) (e + if) (g + ih)=A+ iB, then (a2 + b2)(c2
+ d2)(e2 + f2)(g2 + h2)
=

a)
b)
c)
d)

5) z2 + (p + iq) z+ r + is=0 where p, q, r, s are non-zero has real
roots, then

a)
b)
c)
d)

6) If the amplitude of z -2 -3i=,
then the locus of z is

a) x-y=2
b) x+y=0
c) x+y=1
d) x-y+1=0

7) If is a cube root of unity
and (1+)7= A+B
then A and

a) 0,1
b) 1,1
c) 1,0
d) -1,1

8) Let z and be two non-zero complex
numbers such that and Arg z + Arg
=
then z equals

a)
b)
c)
d)

9) The argument of the complex number
+ i is

a)
b)
c)
d)

10) If
then the locus of z is

a)
b)
c)
d) none of these

11) Let be two complex numbers
such that Then

a)
b)
c)
d) none of these

12) If are two pairs of conjugate
complex numbers, then arg equals

a) 0
b)
c)
d)

13) If m, n, p, q are consecutive integers, then the value of
is

a) 1
b) 4
c) 0
d) none of these

14) If then the value of
is

a) -27
b) 27
c) 3
d) 0

15) Value of arg x when x < 0 is
a) 0
b)
c)
d) none of these

16) The value of is
a) 1
b) -1
c) i
d) -i

17) If is a purely imaginary number,
then is equal to

a) 0
b) 1
c) 2
d) none of these

18) If is an imaginary cube root
of unity, then the value of is

a) -2
b) -1
c) 1
d) 0

19) If , then the value of
is

a) n
b)
c) 0
d) none of these

20) If for complex numbers is
equal to

a)
b)
c)
d) 0

21) If
a) C + iS
b) C – iS
c) S + iC
d) S – i C

22) The number of solutions of the system of equations Re
is

a) 4
b) 3
c) 2
d) 1

23) The closest distance of the origin from a curve given as
is (a is a complex number)

a) 1
b)
c)
d)

24)
a) an equilateral triangle
b) a right angled triangle
c) an isosceles triangle
d) none of these

25) If and locus of 5z – 1 is
the circle having radius a and
then

a) a : 1
b) 2a : 1
c) a : 10
d) none of these

26) If and m and n be the least
and greatest values of and K be
the least value of on the interval
then K =

a) n
b) m
c) n+m
d) none of these

27) If is an imaginary cube root
of unity, then the value of

a)
b)
c)
d)

28) Common roots of the equations z3+2z2+2z+1=0 and z1985
+ z100 +1=0 are

a)
b)
c)
d) none of these

29) The value of is
a) -1
b) -2
c) -3
d) -4

30) If are four complex numbers
represented by the vertices of a quadrilateral taken in this order such that

a) a square
b) a rectangle
c) rhombus

31) If are three distinct complex
numbers and a, b, c are three +ve real numbers such that
then

a)
b)
c)
d) none of these

32) Consider which of the following
hold(s) good

a)
b)
c)
d)

33)
a)
b)
c)
d) none of these

34) If n > 1, then the roots of
lie on a

a) circle
b) straight line
c) parabola
d) none of these

35) For complex number where
then the angle between the roots of the equation
is

a)
b)
c)
d) none of these

36) If , where a, b, c are real
and is a non-real cube root of unity,
then

a)
b) abc + bcd + and + acd = 4
c) a + b + c + d = -2a
d)

37) If
a)
b)
c)
d)

38) If
a) 0
b)
c)
d) none of these

39) If is equal to
a)
b) 4
c)
d) none of these

40) If
a)
b)
c) 1
d) none of these

41) If are conjugate complex numbers
and are also conjugate, then

a)
b)
c)
d)

42) If , then the locus of z is
a) an ellipse
b) a circle
c) a parabola
d) a st. line

43) The complex number which satisfies the equation
is

a) 2+i
b) -2+i
c) -2-i
d) 2-i

44) For any complex number x + iy, put
Then which one of the following is true

a)
b) a>b
c)
d) a = b + 2

45) If and ,
then K =

a) 3
b) 4
c) 2
d) none of these

46) If , then
is equal to

a)
b)
c)
d) none of these

47) is equal to
a)
b)
c)
d) none of these

48) is equal to
a)
b)
c)
d) none of these

49) If are two complex numbers such
that and
where then the angle between
and is

a)
b)
c)
d)

50) If at least one value of the complex number z = x + iy satisfy the condition
and the inequality ,
then

a) a>2
b) a=2
c) a<2
d) none of these

51) Given z is a complex number with modulus 1. Then the equation
has

a) all roots, real and distinct
b) two roots real and two imaginary
c) three roots real and one imaginary
d) one root real and three imaginary

52)

The locus of the complex number z in an argand plane satisfying the inequality
is

(where )

a) a circle
b) an interior of a circle
c) the exterior of the circle
d) none of these

53) If are the nth roots of unity,
then is equal to

a)
b)
c)
d) none of these

54) If p, q are two real numbers lying between 0 and 1 such that
and form an equilateral triangle,
then (p, q) =

a)
b)
c)
d) none of these

55) The origin and the roots of the equation
form an equilateral triangle if

a)
b)
c)
d)

56) Let z be a complex number such that
and arg , then z is equal to

a)
b)
c)
d)

57) Let 3-i and 2+i be affixes of two points A and B in the Argand plane and
P represents the complex number z = x + iy such that
Then the locus of P is

a) a circle on AB as diameter
b) the line AB
c) the perpendicular bisector of AB
d) none of these

58) The locus of the points representing the complex number z for which
is

a) a circle with centre at the origin
b) a st. line passing through the origin
c) the single point (0.-2)
d) none of these

59) Let z be a complex number (not lying on x-axis) of maximum modulus such
that . Then

a) Im(z)=0
b) Re(z)=0
c)
d) none of these

60) The value of a so that the sum of the squares of the roots of
the equation
assume
the least value, is

a) 2
b) 0
c) 3
d) 1

61) The number of values of a for which
is an identity in x is

a) 0
b) 2
c) 1
d) infinite

62) The number of values of the pair (a, b) for which
is an identity in x is

a) 0
b) 1
c) 2
d) infinite

63) The number of values of the triplet (a, b, c) for which
is satisfied by all real x is

a) 0
b) 2
c) 3
d) infinite

64) The polynomial
has

a) four real zeros
b) at least two real zeros
c) at most two real zeros
d) no real zeros

65) The number of real solutions of the equation
is

a) one
b) two
c) four
d) infinite

66) The sum of the real roots of the equation
is

a) 4
b) 0
c) -1
d) none of these

67) The number of real solutions of the equation
is

a) 0
b) 1
c) 2
d) infinitely many

68) The number of real solutions of
is

a) 0
b) 1
c) 2
d) 4

69) The equation
has

a) one real solution
b) no real solution
c) infinitely many real solution
d) none of these

70) The product of all the solutions of the equation
is

a) 2
b) -4
c) 0
d) none of these

71) The number of real solutions of the equation
is

a) two
b) one
c) zero
d) none of these

72) The number of real solutions of
is

a) one
b) two
c) three
d) none of these

73) If x is an integer satisfying
and
then the number of possible values of x is

a) 3
b) 4
c) 6
d) none of these

74) If a, b, c are nonzero, unequal
rational numbers then the roots of the equation

are

a) rational
b) imaginary
c) irrational
d) none of these

75) If l, m, n are real and
then the roots of the equation
are

a) real and equal
b) nonreal complex
c) real and unequal
d) none of these

76) If a, b, c, d are four consecutive
terms of an increasing AP then the
roots of the equation (x-a)(x-c)+2(x-b)(x-d)
= 0 are

a) real and distinct
b) nonreal complex
c) real and equal
d) integers

77) If a, b, c are three distinct positive
real numbers then the number of real
roots of
is

a) 4
b) 2
c) 0
d) none of these

78) The equation
has

a)
b)
c)
d)

79) The roots of ,
where
and coefficients are real, are nonreal complex and a + c < b. Then

a) 4a+c>2b
b) 4a+c<2b
c) 4a+c=2b
d) none of these

80) If the roots
of the equation

are less than
3 then

a) a<2
b)
c)
d) a>4

81) If one root of the equation
is reciprocal of the other then k has the value

a)
b)
c) 1
d) both (a) and (b)

82) If the ratio of the roots of
is equal to the ratio of the roots of
then
are in

a) AP
b) GP
c) HP
d) none of these

83) p, q, r and s are integers. If the AM of the roots
of
and GM of the roots of
are equal then

a) q is an odd integer
b) r is an even integer
c) p is an even integer
d) s is an odd integer

84) If
are roots of the equation
then the roots of the equation
are

a) a, c
b) b, c
c) a, b
d) a+c, b+c

85) If the roots of
differ by unity then the negative value of k is

a) 3
b)
c)
d) none of these

86) If the product of the roots of the equation
is 8 then
is

a)
b)
c) 3
d) none of these

87) If the roots of
are ,
and those of
are
such that
then

a)
b)
c)
d) none of these

88) If
are the roots of
then the equation
in y has the roots

a)
b)
c)
d)

89) If the roots of
change by the same quantity then the expression in a, b, c that
does not change is

a)
b)
c)
d) none of these

90) If
are the roots of
then the product of the roots of the quadratic equation
whose roots are
and
is

a)
b)
c)
d) none of these

91) If the sum of the roots of the quadratic equation
is equal to the sum of the squares of their reciprocals then
is equal to

a) 2
b) -2
c) 1
d) -1

92) If the absolute value of the difference of roots of the equation
exceeds

then
a) p<-1 or p>4
b) p>4
c) -1d)

93) If
are roots of
and
are the roots of
then
is equal to

a) q+r
b) q-r
c) -(q+r)
d) -(p+q+r)

94) If
are roots of
and
then
is

a)
b)
c)
d) none of these

95) If a and b are rational and b is not a perfect square
then the quadratic equation with rational coefficients
whose one root is
is

a)
b)
c)
d) none of these

96) If
is a root of
, where a, b are real, then

a) a=25, b=-8
b) a=25, b=16
c) a=5, b=4
d) none of these

97) If
be the roots of the equation
then the value of
is

a) 3
b)
c)
d) none of these

and
have

a)
b)
c)
d) none of these

99) If
are three quadratic equations of which each pair has exactly
one root common then the number of solutions of the triplet

is

a) 2
b) 1
c) 9
d) 27

100) The set of possible values of
for which

has roots whose sum and product are both less than 1
is

a)
b) (1, 4)
c)
d) none of these