# Complex Numbers &Quadratic Equations MCQs Part VI

1) If
is a root of the equation
then the other root is

a)
b)
c) /5
d) none of these

2) If
are the real roots of
and are
the roots of
then the equation
has always

a) two real roots
b) two negative roots
c) two positive roots
d) none of these

3) If a,b,c are rational and no two of them are equal then the equations
and

a) have rational roots
b) will be such that at least one has rational roots
c) have exactly one root common
d) both a and c

4) The equations
and have
one and only one root common. Then

a) a-b=0
b) a-b+2=0
c)
d) All the above

5) If
has no real roots and p, q,r are real such that
then

a) p-q+r<0
b) p-q+r>0
c) p+r=q
d) all of these

6) For the equation
a) roots are rational
b)
c) roots are irrational
d) None of the above

7)
is a factor of
if

a)
b)
c)
d) All the above

8) If a, b are the real roots
and c, d are the real roots of
then (a-c)(b-c)(a+d)(b+d) is divisible by

a) a+b+c+d
b) a+b-c-d
c) a-b+c-d
d) Both a and b

9) A quadratic equation whose roots are
and ,
where
are the roots of ,
is

a)
b)
c)
d) none of these

10)

If  z1 and z2 are
non zero complex numbers such that | z1 + z2
=  | z1 |  + | z 2 |;  then Arg  z1
– Arg z
a)

p

b)

-( p/2 )

c) 0
d)

p/2

11)
a) -1
b) 0
c) -i
d) i

12)
a) A Circle
b) A Straight Line
c) An Ellipse
d) None of the given

13)
a) |z| = 1
b) |z| > 1
c) |z| < 1
d) None of the given

14)

The equation of the right bisector of the line
joining  z1  and z2  is

a)
b)
c)
d) None of the given

15)
a) n – 1
b) n
c) -1
d) 1

16)
a) 1
b) 2
c) 3
d) 4

17)
a) a = 2, b = -1
b) a = 1, b = 0
c) a = 0, b = 1
d) a = – 1 , b = 2

18)
a) -3
b) -2
c) -1
d) 0

19) If the area of the triangle on the complex plane formed by the points z , z + iz and iz is 50, then | z | is
a) 1
b) 5
c) 10
d) None of the given

20)
a) 1
b) 2
c) < 1
d) > 1

21)
a) a
b) b
c) c
d) d

22)
a) Zero
b) real and positive
c) real and negative
d) purely imaginary

23)
a) 4 , 7
b) -4, -7
c) -4, 7
d) 4, -7

24)
a)
b)
c)
d)

25) The inequality | z – 2 | < | z - 4 | represents the half plane
a)
b) Re (z) = 3
c)
d) None of the given

26)

If n is a positive integer greater than unity,
and z is a complex number satisfying the equation zn  = ( z + 1
)n , then

a) Re (z) < 0
b) Re (z) > 0
c) Re (z) = 0
d) None of the given

27)
a) all roots real and distinct

b) two real and two imaginary roots
c) three roots real and one imaginary
d) one roots real and three imaginary

28)
represent points P,Q on the locus
and the line segment PQ subtends an angle
at the point z=1 then
is equal to

 (i) (ii) (iii) (iv)

a) i and iii
b) ii and iii
c) iii and iv
d) none of these

29) If
then the equation

has

a) one positive root and one negative root
b) both roots positive
c) both roots negative
d) nonreal roots

30)
a) a circle of radius 1
b)
c) a line through the origin
d) a circle on the line joining 1 and -i as diameter

31) If
has equal integral roots then

a) b and c are integers
b) b and c are even integers
c) b is an even integer and c is a perfect square of a positive integer
d) none of these

32) If z = 1 + I, then the multiplicative inverse of
is

a) 1 – i
b)
c) -i/2
d) 2 i

33)
a) a right angled triangle
b) an equilateral triangle
c) isosceles triangle
d) scalene triangle

34) The inequality represents
the half plane

a)
b) Re(z)>3
c) Re(z)=3
d) none of these

35) If 2a+3b+6c=0, then at least one root of the equation
a) (0,1)
b) (1,2)
c) (2,3)
d) none of these

36) all
are real, then the quadratic equation

a) at least one root in (0,1)
b) one root in (1,2) and the other in (3,4)
c) one root in (-1,1) and the other in (-5,-2)
d) both roots imaginary

37) If
then the number of values of x is

a) 2
b) 4
c) 1
d) none of these

38) If a>1, roots
of the equation
are

a) non positive and one negative
b) both negative
c) both positive
d) both nonreal complex

39) Let a, b, c be three real numbers such that 2a+3b+6c=0.
has

a) imaginary roots
b) at least one root in (0, 1)
c) at least one root in (-1, 0)
d) both roots in (1,2)

40)

If a and b
are the roots of x2 + px + q= 0 , then the equation has

a) two real roots
b) two positive roots
c) two negative roots
d) one positive and one negative roots

41)

and ax2 + 2bx + c = 0  (b ¦ c)
have a common root, then a+ 4b+4c  is equal to

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

42)

If Cos a, Sin a
be the roots of the equations ax2+bx+c = 0,á then

a)

a2-b2+2ac =
0

b)

(a+c)2 = b2+c2

c)

a2+b2-2 ac =
0

d)

(a-c)2=b2+c2

43)

If the difference of the corresponding roots of
the equationá x2+px+q = 0 and x2+qx+p=0
be the same then

a) p+q+4=0
b) p+q-4=0
c) p-q-4=0
d) None of the given

44)

If a, b, c Î R ,
then the roots of the quadratic equation a (x-b) (x-c) + b (x-a) (x-c) + c
(x-a) (x-b) = 0 are

a) real
b) imaginary
c) in determinate
d) none of these

45)

The sum of the roots of the quadratic equation
ax2+bx+c = 0  is equal to the sum of the squares of their
reciprocals,  then  a/c , b/a , c/b  are  in

a) AP
b) GP
c) HP
d) None of the given

46)

If the equation  (3X)2  +
(27
x
31/p – 15) X  + 4  = 0  has equal roots, then p is
equal

a) -1/2
b) 1/2
c) -1/3
d) 1/3

47) If a, b, c are real and a + b + c=0, then the quadratic equation 4ax+3bx+2c = 0 has
a) one positive and one negative root
b) imaginary roots
c) two real roots
d) None of the given

48)

If the roots of the equation á x2-2ax+a2+a-3=0
are real and less than 3, then

a) a < 2
b)

2 ú
a ú
c)

3 <
a ú 4

d) a > 4

49)

If  one root of the equation ax2+
bx + c = 0  is reciprocal of the one root of the equation a1x2+
b1x +c1 = 0, then

a)

(cc1– aa1)2 =
(ba1 – b1c)  (ab1-c1b)

b)

(ab1– a1b)2 =
(bc1-b1c)  (ca1-c1a)

c)

(bc1– b1c)2
= (ca1– a1c)  (ab1– a1b)

d) None of the given

50)

If the ratio of the roots of the equation
x2+px+q=0   be equal to the ratio of the roots of x2+lx+m=0,
then

a)

p2m = q2 l

b)

pm2 = q2 l

c)

p2 l = q2m

d)

p2m = l2 q

51)
a) a > 0
b)

b2 < 4 ac

c) c > 0
d) a and b are of opposite signs