Complex numbers and quadratic equations is a segment of maths that deals
with crucial theorems and concepts along with various formulae. It
comprises of linear and quadratic equations along with roots related to
the complex number's set (known as complex roots)..

**Q1.**The number of integral values of a for which the quadratic equation (x+a)(x+1991)+1=0 has integral roots are

Solution

(d)

(x+a)(x+1991)+1=0

⇒(x+a)(x+1991)=-1

⇒(x+a)=1 and x+1991=-1

⇒a=1993

or x+a=-1 and x+1991=1 ⇒a=1989

(d)

(x+a)(x+1991)+1=0

⇒(x+a)(x+1991)=-1

⇒(x+a)=1 and x+1991=-1

⇒a=1993

or x+a=-1 and x+1991=1 ⇒a=1989

**Q2.**Let z and Ï‰ be two complex numbers such that |z| ≤ 1, |Ï‰| ≤ 1 and |z - iÏ‰|= |z - i¯Ï‰|=2 then z equals

Solution

(c)

We have, 2 = |z-iÏ‰| ≤ |z| + |Ï‰| (∵|z

∴|z| + |Ï‰| ≥ 2 (i)

But given that |z|≤1 and |Ï‰|≤1. Hence

⇒|z| + |Ï‰|≤2 (ii)

From (i) and (ii),

|z|=|Ï‰|=1

Also, |z+iÏ‰|=|z-i¯Ï‰|

⇒|z-(-iÏ‰)|=|z-i¯Ï‰|

Hence, z lies on perpendicular bisector of the line segment joining (-iÏ‰) and (i¯Ï‰), which is a real axis, as (-iÏ‰) and (i¯Ï‰) are conjugate to each other. For z,Im(z)=0. If z= x, then

|z| ≤ 1 ⇒ x

⇒ -1 ≤ x ≤ 1

(c)

We have, 2 = |z-iÏ‰| ≤ |z| + |Ï‰| (∵|z

_{1}+z_{2}|≤|z_{1}|+|z_{2}|)∴|z| + |Ï‰| ≥ 2 (i)

But given that |z|≤1 and |Ï‰|≤1. Hence

⇒|z| + |Ï‰|≤2 (ii)

From (i) and (ii),

|z|=|Ï‰|=1

Also, |z+iÏ‰|=|z-i¯Ï‰|

⇒|z-(-iÏ‰)|=|z-i¯Ï‰|

Hence, z lies on perpendicular bisector of the line segment joining (-iÏ‰) and (i¯Ï‰), which is a real axis, as (-iÏ‰) and (i¯Ï‰) are conjugate to each other. For z,Im(z)=0. If z= x, then

|z| ≤ 1 ⇒ x

^{2}≤1⇒ -1 ≤ x ≤ 1

**Q3.**z

_{1}and z

_{2}lie on a circle with centre at the origin. The point of intersection z

_{3}of the tangents at z

_{1}and z

_{2}is given by

**Q4.**If Î± and Î² be the roots of the equation x

^{2}+px-1/(2p

^{2})=0 where p∈R. Then the minimum value of Î±

^{4}+Î²

^{4}is

**Q5.**If Î±,Î² are the roots of ax

^{2}+ c = bx, then the equation (a+ cy)

^{2}= b

^{2}y in y has the roots

**Q6.**If (cosÎ¸+i sinÎ¸ )(cos2Î¸+i sin2Î¸ )⋯(cosnÎ¸+i sinnÎ¸ ) = 1, then the value of Î¸ is, m∈N

**Q7.**The roots of the cubic equation(z + ab)

^{3}=a

^{3}, such that a ≠ 0, represent the vertices of a triangle of sides of length

**Q8.**A quadratic equation whose product of roots x

_{1}and x

_{2}is equal to 4 and satisfying the relation x

_{1}/(x

_{1}-1) + x

_{2}/( x

_{2}-1)=2 is

**Q9.**If the equation cot

^{4}x -2cosec

^{2}x + a

^{2}= 0 has at least one solution then, sum of all possible integral values of a is equal to

**Q10.**The number of irrational roots of the equation 4x/(x

^{2}+x+3)+5x/(x

^{2}-5x+3)=-3/2 is

Solution

(d)

Here, x=0 is not a root. Divide both the numerator and denominator by x and put x+3/x=y to obtain

4/(y+1)+5/(y-5)=-3/2⇒y=-5,3

x+3/x=-5 has two irrational roots and x+3/x=3 has imaginary roots

(d)

Here, x=0 is not a root. Divide both the numerator and denominator by x and put x+3/x=y to obtain

4/(y+1)+5/(y-5)=-3/2⇒y=-5,3

x+3/x=-5 has two irrational roots and x+3/x=3 has imaginary roots