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.**Statement 1: If a,b,c∈Z and ax

^{2}+bx+c=0 has an irrational root, then |f(Î»)|≥1/q

^{2}, where Î»∈(Î»=p/q;p,q∈Z) and f(x)=ax

^{2}+bx+c

Statement 2: If a,b,c∈Q and b

^{2}-4ac is positive but not a perfect square, then roots of equation ax

^{2}+bx+c=0 are irrational and always occur in conjugate pair like 2+√3 and 2-√3

**Q2.**Statement 1: If equations ax

^{2}+bx+c=0 and x

^{2}-3x+4=0 have exactly one root common, then at least one of a,b,c is imaginary

Statement 2: If a,b,c are not all real, then equation ax

^{2}+bx+c=0 can have one root real and one root imaginary

**Q3.**Statement 1: Locus of z, satisfying the equation |z-1|+|z-8|=5 is an ellipse

Statement 2: Sum of focal distances of any point on ellipse is constant

**Q4.**Statement 1: If cos

^{2}Ï€/8 is a root of the equation x

^{2}+ax+b=0 where a,b∈Q, then ordered pair (a,b) is [-1,(1/8)]

Statement 2: If a+mb=0 and m is irrational, then a,b=0

**Q5.**Consider the function f(x)=log

_{e}(ax

^{3}+(a+b)x

^{2}+(b+c)x+c)

Statement 1: Domain of the functions is (-1,∞)~{-(b/2a)}, where a>0,b

^{2}-4ac=0

Statement 2: ax

^{2}+bx+c=0 has equal roots when b

^{2}-4ac=0

**Q6.**If z

_{1}≠-z

_{2}and |z

_{1}+z

_{2}|=|(1/z

_{1})+(1/z

_{2})| then

Statement 1: z

_{1}z

_{2}is unimodular

Statement 2: z

_{1}and z

_{2}both are unimodular

**Q7.**Statement 1: If z

_{1}+z

_{2}=a and z

_{1}z

_{2}=b, where a=¯a and b=¯b, then arg(z

_{1}z

_{2})=0

Statement 2: The sum and product of two complex numbers are real if and only if they are conjugate of each other

**Q8.**Statement 1: If all real values of x obtained from the equation 4

^{x}-(a-3) 2

^{x}+(a-4)=0 are non-positive, then a∈(4,5]

Statement 2: If ax

^{2}+bx+c is non-positive for all real values of x, then b

^{2}-4ac must be negative or zero and ‘a’ must be negative

**Q9.**Statement 1: If px

^{2}+qx+r=0 is a quadratic equation (p,q,r∈R) such that its roots are Î±,Î² and p+q+r<0,p-q+r<0 and r>0, then [Î±]+[Î²]=-1, where [∙] denotes greatest integer function

Statement 2: If for any two real numbers a and b, function f(x) is such that f(a)f(b)<0⇒f(x) has at least one real root lying in (a,b)

**Q10.**Statement 1: If a>0 and b

^{2}-ac<0, then domain of the function f(x)=√(ax

^{2}+2bx+c) is R

Statement 2: If b

^{2}-ac<0, then ax

^{2}+2bx+c=0 has imaginary roots