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 |z

_{1}|=1,|z

_{2}|=2,|z

_{3}|=3 and |z

_{1}+2z

_{2}+3z

_{3}|=6, then the value of |z

_{2}z

_{3}+8z

_{3}z

_{1}+27z

_{1}z

_{2}| is 36

Statement 2: | z

_{1}+z

_{2}+z

_{3}|≤|z

_{1}|+|z

_{2}|+|z

_{3}|

**Q2.**Statement 1: If f(x) is a quadratic polynomial satisfying f(2)+f(4)=0. If unity is a root of f(x)=0, then the other root is 3.5

Statement 2: If g(x)=px

^{2}+qx+r=0 has roots Î±,Î², then Î±+Î²=-q/p and Î±Î²=(r/p)

**Q3.**Statement 1: If both roots of the equation 4x

^{2}-2x+a=0,a∈R lie in the interval (-1,1), then-2<a≤1/4.

Statement 2: If f(x)=4x

^{2}-2x+a,then D≥0,f(-1)>0 and f(1)>0⇒-2<a≤1/4.

**Q4.**Statement 1: If a

^{2}+b

^{2}+c

^{2}<0, then if roots of the equation ax

^{2}+bx+c=0 are imaginary, then they are not complex conjugates

Statement 2: equation ax

^{2}+bx+c=0 has complex conjugate roots when a,b,c are real

**Q5.**Statement 1: The equation x

^{2}+(2m+1)x+(2n+1)=0, where m and n are integer cannot have any rational roots

Statement 2: The quantity (2m+1)

^{2}-4(2n+1), where m,n∈I can never be a perfect square

**Q6.**Statement 1: If n is an odd integer greater than 3 but not a multiple of 3, then (x+1)

^{n}-x

^{n}-1 is divisible by x

^{3}+x

^{2}+x

Statement 2: If n is an odd integer greater than 3 but not a multiple of 3, we have 1+Ï‰

^{n}+Ï‰

^{2n}=3

**Q7.**Statement 1: If x+(1/x)=1 and p=x

^{4000}+(1/x

^{4000}) and q be the digit at unit place in the number 2

^{(2n)}+1,n∈N and n>1, then the value of p+q=8

Statement 2: If Ï‰,Ï‰

^{2}are the roots of x+1/x=-1, then x

^{2}+1/x

^{2}=-1,x

^{3}+(1/x

^{3})=2

**Q8.**Statement 1: If 0<Î±<(Ï€/4), then the equation (x-sinÎ± )×(x-cosÎ± )-2=0 has both roots in (sinÎ±,cosÎ±)

Statement 2: If f(a) and f(b) possess opposite signs, then there exists at least one solution of the equation f(x)=0 in open interval (a,b)

**Q9.**Statement 1: Let z

_{1}and z

_{2}are two complex numbers such that |z

_{1}-z

_{2}|=|z

_{1}+z

_{2}| then the orthocenter of ∆AOB is [(z

_{1}+z

_{2})/2] (where O is origin)

Statement 2: In case of right-angled triangle, orthocenter is that point at which the triangle is right angled

**Q10.**Statement 1: If |(zz

_{1}-z

_{2})/(zz

_{1}+z

_{2})|=k,(z

_{1},z

_{2}≠0), then the locus of z is circle

Statement 2: As |(z-z

_{1})/(z-z

_{2})|=Î» represents a circle, if Î»∉{0,1}