Complex Number

Q: Roots of the equation xⁿ –1 = 0, n ∈ I,

(A) are collinear (B) lie on a circle.

(C) form a regular polygon of unit circum-radius .

(D) are non-collinear.

Sol. Clearly, roots are 1, α, α² , . . . α^n-1 , where $\large \alpha = cos\frac{2\pi}{n} + i sin\frac{2\pi}{n}$ .

The distance of the complex numbers represented by these roots from origin is 1 i.e. all these points lie on a circle.

⇒ They are non-collinear.

⇒ They form a regular polygon of unit circum-radius.

Hence (B), (C) and (D) are the correct answers.

Q: Let ‘z’ be a complex number and ‘a’ be a real parameter such that z² + az + a² = 0, then

(A) locus of z is a pair of straight lines

(B) locus of z is a circle

(C) arg(z) = ± 2π/3

(D) |z| =|a|

Q: If f(x) and g(x) are two polynomials such that the polynomial $h(x) = x f(x^3) + x^2 g(x^6)$ is divisible by x² +x +1 , then

(A) f(1) = g(1)

(B) f(1) = – g( 1)

(C) f(1) = g(1) ≠ 0

(D) f(1) = -g(1) ≠ 0

Q: If the reflection of the line $\bar{a}z + a \bar{z}= 0 $ in the real axis is $\bar{\alpha}z + \alpha \bar{z}= 0 $ in the simplest form, then

(A) α + a is purely real

(B) $\bar{\alpha} – a $ is purely real

(C) $ \alpha -\bar{a}$ is purely real

(D) $ \bar{\alpha} -\bar{a}$ is purely imaginary