Question

Column I		Column II
A	Let w be a non real cube root of unity then the number of distinct elements in the set $\left\{{\left(1+w+w^2+\dots \dots +w^n\right)}^m\mid m,n\in N\right\}$ is	P	4
B	Let $1,w,w^2$ be the cube root of unity. The least possible degree of a polynomial with real coefficients having roots $2w,(2+3w),\left(2+3w^2\right),\left(2-w-w^2\right)$, is	Q	5
C	$\alpha =6+4i$ and $\beta =(2+4i)$ are two complex numbers on the complex plane. A complex number z satisfying amp $\left(\frac{z-\alpha }{z-\beta }\right)=\frac{\pi }{6}$ moves on the major segment of a circle whose radius is	R	6
		S	7

• A. (A) R; (B) Q; (C) P
• B. (A) R; (B) Q; (C) Q
• C. (A) R; (B) Q; (C) R
• D. (A) R; (B) Q; (C)S

Answer 1

Answer: (A)

(A)
${\left(-w^2\right)}^m\in \left\{-1,w,w^2,-w^2\right\}$ depending upon the value of $m$.
$\Rightarrow$ answer is $6;\left\{0,1,-1,w,-w^2,w^2\right\}\Rightarrow$(R)
(B) ${\alpha }_1=2w;{\alpha }_2=2w^2\text{ (conjugate roots) }$
${\beta }_1=2+3w^2;{\beta }_2=2+3w^2\text{ (conjugate roots) }$
${\gamma }_1=2+3w^2;{\gamma }_2=2+3w$
${\delta }_1=3$
$\Rightarrow \text{ Roots are }2w;2w^2;2+3w;2+3w^2;3\Rightarrow (Q)$
(C)
Refer to the figure
$\sin 30^{\circ }=\cos 60^{\circ }=\frac{2}{R}$
$\Rightarrow R=4\text{ Ans. }\Rightarrow (P)$
$\therefore \mathrm{Im}(z)=0$