1. Tardigrade
  2. Question
  3. Mathematics
  4. Column I Column II A Let w be a non real cube root of unity then the number of distinct elements in the set (1+ w + w 2+ ldots ldots+ w n ) m mid m , n ∈ N 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 2 w,(2+3 w),(2+3 w2),(2-w-w2), is Q 5 C α=6+4 i and β=(2+4 i ) are two complex numbers on the complex plane. A complex number z satisfying amp ((z-α/z-β))=(π/6) moves on the major segment of a circle whose radius is R 6 S 7

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Q.
Column I Column II
A Let w be a non real cube root of unity then the number of distinct elements in the set $\left\{\left(1+ w + w ^2+\ldots \ldots+ w ^{ n }\right)^{ m } \mid m , n \in N \right\}$ 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 $2 w,(2+3 w),\left(2+3 w^2\right),\left(2-w-w^2\right)$, is Q 5
C $\alpha=6+4 i$ and $\beta=(2+4 i )$ 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

Complex Numbers and Quadratic Equations

Solution:

(A)image
$\left(- w ^2\right)^{ m } \in\left\{-1, w , w ^2,- w ^2\right\}$ 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=2 w ; \alpha_2=2 w ^2 \text { (conjugate roots) } $
$\beta_1=2+3 w^2 ; \beta_2=2+3 w^2 \text { (conjugate roots) }$
$\gamma_1=2+3 w^2 ; \gamma_2=2+3 w$
$\delta_1=3$
$\Rightarrow \text { Roots are } 2 w ; 2 w^2 ; 2+3 w ; 2+3 w^2 ; 3 \Rightarrow (Q)$
(C) image
Refer to the figure
$\sin 30^{\circ}=\cos 60^{\circ}=\frac{2}{R} $
$\Rightarrow R=4 \text { Ans. } \Rightarrow (P)$
$\therefore \operatorname{Im}(z)=0$