Question

Column I		Column II
A	If the equations $(x-2)^4-(x-2)=0$ and $x^2-k x+k=0, k \in R$ have two roots in common then $k$ equals	P	0
B	Let $Z =\left(\cos 12^{\circ}+i \sin 12^{\circ}+\cos 48^{\circ}+i \sin 48^{\circ}\right)^6$ then $\operatorname{Im}( z )$ is equal to	Q	1
C	If $w$ is one of the imaginary cube root of unity then the sum $1(2-w)\left(2-w^2\right)+2(3-w)\left(3-w^2\right)+$ $(n-1)(n-w)\left(n-w^2\right)=220$ The value of $n$ equals	R	3
D	If $|z-2|=\min \{|z|,|z-4|\}$ then possible value(s) of $\operatorname{Re}(z)$ will be	S	5

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

Answer 1

Answer: (C)

(A) Roots of $1^{1 t}$ are $2,3,(2+w),\left(2+w^2\right)$ where $w$ is the cube roots of unity.
2 and 3 can not be the common root
$\therefore$ common roots are $2+ w , 2+ w ^2$
equation of $x^2-3 x+3=0 \Rightarrow k=3$
(B)$Z =\cos 12^{\circ}+\cos 48^{\circ}+i\left(\sin 48^{\circ}+\sin 12^{\circ}\right) \cos 30^{\circ} \cdot \cos 18^{\circ}+2 i \sin 30^{\circ} \cdot \cos 18^{\circ} $
$=2 \cos 18^{\circ}\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)$
$ =2 \cos 18^{\circ}[\cos (\pi / 6)+i \sin (\pi / 6)]$
$Z ^6 =2^6 \cdot \cos ^6 18^{\circ}(\cos \pi+i \sin \pi) $
$ =-2^6\left(\cos ^6 18^{\circ}\right)^6+0 i$
$\operatorname{Im} Z ^6=0$
(C)$\text { sum }=\displaystyle\sum_{n=1}^n(n-1)(n-w)\left(n-w^2\right)=\displaystyle\sum_{n=1}^n\left(n^3-1\right)=\left(\sum_{n=1}^n n^3\right)-n=220 $
$\text { or } \left[\frac{n(n+1)}{2}\right]^2-n=220$
if $n =5$, then LHS $=225-5=220$
Hence $n =5$
(D) Case-I: $|z|<|z-4| \ldots .(1)$
then $|z-2|=|z| \Rightarrow \operatorname{Re}(z)=1$
which satisfy equation (1)
$\text { Case-II: } |z|>|z-4| \ldots .(2) $
$ \text { then }|z-2|=|z-4| \Rightarrow \operatorname{Re}(z)=3 $
$ \text { which satisfy equation (2) } $
$ \text { so } \operatorname{Re}(z) \text { can be } 1 \text { and } 3 $