Question

Column I		Column II
A	If the equations $(x-2{)}^4-(x-2)=0$ and $x^2-kx+k=0,k\in R$ 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 $\mathrm{Im}(z)$ 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 $\mathrm{Re}(z)$ 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 $1^{1t}$ 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 $x^2-3x+3=0\Rightarrow k=3$
(B)$Z=\cos 12^{\circ }+\cos 48^{\circ }+i\left(\sin 48^{\circ }+\sin 12^{\circ }\right)\cos 30^{\circ }\cdot \cos 18^{\circ }+2i\sin 30^{\circ }\cdot \cos 18^{\circ }$
$=2\cos 18^{\circ }\left(\cos 30^{\circ }+i\sin 30^{\circ }\right)$
$=2\cos 18^{\circ }[\cos (\pi /6)+i\sin (\pi /6)]$
$Z^6=2^6\cdot {\cos }^{6}18^{\circ }(\cos \pi +i\sin \pi )$
$=-2^6{\left({\cos }^{6}18^{\circ }\right)}^6+0i$
$\mathrm{Im}{Z}^6=0$
(C)$\text{ sum }=\sum _{n=1}^n(n-1)(n-w)\left(n-w^2\right)=\sum _{n=1}^n\left(n^3-1\right)=\left(\sum _{n=1}^n{n}^3\right)-n=220$
$\text{ or }{\left[\frac{n(n+1)}{2}\right]}^2-n=220$
if $n=5$, then LHS $=225-5=220$
Hence $n=5$
(D) Case-I: $|z|<|z-4|\dots .(1)$
then $|z-2|=|z|\Rightarrow \mathrm{Re}(z)=1$
which satisfy equation (1)
$\text{ Case-II: }|z|>|z-4|\dots .(2)$
$\text{ then }|z-2|=|z-4|\Rightarrow \mathrm{Re}(z)=3$
$\text{ which satisfy equation (2) }$
$\text{ so }\mathrm{Re}(z)\text{ can be }1\text{ and }3$