If $(a + ib)(c + id)(e + if)(g + ih)$$ = A + iB,$ then $({a^2} + {b^2})({c^2} + {d^2})({e^2} + {f^2})({g^2} + {h^2})$ =

If $z = x + iy,\,{z^{1/3}} = a – ib$ and $\frac{x}{a} – \frac{y}{b} = k\,({a^2} – {b^2})$ then value of $k$ equals

Solve the equation $27 x^{2}-10 x+1=0$

For two non-zero complex number $z_1$ and $z_2$, if $\operatorname{Re}\left(z_1 z_2\right)=0$ and $\operatorname{Re}\left(z_1+z_2\right)=0$, then which of the following are possible ?

$(A)$ $\operatorname{Im}\left(z_1\right) > 0$ and $\operatorname{Im}\left(z_2\right) > 0$

$(B)$ $\operatorname{Im}\left(z_1\right) < 0$ and $\operatorname{Im}\left(z_2\right) > 0$

$(C)$ $\operatorname{Im}\left(z_1\right) > 0$ and $\operatorname{Im}\left(z_2\right) < 0$

$(D)$ $\operatorname{Im}\left( z _1\right) < 0$ and $\operatorname{Im}\left( z _2\right) < 0$

Choose the correct answer from the options given below :

If ${i^2} = – 1$, then the value of $\sum\limits_{n = 1}^{200} {{i^n}} $ is