If $x+iy = (1+i)^6 - (1-i)^6$,then which one of the following is true?

If $e^{i x}$ is a solution of the equation $z^n+p_1 z^{n-1}+p_2 z^{n-2}+\ldots+p_n=0$,where $p_i$ are real $(i=1, 2, \ldots, n)$,then $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x + \sin(0) = $ (Note: The constant term in the equation is $p_n$ and the coefficient of $z^0$ is $1$ if we normalize,but here the equation is given as $z^n + p_1 z^{n-1} + \ldots + p_n = 0$. Let us assume the constant term is $p_n$. The expression to evaluate is $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x + \sin(0)$). Given the standard form,find the value of $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x$.

Let $z$ and $w$ be two complex numbers such that $|z| \le 1$,$|w| \le 1$ and $|z + iw| = |z - i\overline{w}| = 2$. Then $z$ is equal to

If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3}$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

The complex numbers $\sin x + i\cos 2x$ and $\cos x - i\sin 2x$ are conjugate to each other for

$(r, \theta)$ denotes $r(\cos \theta + i \sin \theta)$. If $x = (1, \alpha)$,$y = (1, \beta)$,$z = (1, \gamma)$ and $x + y + z = 0$,then $\sum \cos (2\alpha - \beta - \gamma) = $