Odisha Board Solutions for Chapter: Complex Numbers and Quadratic Equations, Exercise 1: EXERCISES 6 (a)

Express with rational denominator: $\frac{3+2i}{2-5i}+\frac{3-2i}{2+5i}$

Find the square of $\sqrt{9+40i}+\sqrt{9-40i}$.

Express the given complex number in the $r,\theta$ form.

$1+i\sqrt{2}$

Express the given complex number in the $r,\theta$ form :

$i\sqrt{3}$

If $1,\omega ,{\omega }^2$ are the three cube roots of unity, prove that ${\left(2+5\omega +2{\omega }^2\right)}^6={\left(2+2\omega +5{\omega }^2\right)}^6=729$.

If $1,\omega ,{\omega }^2$ are the three cube roots of unity, prove that $\left(1-\omega +{\omega }^2\right)\left(1-{\omega }^2+{\omega }^4\right)\left(1-{\omega }^4+{\omega }^8\right)\dots$ to $2n$ factors $=2^{2n}$

If $\omega$ is one imaginary cube root of unity then prove that $x^3+y^3+z^3-3xyz=(x+y+z)\left(x+\omega y+{\omega }^{2}z\right)\left(x+y{\omega }^2+z\omega \right)$

If $ax+cy+bz=\mathrm{X}, cx+by+az=\mathrm{Y}, bx+ay+cz=\mathrm{Z}$ show that
$\left(a^2+b^2+c^2-ab-bc-ca\right)\left(x^2+y^2+z^2-xy-yz-zx\right)={\mathrm{X}}^2+{\mathrm{Y}}^2+{\mathrm{Z}}^2-\mathrm{YZ}-\mathrm{ZX}-\mathrm{XY}$