Basic Properties

Q.

How do you find the exact value for $\cos (240°)$?

Q.

Prove that $\tan (180-x)=-\tan \left(x\right)$

Q. The conjugate point of 1 + i with respect to the circle | z - 1 | = 2 is

1. 1 - i
2. 1 + 4i
3. 1 + 2i
4. -1 - i

Q. Given two complex number
$Z_1=\frac{7}{\sqrt{3}}+7iand{Z}_2=9+\left(9\sqrt{3}\right)i$
The argument of $Z_1{Z}_2$ (in degree) is

1. 90
2. 150
3. 120
4. 180

Q. A complex function
$f\left(z\right)=u+iv$ is defined as $f\left(z\right)=\frac{z-i}{z+1}$ where $z=x+iy$ then the imaginary part of f(z) is

1. $\frac{x^2+y^2+x-y}{x^2+y^2+2x+1}$
2. $\frac{y-x-1}{x^2+y^2+2x+1}$
3. $\frac{-1}{\sqrt{x^2+y^2}+1}$
4. $\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2}+1}$

Q. The modulus of the complex number is $\left(\frac{6+8i}{1-2i}\right)$

1. $2\sqrt{5}$
2. $\sqrt{5}$
3. $\frac{\sqrt{5}}{2}$
4. $\frac{2}{\sqrt{5}}$

Q. $\text{The smallest positive integer n for which}{\left(\frac{1+i}{1-i}\right)}^n=1\text{is\_\_\_\_\_.}$

1. 4

Q. The complex number 2(1 + i) can be represented in polar form as,

1. $2\sqrt{2}{e}^{i\pi /4}$
2. $\frac{\pi }{4}{e}^{i2\sqrt{2}}$
3. $\sqrt{2}{e}^{i\pi /4}$
4. $\frac{\pi }{4}{e}^{i\sqrt{2}}$

Q. If a complex number then |z| would be
$z=\frac{20}{\sqrt{3}-\sqrt{-2}}+\frac{30}{3\sqrt{-2}-2\sqrt{3}}-\frac{14}{2\sqrt{3}-\sqrt{-2}}$

1. 0
2. 1
3. $\frac{1}{210}$
4. none

Q. The value $x$ and $y$ where $\frac{(\cos \theta +i\sin \theta {)}^7}{(\sin \theta +i\cos \theta {)}^4}=x+iy$ is

1. $x=\sin 11\theta ,y=\cos 11\theta$
2. $x=\sin 3\theta ,y=\cos 3\theta$
3. $x=\cos 3\theta ,y=\sin 3\theta$
4. $x=\cos 11\theta ,y=\sin 11\theta$

Q. The modulus of the complex number$\left(\frac{6i-3}{12i+9}\right)$ is

1. $5$
2. $\sqrt{5}$
3. $\frac{1}{5}$
4. $\frac{1}{\sqrt{5}}$