Modulus of a Complex Number

Q.

If ω is a complex cube root of unity and if 1, ω and ω² are the cube roots of unity and $\left[\begin{array}{cc}\omega & 0\\ 0& \omega \end{array}\right]$, then $A^{50}=$

1. ${\omega }^{2}A$

2. $\omega A$

3. $A$

4. $0$

Q.

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$is defined by $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3$, then ${\mathrm{f}}^{-1}\left(8\right)$ is equal to

1. $\left\{2\right\}$

2. $\left\{2,2\mathrm{\omega },2{\mathrm{\omega }}^2\right\}$

3. $\left\{2,-2\right\}$

4. $\left\{2,2\right\}$

Q. If $z=x+iy$ and $|z-ai|=|z+ai|$, then the locus of $z$ is

1. real axis
2. imaginary axis
3. $x=y$
4. $x^2+y^2=1$

Q.

If $\omega$ is an imaginary cube root of unity and if $\left|\begin{array}{ccc}x+{\omega }^2& \omega & 1\\ \omega & {\omega }^2& 1+x\\ 1& x+\omega & {\omega }^2\end{array}\right|=0$ then one of the value of $x$ is

1. $1$

2. $0$

3. $-1$

4. $2$

Q. The locus of the centre of a circle which touches the circles $|z-z_1|=aand|z-z_2|=b$ externally where $z,z_1,and{z}_2$ are complex numbers is

1. an ellipse
2. a hyperbola
3. a circle
4. none of these

Q.

If $1,\omega and{\omega }^2$ are the cube roots of $1$ then $\left(1+\omega \right)\left(1+{\omega }^2\right)\left(1+{\omega }^4\right)\left(1+{\omega }^8\right)$ equals

1. $1$

2. $0$

3. $-1$

4. $2$

Q. The value of $\sec \left({\sec }^{-1}5\right)$ is

Q. If $x+y+z=1$, $x,y,z>0$. Then greatest value of $x^2{y}^3{z}^4$ is

1. $\frac{2^9}{3^5}$
2. $\frac{2^{10}}{3^{15}}$
3. $\frac{2^{15}}{3^{10}}$
4. $\frac{2^{10}}{3^{10}}$

Q. If complex number $z$ satisfies $|z|+z=2+i$, then $z$ is

1. $z=\frac{3}{4}+2i$
2. $z=\frac{3}{4}+i$
3. $z=1+i$
4. $z=\frac{3}{5}+i$

Q. If $z=\sqrt{\frac{1+i}{1-i}},$ then the modulus of $z$ is

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

Q.

Both the roots of the equation (x - a) ( x - b) + (x - b) (x - c) + (x - c) (x - a) = 0 are always

1. Positive

2. Negative

3. Real

4. Imaginary

Q. If $z_1=(2-i)$ and $z_2=(1+i),$ then $\left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|$ is equal to

1. $2\sqrt{2}$
2. $4\sqrt{2}$
3. $\sqrt{2}$
4. $8\sqrt{2}$

Q.

If $i{z}^3+z^2-z+i=0$, then $|z|$ equals

1. 4

2. 3

3. 2

4. 1

Q. If $z=\sqrt{\frac{1+i}{1-i}},$ then the modulus of $z$ is

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

Q. Let z and w be two complex numbers such that $|z|\le 1,|\omega |\le 1and|z+i\omega |=|z-i\overline{\omega }|=2,$ then z equals

1. 1 or i
2. i or -i
3. 1 or -1
4. i or -1

Q. The principal amplitude of $(2-i)(1-2i{)}^2$ is in the interval :

1. $(0,\frac{\pi }{2})$
2. $(-\frac{\pi }{2},0)$
3. $(-\pi ,-\frac{\pi }{2})$
4. $(\frac{\pi }{2},\pi )$

Q. If $z(2-2\sqrt{3}i{)}^2=i(\sqrt{3}+i{)}^4$, then $\arg \left(z\right)$ is

1. $\frac{5\pi }{6}$
2. $\frac{\pi }{6}$
3. $-\frac{\pi }{6}$
4. $\frac{2\pi }{3}$

Q. If $a=1+i^2+i^4+\cdots +i^{2n}$ and $b={\cos }^{-1}\left(\left|\frac{1}{1+i}\right|\right)$, where $i=\sqrt{-1}$, then the value of $\tan \left(\frac{b}{a}\right)$ is

1. $0$
2. $1$
3. cannot be determined
4. depends on the value of $b$

Q. If $z$ is a complex number satisfying $\arg (z+a)=\frac{\pi }{6}$ and $\arg (z-a)=\frac{2\pi }{3},a\in R^{+}$, then

1. $|z|=a$
2. $|z|=2a$
3. $\arg \left(z\right)=\frac{\pi }{2}$
4. $\arg \left(z\right)=\frac{\pi }{3}$

Q. If $z_1=(2-i)$ and $z_2=(1+i),$ then $\left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|$ is equal to

1. $2\sqrt{2}$
2. $4\sqrt{2}$
3. $\sqrt{2}$
4. $8\sqrt{2}$

Q. Given that $|z-1|=1$, where $z$ is a non zero point on the complex plane, then $\frac{z-2}{z}$ is equal to :

1. $i\tan \left(\arg z\right)$
2. $i\cot \left(\arg z\right)$
3. $\cot \left(\arg z\right)$
4. $\tan \left(\arg z\right)$

Q. If $z_1\ne 0$ and $z_2$ be two complex numbers such that $\frac{z_2}{z_1}$ is a purely imaginary number, then the value of $\left|\frac{2z_1+3z_2}{2z_1-3z_2}\right|$ is

1. $1$
2. $3$
3. $5$
4. $2$

Q.

$If|z|=1and\omega =\frac{z-1}{z+1}(where,z\ne -1),thenRe\left(\omega \right)is$

1. 0

2. $\frac{1}{|z+1{|}^2}$

3. $\left|\begin{array}{c}\frac{1}{z+1}\end{array}\right|.\frac{1}{|z+1{|}^2}$

4. $\frac{\sqrt{2}}{|z+1{|}^2}$

Q. If $z$ is a complex number of unit modulus and argument $\theta$, then $\text{arg}\left(\frac{1+z}{1+\overline{z}}\right)=$

1. $\frac{\pi }{4}$
2. $\theta$
3. $\frac{\pi }{2}-\theta$
4. $2\theta$

Q. If $z$ is a complex number of unit modulus and argument $\theta$, then $\text{arg}\left(\frac{1+z}{1+\overline{z}}\right)=$

1. $\frac{\pi }{4}$
2. $\theta$
3. $\frac{\pi }{2}-\theta$
4. $2\theta$

Q. The number of integral solutions of the expression $|1-i{|}^x=2^x$ is

Q. Given that $|z-1|=1$, where $z$ is a non zero point on the complex plane, then $\frac{z-2}{z}$ is equal to :

1. $i\tan \left(\arg z\right)$
2. $i\cot \left(\arg z\right)$
3. $\cot \left(\arg z\right)$
4. $\tan \left(\arg z\right)$

Q. For the complex number $z=\frac{3+\sqrt{-1}}{2-\sqrt{-1}}$, the correct option(s) is/are

1. $\text{Re}\left(z\right)=1$
2. $\text{Im}\left(z\right)=1$
3. $\text{Im}\left(z\right)=0$
4. $|z|=\sqrt{2}$

Q. The expression $(x^2+y^2{)}^4$ is equal to

1. $(x^4-2x^2{y}^2+y^4{)}^2+(4x^{3}y+4x{y}^3{)}^2$
2. $(x^4+6x^2{y}^2-y^4{)}^2+(4x^{3}y-4x{y}^3{)}^2$
3. $(x^4-6x^2{y}^2+y^4{)}^2+(4x^{3}y-4x{y}^3{)}^2$
4. None of these

Q. If $z=\frac{3}{2+\cos \theta +i\sin \theta }$, then locus of $z$ is

1. $x^2+y^2+4x+3=0$
2. $x^2+y^2+4x-3=0$
3. $x^2+y^2-4x-3=0$
4. $x^2+y^2-4x+3=0$