Modulus of a Complex Number

Q.

Derivative of mod $x$ is

Q.

Find the domain of the real function, $f\left(x\right)=\frac{1}{\sqrt{x+|x|}}$

Q. If $z_1$ and $z_2$ are complex numbers such that $Re\left(z_1\right)=|z_1-2|,Re\left(z_2\right)=|z_2-2|$ and $\arg (z_1-z_2)=\frac{\pi }{3},$ then the value of $Im(z_1+z_2)$ is

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

Q.

The modulus and amplitude of ${\left(1+i\sqrt{3}\right)}^8$ are respectively,

1. $256$ and $\frac{\mathrm{\pi }}{3}$

2. $256$ and $\frac{2\mathrm{\pi }}{3}$

3. $2$ and $\frac{2\mathrm{\pi }}{3}$

4. $256$ and $\frac{8\mathrm{\pi }}{3}$

Q. A ray of light coming from the point (1, 2) is reflected at a point A on the X-axis and then passes through the point (5, 3). Find the coordinates of the point A.

Q.

If $z=\frac{{\left(\sqrt{3}+i\right)}^3{\left(3i+4\right)}^2}{{\left(8+6i\right)}^2}$, then $\left|z\right|$ is equal to

Q.

If z be a complex number satisfying $z^4$ + $z^3$ +$2z^2$ + z + 1 = 0 Then the value of $\left|z\right|$.

__

Q. Find the modulus and argument of the complex number .

Q.

If a circle passes through the point $(0,0),(a,0),(0,b)$, then its centre is

1. $(a,b)$

2. $(b,a)$

3. $\left(\frac{a}{2},\frac{b}{2}\right)$

4. $\left(\frac{b}{2},\frac{-a}{2}\right)$

Q. If $z=x+iyandw=\frac{1-iz}{z-i},$ then $|w|=1$ implies that, on the complex plane

1. z lies on the real axis
2. None of the above
3. z lies on the imaginary axis
4. z lies on the unit circle

Q.

Draw the graph of the identity function $f:R\to R:f\left(x\right)=x$ for all $xϵR$

Q. In the xy plane, the segment with end points (3, 8) and (–5, 2) is the diameter of the circle. The point (k, 10) lies on the circle for

1. exactly one non integral k
2. no value of k
3. exactly one integral k
4. two real values of k

Q.

Find the modulus and the arguments of each of the complex numbers
$z=-\sqrt{3}+i$

Q.

Find the number of non-zero integral solutions of the equation $|1-i{|}^x=2^x$.

Q.

A circle touches the y-axis at the point (0, 4) and cuts the x-axis in a chord of length 6 units. The radius of the
circle is

1. 3

2. 4

3. 5

4. 6

Q.

Find the equation of straight line which is parallel to $2x+3y+11=0$ and sum of its intercepts on the axes is 15.

Q.

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

1. 4

2. 3

3. 2

4. 1

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. Function $f\left(x\right)=|x-1|+|x-2|+\cos x$ where $x\in [0,4]$ is not continuous at number of points

1. 1
2. 2
3. 3
4. 0

Q. The value of Z satisfying the equation $logZ+log{z}^2+log{z}^3+.....+log{z}^n=0$ is

1. 

2. 

3. 

4. 

Q.

The region of argand plane defined by |z+1|+|z-1|$\le$4 is

1. Interior and boundary of ellipse

2. Interior and boundary of parabola

3. Interior and boundary of a hyperbola

4. Interior and boundary of circle

Q. If $|z^2-1|=|z{|}^2+1$, then z lies on

1. the real axis
2. an ellipse
3. a circle
4. the imaginary axis

Q.

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

1. 4

2. 3

3. 2

4. 1

Q. If $z=x+iyandw=\frac{1-iz}{z-i},$ then $|w|=1$ implies that, on the complex plane

1. z lies on the imaginary axis
2. z lies on the real axis
3. z lies on the unit circle
4. None of the above

Q. If ( a + ib ) ( c + id ) ( e + if ) ( g + ih ) = A + i B, then show that ( a 2 + b 2 ) ( c 2 + d 2 ) ( e 2 + f 2 ) ( g 2 + h 2 ) = A 2 + B 2 .

Q.

The distances of the roots of the equation $|\sin {\theta }_1|z^3+|\sin {\theta }_2|z^2+|\sin {\theta }_3|z+|\sin {\theta }_4|=3$, from $z=0$, are

1. Greater than $\frac{2}{3}$

2. less than $\frac{2}{3}$

3. greater than $|\sin {\theta }_1|+|\sin {\theta }_2|+|\sin {\theta }_3|+|\sin {\theta }_4|$

4. les than $|\sin {\theta }_1|+|\sin {\theta }_2|+|\sin {\theta }_3|+|\sin {\theta }_4|$

Q. The number of complex numbers z such that $|z-1|=|z+1|=|z-i|$ equals

1. $\infty$
2. 2
3. 1
4. 0

Q. Modulus of $\left(\frac{3+2i}{3-2i}\right)$ is
[RPET 1996]

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

Q. Length of the line segment joining the points $-1-i$ and $2+3i$ on the complex plane is

1. 5
2. -5
3. 7
4. -7

Q. Find the modulus and the argument of the complex number