Inequality of 2 Complex Numbers

Q.

The statement $(p\to (q\to p\left)\right)\to (p\to (p\vee q\left)\right)$ is:

1. equivalent to $(p\vee q)\wedge (~p)$

2. equivalent to $(p\wedge q)\vee (~p)$

3. a contradiction

4. a tautology

Q. If |z - 25i| $\le$ 15, then |max amp(z) - min.amp(z)| =

1. $co{s}^{-1}\left(\frac{3}{5}\right)$
2. $\pi -2co{s}^{-1}\left(\frac{3}{5}\right)$
3. $\frac{\pi }{2}+co{s}^{-1}\left(\frac{3}{5}\right)$
4. $si{n}^{-1}\left(\frac{3}{5}\right)-co{s}^{-1}\left(\frac{3}{5}\right)$

Q. The locus of the complex number z is an argand plane satisfying the inequality $lo{g}_{\frac{1}{2}}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1(where|z-1|\ne \frac{2}{3})$

1. None of these
2. a circle
3. interior of a circle
4. exterior of a circle

Q.

If $z=x+iy$ and $w=\frac{1-zi}{z-i},$ show that $|W|=1\Rightarrow z$ is purely real.

Q. The locus of z satisfying the inequality $lo{g}_{1/3}|z+1|>lo{g}_{1/3}|z-1|$ is

1. I (z) < 0
2. R (z) < 0
3. R (z) > 0
4. None of these

Q.

Find the position of the circles $x^2+y^2-2x-6y+9=0and{x}^2+y^2+6x-2y+1=0$ with respect to each other.

1. Touch each other externally

2. Intersect each other at two points

3. Touch each other internally

4. One circle lies completely outside the other circle

5. One circle lies completely inside the other circle

Q.

If 'z' be any complex number such that |3z - 2| + |3z + 2| = 4, then locus of 'z' is:

1. A circle

2. An ellipse

3. A line segment

4. None of these

Q. Let $z=a+ib$ (where $a,b\in R$ and $i=\sqrt{-1})$ such that $|2z+3i|=|z^2|.$ Identify the correct statement(s)?

1. $|z{|}_{\text{maximum}}$ is equal to $3$.
2. $|z{|}_{\text{minimum}}$ is equal to $1$.
3. If $|z|\text{is maximum}$, then $a^3+b^3$ is equal to $27$.
4. $|z|\text{is minimum}$, then $(a^2+2b^2)$ is $2$.

Q.

If $z_1=2-i$ and $z_2=1+i$, find $\left|\frac{z_1+z_2+1}{z_1-z_2+1}\right|$

Q.

The locus of z satisfying the inequality $lo{g}_{\frac{1}{3}}$|z+1| > $lo{g}_{\frac{1}{3}}$|z-1| is

1. R(z)<0

2. R(z)>0

3. I(z)<0

4. None of these

Q. Let $z_1$ and $z_2$ be two complex numbers represented by points on the circle $|z_1|=1$ and $|z_2|=2$ respectively. Let $P$ be the point whose coordinates are $(1,1).$ Then

1. the maximum value of $|2z_1+z_2|$ is $4.$
2. the minimum value of $|z_1-z_2|$ is $1.$
3. $|z_2+\frac{1}{z_1}|\le 3$
4. the minimum distance of $P$ from $z_1$ is $2-\sqrt{2}$ unit.

Q.

If $|z-\frac{1}{z}|$ = 1, then:

1. 

2. 

3. 

4. 

Q. If |z - 25i| $\le$ 15, then |max amp(z) - min.amp(z)| =

1. $co{s}^{-1}\left(\frac{3}{5}\right)$
2. $\pi -2co{s}^{-1}\left(\frac{3}{5}\right)$
3. $\frac{\pi }{2}+co{s}^{-1}\left(\frac{3}{5}\right)$
4. $si{n}^{-1}\left(\frac{3}{5}\right)-co{s}^{-1}\left(\frac{3}{5}\right)$

Q. If $|z-2|=min\{|z-1|,|z-5|\}$, where $z$ is a complex number, then possible value(s) of $Re\left(z\right)$ is/are

1. $\frac{3}{2}$
2. $\frac{5}{2}$
3. $\frac{7}{2}$
4. $\frac{9}{2}$

Q. If at least one value of the complex number z = x + iy satisfy the condition $|z+\sqrt{2}|=a^2-3a+2$ and the inequality $|z+i\sqrt{2}|=a^2,$ then

1. a > 2
2. a = 12
3. a < 2
4. None of these

Q. If the imaginary part of $\frac{2z+1}{iz+1}$ is $-2$, then show that the locus of the point representing $z$ in the argand plane is a straight line.

Q.

If $|z-\frac{4}{z}|=2$, then the greatest value of $|z|$ is

1. $1+\sqrt{2}$

2. $2+\sqrt{2}$

3. $\sqrt{3}+1$

4. $\sqrt{5}+1$

Q. Write down the conjugate of each of the following:

(i)$(-5+\sqrt{-1})$

(ii)$(-6-\sqrt{3})$

(iii)$i^3$

(iv)$(4+5i{)}^2$

Q. The locus of complex number $z$ satisfying $\arg [z-(1+i\left)\right]=\{\begin{array}{cc}\frac{3\pi }{4}& ;|z|\le |z-2|\\ -\frac{\pi }{4}& ;|z|>|z-2|\end{array}$ is

1. a straight line passing through $(2,0)$.
2. a straight line passing through $(2,0),(1,1)$.
3. is a line segment.
4. is a set of two rays.

Q.

If $|z-\frac{4}{z}|=2$, then the greatest value of $|z|$ is

1. $1+\sqrt{2}$

2. $2+\sqrt{2}$

3. $\sqrt{3}+1$

4. $\sqrt{5}+1$

Q.

If $z_1=(2-i)$ and $z_2=(1+i),$ find $\left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|$.

Q.

Write the mathematical form of following statement using signs and symbols :

$x$ is multiplied by the sum of $y$ and $3$

Q. From the point $(1,-2,3)$, lines are drawn to meet the sphere $x^2+y^2+z^2=4$ and they are divided internally in the ratio $2:3$. The locus of the point of division is

1. $5x^2+5y^2+5z^2-2xy-3yz-zx-6x+12y+5z+22=0$
2. $5x^2+5y^2+5z^2=0$
3. $5x^2+5y^2+5z^2-6x+12y+2z=0$
4. None of these

Q. The points representing the complex numbers z for which $|z+4{|}^2-|z-4{|}^2=8$ lie on

1. a straight line parallel to x-axis
2. a straight line parallel to y-axis
3. a circle with centre as origin
4. a circle with centre other than the origin