Geometrical Representation of Argument and Modulus

Q. If $z$ be a complex number satisfying $|Re\left(z\right)|+|Im\left(z\right)|=4$, then $|z|$ cannot be:

1. $\sqrt{7}$
2. $\sqrt{10}$
3. $\sqrt{8}$
4. $\sqrt{\frac{17}{2}}$

Q. The imaginary part of $(z-1)(\cos \alpha -i\sin \alpha )+(z-1{)}^{-1}\times (\cos \alpha +i\sin \alpha )$ is zero, if

1. $|z-1|=1$
2. $\arg (z-1)=2\alpha$
3. $\arg (z-1)=\alpha$
4. $|z-1|=2$

Q. The continued product of the four values of ${[\cos \left(\frac{\pi }{3}\right)+i\sin \left(\frac{\pi }{3}\right)]}^{3/4}$ is

1. $-1$
2. $1$
3. $-2$
4. $2$

Q. If ${}^{\prime }{z}^{\prime }$ lies on the circle $|z-2i|=2\sqrt{2}$, then the value of $\arg \left(\frac{z-2}{z+2}\right)$ is equal to

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

Q. The number of complex numbers $z$ that satisfies both the equation $|z-1-i|=\sqrt{2}$ and $|z+1+i|=2$, is

1. $2$
2. $4$
3. infinite
4. $1$

Q. The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|}{\log }_{e}2\right)\ge {\log }_{\sqrt{2}}|5\sqrt{7}+9i|,$ $i=\sqrt{-1},$ is equal to :

1. $2$
2. $3$
3. $8$
4. $\sqrt{5}$

Q. If $z$ is a complex number such that $|3z-2|+|3z+2|=4$, then the locus of $z$ is

1. a line
2. a circle
3. a point
4. a line segment

Q. If $z=-3+4i$ and $zw=-14+2i$ then

1. $\arg w=\frac{\pi }{4}$
2. $\arg w=-\frac{\pi }{4}$
3. $|w|=2\sqrt{2}$
4. $|w|=4$

Q. For a complex number $z,$ if $|z-1+i|+|z+i|=1,$ then the range of the principle argument of $z$ is
$($ Here, principle argument $\in (-\pi ,\pi ])$

1. $[\frac{-\pi }{4},\frac{\pi }{4}]$
2. $[\frac{\pi }{4},\frac{\pi }{2}]$
3. $[\frac{-\pi }{2},\frac{-\pi }{4}]$
4. $[\frac{-\pi }{2},\frac{\pi }{2}]$

Q. Let $z$ be a complex number such that $|2z+\frac{1}{z}|=1$ and $\arg \left(z\right)=\theta$, then minimum value of $8{\sin }^2\theta$ is

1. $0$
2. $8$
3. $7$
4. $5$

Q.

The projection of the line segment joining the points $(-1,0,3)$ and $(2,5,1)$ on the line whose direction ratios are $6,2,3$ is

1. $\frac{10}{7}$

2. $\frac{22}{7}$

3. $\frac{18}{7}$

4. none of these

Q. If $z_1,z_2,z_3,z_4,z_5$ and $z_6$ are vertices in anticlockwise direction of a regular hexagon whose circumcentre is origin and vertex $z_1=2+6i,$ then

1. $z_4=-2+6i$
2. $z_2=(1-3\sqrt{3})+i(3+\sqrt{3})$
3. $z_4=-2-6i$
4. $z_2=(1-3\sqrt{3})-i(3+\sqrt{3})$

Q. If $|z_1|=|z_2|=\cdots |z_n|=1$ then which of the following are true?

1. ${\overline{z}}_1=\frac{1}{z_1}$
2. $|z_1+z_2+\cdots z_n|=|\frac{1}{z_1}+\frac{1}{z_2}+\cdots \frac{1}{z_n}|$
3. centroid of polygon with $2n$ vertices $z_1,z_2,\cdots z_n,\frac{1}{z_1},\frac{1}{z_2},\cdots \frac{1}{z_n}$ (need not be in order) lies on real axis.
4. centroid of polygon with $2n$ vertices $z_1,z_2,\cdots z_n,\frac{1}{z_1},\frac{1}{z_2},\cdots \frac{1}{z_n}$ (need not be in order) lies on imaginary axis.

Q. If $z_1,z_2,z_3\text{and}{z}_4$ are any four complex numbers such that $|z_1|<\frac{1}{\pi },|z_2|=1,|z_3|\le 1$ and $z_3=\frac{z_2(z_1-z_4)}{\overline{z_1}{z}_4-1}.$ Then possible value of $|z_4|$ is /are

1. $2{\text{log}}_{3}2$
2. ${\text{log}}_{3}4$
3. ${\text{log}}_e\pi$
4. ${\text{log}}_{\pi }e$

Q. If $\left|\frac{z_1}{z_2}\right|=1$ and $\arg \left(z_1{z}_2\right)=0$, then

1. $z_1=z_2$
2. $z_1=2z_2$
3. $z_1{z}_2=1$
4. $|z_2{|}^2=z_1{z}_2$

Q. The four lines drawn from the vertices of any tetrahedron to the centroid of the opposite faces meet in a point whose distance from each vertex is k times the distance from each vertex to the opposite face, where k is

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

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

1. $|z|=2\sqrt{2}$
2. $\arg \left(z\right)=\frac{11\pi }{12}$
3. $|z|=2$
4. $\arg \left(z\right)=\frac{13\pi }{12}$

Q.

If $15{\sin }^4\left(\theta \right)+10{\cos }^4\left(\theta \right)=6,$ for some $\theta \in R$ then the value of $27{\sec }^6\left(\theta \right)+8{\mathrm{cosec}}^6\left(\theta \right)$ is equal to:

1. $250$

2. $500$

3. $400$

4. $350$

Q. If ‘z’ be any complex number such that $\mid 3z-2\mid +\mid 3z+2\mid =4$, then locus of ‘z’ is

1. A line segment
2. None of these
3. An ellipse
4. A circle

Q. Polar form of $z=\frac{(1+7i)}{(2-i{)}^2}$ is

1. $\sqrt{2}(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4})$
2. $2(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4})$
3. $\sqrt{2}(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$
4. $2(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$

Q. Let $z=\left(\frac{\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta }\right),\frac{\pi }{4}<\theta <\frac{\pi }{2},$ then $\arg \left(z\right)$ will be

1. $2\theta$
2. $\pi -2\theta$
3. $\theta$
4. $\pi +2\theta$

Q. If $z=(4{\sin }^2\theta -1)+i({\cos }^2\theta +1)$ is purely imaginary number, then the number of value(s) of $\theta \in [0,2n\pi ]$ where $n\in I$ is

1. $n$
2. $2n$
3. $4n$
4. $0$

Q. If $z_1=7+24i$ and $|z_2|=5,$ then

1. Maximum value of $|z_1+z_2|$ is $30$
2. Maximum value of $|z_1+z_2|$
   $=$ Maximum value of $|z_1-z_2|$
3. Minimum value of $\left|\frac{z_1}{z_2+\frac{5}{z_2}}\right|$ is $\frac{25}{6}$
4. Maximum value of $\left|\frac{z_1}{z_2+\frac{5}{z_2}}\right|$ is $5$

Q. If $z_1$ and $z_2$ satisfy $z+\overline{z}=2|z-1|$ and $\arg (z_1-z_2)=\frac{\pi }{4},$ then $Im(z_1+z_2)=$

Q.

If arg ($Z_1$) = $\alpha$ , arg ($Z_2$) = $\beta$, the value of $|Z_1+Z_2{|}^2$ is

1. 

2. 

3. 

4. 

Q. If $z_1$, $z_2$ are the complex numbers such that $|z_1+z_2|=|z_1|+|z_2|$ then arg $z_1-$ arg $z_2$ is

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

Q. If for a complex number $z_1$ and $z_2$, $\arg \left(z_1\right)-\arg \left(z_2\right)=0$, then $|z_1-z_2|$ is equal to

1. $|z_1|+|z_2|$
2. $|z_1+z_2|$
3. $||z_1|-|z_2||$
4. $0$

Q. If $z$ is a complex number such that $|3z-2|+|3z+2|=4$, then the locus of $z$ is

1. a line
2. a circle
3. a point
4. a line segment

Q. Assertion :Statement -1: If $z_1$ and $z_2$ are two complex numbers such that $|z_1|=|z_2|+|z_1-z_2|$, then $Im\left(\frac{z_1}{z_2}\right)=0$ Reason: Statement -2: $arg\left(z\right)=0\Rightarrow z$ is purely real.

1. Statement -1 is true, Statement -2 is true ; Statement -2 is a correct explanation for Statement -1
2. Statement-1 is true, Statement-2 is true ; Statement-2 is not a correct explanation for Statement-1
3. Statement -1 is true, Statement -2 is false
4. Statement -1 is false, Statement -2 is true

Q.

$A\left(z_1\right)$ and $B\left(z_2\right)$ are the points on the circles |z| = 1 and |z| = 3 respectively, then:

1. 

2. 

3. 

4.