Properties of Modulus

Q. Let $z_1,z_2,z_3$ be three complex numbers such that $|z_1|=|z_2|=|z_3|=1$ and $\frac{z_1^2}{z_2{z}_3}+\frac{z_2^2}{z_1{z}_3}+\frac{z_3^2}{z_1{z}_2}=-1.$ Then the possible value(s) of $|z_1+z_2+z_3|$ is/are

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

Q.

Let $f\left(x\right)$ be a function whose domain is $\left[-5,7\right]$. Let $g\left(x\right)=\left|2x+5\right|$. Then what is the domain of $\left(fog\right)\left(x\right)$.

1. $\left[-4,1\right]$

2. $\left[-5,1\right]$

3. $\left[-6,1\right]$

4. None of these

Q.

If $-\pi <\text{arg}\left(z\right)<-\frac{\pi }{2}$, then $\text{arg}\left(\overline{\mathrm{z}}\right)-\text{arg}\left(-\overline{\mathrm{z}}\right)$ is equal to

1. $\pi$

2. $-\pi$

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

4. $-\frac{\pi }{2}$

Q.

Let $A=\left\{\theta :\tan \theta +sec\theta =\sqrt{2}sec\theta \right\}$ and $B=\left\{\theta :sec\theta -\tan \theta =\sqrt{2}\tan \theta \right\}$ be $2$ sets. Then:

1. $A=B$

2. $A\subset B$

3. $A\ne B$

4. $B\subset A$

Q. If $z_1=5+12i$ and $|z_2|=4$, then

1. Maximum value of $|z_1+i{z}_2|$ is $17$
2. Minimum value of $|z_1+(1+i)z_2|$ is $13-4\sqrt{2}$
3. Minimum value of $\left|\frac{z_1}{z_2+\frac{4}{z_2}}\right|$ is $\frac{13}{4}$
4. Maximum value of $\left|\frac{z_1}{z_2+\frac{4}{z_2}}\right|$ is $\frac{13}{3}$

Q.

If ${\log }_{10}\left(x^3+y^3\right)-{\log }_{10}\left(x^2+y^2-xy\right)\le 2$, the maximum value of $xy$, for all $x\ge 0,y\ge 0$ is

1. $2500$

2. $3000$

3. $1200$

4. $3500$

Q. If $z_1$ and $z_2$ are any two complex numbers, then $|z_1+\sqrt{z_1^2-z_2^2}|+|z_1-\sqrt{z_1^2-z_2^2}|$ is equal to

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

Q. If $f_1\left(x\right)=\frac{x}{x-1}$ and $f_n\left(x\right)=f_1\left(f_{n-1}\right(x\left)\right)$ for $n\ge 2$, then the integral value of $x$ that satisfies $f_{101}\left(x\right)=3x$ is

Q.

Write the value of ${30}^3+{20}^3-{50}^3.$

Q. Let $z_1,z_2,z_3$ be three complex numbers such that $|z_1|=|z_2|=|z_3|=1$ and $\frac{z_1^2}{z_2{z}_3}+\frac{z_2^2}{z_1{z}_3}+\frac{z_3^2}{z_1{z}_2}=-1.$ Then the possible value(s) of $|z_1+z_2+z_3|$ is/are

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

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

1. $|z|=|w|=\frac{1}{2}$
2. $|z|=\frac{1}{2},|w|=\frac{3}{4}$
3. $|z|=|w|=\frac{3}{4}$
4. $|z|=|w|=1$

Q. Number of turning points for the modulus function given as
is

Q. If Z is a complex number such that $\left|z\right|$ greater than or equal to 2, then the minimum value of $|z+\frac{1}{2}|$.

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

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

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

4. Less than $\frac{2}{3}$

Q. Let $x_n,y_n,z_n,w_n$ denote $n^{th}$ terms of four different arithmetic progressions with positive terms. If $x_4+y_4+z_4+w_4=8$ and $x_{10}+y_{10}+z_{10}+w_{10}=20$ then the maximum value of $x_{20}\cdot y_{20}\cdot z_{20}\cdot w_{20}$ is

1. ${10}^8$
2. ${10}^4$
3. ${10}^6$
4. ${10}^{10}$

Q. Let $z_1$ be a complex root of the equation $a_n{z}^n+a_{n-1}{z}^{n-1}+\cdots +a_{1}z+a_0=3(|z|<1)$, where $|a_i|<2$ for $i=0,1,2,\dots ,n$. Then

1. $\frac{1}{3}<|z_1|<1$
2. $|z_1|<\frac{1}{3}$
3. $\frac{1}{2}<|z_1|<1$
4. $|z_1|<\frac{1}{2}$

Q. If $|Z_1+Z_2|$ = $\left|Z_1\right|+\left|Z_2\right|$, then find the value of arg $\left(\frac{Z_1}{Z_2}\right)$

1. $\frac{\pi }{3}$

2. $\frac{\pi }{4}$

3. $\pi$

4. 0

Q.

If z is a complex number, then the minimum value of |z|+|z-1| is

1. 1

2. 0

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

4. None of these

Q. If $z_1$ and $z_2$ are complex numbers and $u=\sqrt{z_1{z}_2},$ then $|\frac{z_1+z_2}{2}+u|+|\frac{z_1+z_2}{2}-u|$ is equal to

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

Q. Let complex numbers $\alpha$ and $\frac{1}{\overline{\alpha }}$ lie on circles $(x-x_0{)}^2+(y-y_0{)}^2=r^{2}and(x-x_0{)}^2+(y-y_0{)}^2=4r^2$, respectively.
If $z_0=x_0+i{y}_0$ satisfies the equation $2|z_0{|}^2=r^2+2$, then $|\alpha |$ is equal to

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

Q. If $|z_1|=1,|z_2|=2,|z_3|=3$ and $|9z_1{z}_2+4z_1{z}_3+z_2{z}_3|=12,$ then the value of $|z_1+z_2+z_3|$ is

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

Q. If $-3$ and $|5-4p|$ have the same absolute value and the sum of all possible values of $p$ is $k$, then the value of $2k$ is

Q. The distance 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 $z=0$ is

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

Q. If $z_1=24+7i$ and $|z_2|=6$ then $|z_1+z_2|$ lies in

1. $[12,25]$
2. $[19,31]$
3. $[6,25]$
4. $[13,37]$

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

1. $|z|=|w|=\frac{1}{2}$
2. $|z|=\frac{1}{2},|w|=\frac{3}{4}$
3. $|z|=|w|=\frac{3}{4}$
4. $|z|=|w|=1$

Q. If $f:R\to R,g:R\to R,h:R\to R$ be three functions given by $f\left(x\right)=x^2-1,g\left(x\right)=\sqrt{x^2+1},$ $h\left(x\right)=\{\begin{array}{cc}0,& x\le 0\\ x,& x>0\end{array}$ then $\left(hofog\right)\left(x\right)=$

1. $\{\begin{array}{cc}{x}^2+1,& x>0\\ x^2-1,& x\le 0\end{array}$
2. $\{\begin{array}{cc}{x}^2+1,& x>0\\ 0,& x\le 0\end{array}$
3. $\{\begin{array}{cc}-x^2,& x<0\\ 0,& x=0\\ x^2& x>0\end{array}$
4. $\{\begin{array}{cc}{x}^2,& x\ne 0\\ 0,& x=0\end{array}$

Q. For any complex number $z$, the minimum value of $|z|+|z-3i|$ is

Q. Let $z$ be a complex number such that $|z|=z+32-24i$. Then $Re\left(z\right)+Im\left(z\right)$ is equal to

1. $17$
2. $31$
3. $25$
4. $-17$

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

1. number of solutions for $z$ is $1$
2. number of solutions for $z$ is $2$
3. number of solutions for $w$ is $1$
4. number of solutions for $w$ is $2$

Q. For any two complex numbers $z_1$ and $z_2,$ if $|z_1|=2$ and $|z_2|=3$, then the value of $|3z_1+2z_2{|}^2+|3z_1-2z_2{|}^2$ is

Q. Let $a,b\in R$ and $a^2+b^2\ne 0$. Suppose $S=\{z\in C:z=\frac{1}{a+ibt},t\in R,t\ne 0\},$ where $i=\sqrt{-1}$. If $z=x+iy$ and $z\in S,$ then $(x,y)$ lies on

1. the circle with radius $\frac{1}{2a}$ and centre $(\frac{1}{2a},0)$for $a>0,b\ne 0$
2. the circle with radius $-\frac{1}{2a}$ and centre $(-\frac{1}{2a},0)$for $a<0,b\ne 0$
3. the $x$-axis for $a\ne 0,b=0$
4. the $y$-axis for $a=0,b\ne 0$