Properties of Argument

Q. Let ${\theta }_1,{\theta }_2,\cdots ,{\theta }_{10}$ be positive valued angles (in radian) such that ${\theta }_1+{\theta }_2+\cdots +{\theta }_{10}=2\pi$. Define the complex numbers $z_1=e^{i{\theta }_1},z_k=z_{k-1}{e}^{i{\theta }_k}$ for $k=2,3\cdots 10,$ where $i=\sqrt{-1}$. Consider the statements $P$ and $Q$ given below:
$P:|z_2-z_1|+|z_3-z_2|+\cdots +|z_{10}-z_9|+|z_1-z_{10}|\le 2\pi$
$Q:|z_2^2-z_1^2|+|z_3^2-z_2^2|+\cdots +|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|\le 4\pi$

1. $P$ is TRUE and $Q$ is FALSE
2. $Q$ is TRUE and $P$ is FALSE
3. Both $P$ and $Q$ are TRUE
4. Both $P$ and $Q$ are FALSE

Q.

The maximum value of $\sin \left(x+\frac{\mathrm{\pi }}{6}\right)+\cos \left(x+\frac{\mathrm{\pi }}{6}\right)$ in the interval $\left(0,\frac{\mathrm{\pi }}{2}\right)$ is attained at

1. $x=\frac{\mathrm{\pi }}{12}$

2. $x=\frac{\mathrm{\pi }}{6}$

3. $x=\frac{\mathrm{\pi }}{3}$

4. $x=\frac{\mathrm{\pi }}{2}$

Q.

The argument of $\frac{1-i}{1+i}$ is

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

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

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

4. $\frac{5\pi }{2}$

Q. For a non-zero complex number $z$, let $\arg \left(z\right)$ denote the principal argument with $-\pi <\arg \left(z\right)\le \pi$. Then which of the following statement(s) is (are) $\text{FALSE}$?

1. $\arg (-1-i)=\frac{\pi }{4},$ where $i=\sqrt{-1}$
2. The function $f:R\to (-\pi ,\pi ]$, defined by $f\left(t\right)=\arg (-1+it)$ for all $t\in R$, is continuous at all points of $R$, where $i=\sqrt{-1}$
3. For any two non-zero complex numbers $z_1$ and $z_2$, $\arg \left(\frac{z_1}{z_2}\right)-\arg \left(z_1\right)+\arg \left(z_2\right)$ is an integer multiple of $2\pi$.
4. For any three given distinct complex numbers $z_1,z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $\arg \left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right)=\pi$,
   lies on a straight line.

Q.

The amplitude of $\sin \frac{\pi }{5}+i\left(1-\cos \frac{\pi }{5}\right)$ is

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

2. $\frac{\mathrm{\pi }}{15}$

3. $\frac{\mathrm{\pi }}{10}$

4. $\frac{\mathrm{\pi }}{5}$

Q. Let $z$ and $w$ be two non- zero complex numbers such that $|z|=|w|$ and $\arg \left(z\right)+\arg \left(w\right)=\pi$. Then the value of $(z+\overline{w}{)}^{10}$ is

Q.

Find the principal argument of $(1+i\sqrt{3})$.

Q. If $x=9^{1/3}\cdot 9^{1/9}\cdot 9^{1/27}\cdots \infty ;y=4^{1/3}\cdot 4^{-1/9}\cdot 4^{1/27}\cdots \infty$ and $z=\sum _{r=1}^{\infty }(1+i{)}^{-r}$ and the principal argument of the complex number $p=x+yz$ is $-{\tan }^{-1}\frac{\sqrt{a}}{b}$, then the value of $a^2+b^2$ is ($a$ and $b$ are coprime natural numbers)

Q. If $z=\frac{(1+i)(1+2i)(1+3i)\dots \dots \dots (1+ni)}{(1-i)(2-i)(3-i)\dots \dots \dots (n-i)}$, where $i=\sqrt{-1},n\in N$, then principal argument of $z$ can be -

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

Q.

Convert $\frac{\left(i+1\right)}{\cos \left({\displaystyle \frac{\pi }{4}}\right)-i\sin \left({\displaystyle \frac{\pi }{4}}\right)}$ in polar form

1. $\cos \left(\frac{\pi }{4}\right)+i\sin \left(\frac{\pi }{4}\right)$

2. $\cos \left(\frac{\pi }{2}\right)-i\sin \left(\frac{\pi }{2}\right)$

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

4. $2\left(\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)\right)$

Q.

The differential equation of the systems of all circles of radius $r$ in the $xy$-plane is

1. ${\left[1+{\left(\frac{dy}{dx}\right)}^3\right]}^2=r^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$

2. ${\left[1+{\left(\frac{dy}{dx}\right)}^3\right]}^2=r^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^3$

3. ${\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^3=r^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$

4. ${\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^3=r^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^3$

Q.

The principal argument of $-2\sqrt{3}-2i$ is

1. $-\frac{5\mathrm{\pi }}{6}$

2. $\frac{5\mathrm{\pi }}{6}$

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

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

Q.

The value of ${\left(1+\sqrt{3}i\right)}^4+{\left(1-\sqrt{3}i\right)}^4$ is

1. $-16$

2. $16$

3. $14$

4. $-14$

Q. Let $\frac{\sin A}{\sin B}=\frac{\sin (A-C)}{\sin (C-B)}$, where $A,B,C$ are angles of a triangle $ABC$. If the lengths of the sides opposite these angles are $a,b,c$ respectively, then

1. $a^2,b^2,c^2$ are in A.P.
2. $b^2-a^2=a^2+c^2$
3. $b^2,c^2,a^2$ are in A.P.
4. $c^2,a^2,b^2$ are in A.P.

Q. The principal argument of $\frac{\left[\right(1+i{)}^5(1+\sqrt{3}i{)}^2]}{-2i[\sqrt{3}-1-(\sqrt{3}+1\left)i\right]}$ is

1. $\frac{7\pi }{12}$
2. $\frac{\pi }{2}$
3. $\frac{5\pi }{6}$
4. $-\frac{5\pi }{12}$

Q. The point of intersection of the curves $\arg (z-3i)=\frac{3\pi }{4}$ and $\arg (2z+1-2i)=\frac{\pi }{4}$ is

1. Given curves do not intersect
2. $(\frac{3}{2},\frac{3}{2})$
3. $(\frac{3}{4},\frac{9}{4})$
4. $(\frac{9}{4},\frac{3}{4})$

Q. If cos $\alpha +2\cos \beta +3\cos \gamma =\sin \alpha +2\sin \beta +3sin\gamma =0,$then the value of sin 3$\alpha$ + 8sin 3$\beta$ + 27 sin 3$\gamma$ is

1. 3sin (a + + )
2. 18 sin (+ + )
3. sin (+ + 3 )
4. sin (a + b + )

Q.

If $\sqrt{x}+\frac{1}{\sqrt{x}}=2\cos \theta$, then $x^6+\frac{1}{x^6}=$

1. $2\cos 6\theta$

2. $2\cos 12\theta$

3. $2\cos 3\theta$

4. $2\sin 3\theta$

Q. The principal argument of $z=-3+3i$ is:

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

Q. A point $P\left(z_1\right)$ lies on the curve $|z|=2$. A pair of tangents from the point $P$ is drawn to the curve $|z|=1$ meeting it at points $Q\left(z_2\right)$ and $R\left(z_2\right)$. Then

1. $(\frac{4}{z_1}+\frac{1}{z_2}+\frac{1}{z_3})(\frac{4}{\overline{z_1}}+\frac{1}{\overline{z_2}}+\frac{1}{\overline{z_3}})=9$
2. $\arg \left(\frac{z_2}{z_3}\right)=\frac{2\pi }{3}$
3. point with complex representation $\left(\frac{z_1+z_2+z_3}{3}\right)$ lies on the curve $|z|=1$
4. point with complex representation $\left(\frac{z_1}{2}\right)$ lies on the curve $|z|=1$

Q. If $z_1$ and $z_2$ are conjugate to each other, and $\arg (-z_1{z}_2)=k\pi$, then $k=$

Q. If $\arg \left(z\right)$ denotes the principal argument of a complex number $z$, then the value of the expression $\arg (-i{e}^{\frac{i\pi }{9}}.z^2)+2\arg (2i{e}^{\frac{-i\pi }{18}}.\overline{z})$ is

1. $0$
2. $\frac{\pi }{2}$
3. $\pi$
4. $\arg \left(z\right)$

Q. If $z_1$ and $z_2$ are non zero solutions of equation $z^2+z=i\overline{z}$ where $i=\sqrt{-1}$ , then the value of $|z_1+z_2|$ is

Q. If $z=\sqrt{3}+i$ and $w=3i$
Then $\arg \left(zw\right)$ and $\arg \frac{w}{z}$ is

1. $\arg \left(zw\right)=\frac{2\pi }{3},\arg \left(\frac{w}{z}\right)=\frac{\pi }{3}$
2. $\arg \left(zw\right)=\frac{2\pi }{3},\arg \left(\frac{w}{z}\right)=\frac{2\pi }{3}$
3. $\arg \left(zw\right)=\frac{\pi }{3},\arg \left(\frac{w}{z}\right)=\frac{\pi }{3}$
4. $\arg \left(zw\right)=\frac{\pi }{3},\arg \left(\frac{w}{z}\right)=\frac{\pi }{5}$

Q. If $\arg \left(z_1\right)={170}^{\circ }$ and $\arg \left(z_2\right)={70}^{\circ }$, then the principal argument of $z_1{z}_2$ is

1. ${120}^{\circ }$
2. $-{120}^{\circ }$
3. $-{240}^{\circ }$
4. ${240}^{\circ }$

Q.

${\int }_0^1\frac{d\mathrm{x}}{\mathrm{x}+\sqrt{1-{\mathrm{x}}^2}}=$

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

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

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

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

Q. For two complex numbers $z_1$ and $z_2;(a{z}_1+b\overline{z_1})(c{z}_2+d\overline{z_2})=(c{z}_1+d\overline{z_1})(a{z}_2+b\overline{z_2})$ $b\ne 0,d\ne 0$ if

1. $\frac{a}{b}=\frac{c}{d}$
2. $\frac{a}{d}=\frac{b}{c}$
3. $|z_1|=|z_2|$
4. $\arg \left(z_1\right)=\arg \left(z_2\right)$

Q. In $\Delta ABC$, if $a=2,b=3$ and $\sin A=\frac{2}{3},$ then $\cos C$ is equal to:

1. $\frac{1}{2}$
2. $\frac{2}{3}$
3. $\frac{2}{\sqrt{13}}$
4. $\frac{1}{\sqrt{13}}$

Q. Let $z$ be a complex number lying in first or fourth quadrant of Argand plane satisfying $|z-1|=1.$ If $\arg (z-1)=k\arg \left(z\right),$ then the value of $k$ is

Q. If $\alpha$ and $\beta$ are two distinct complex numbers satisfying $|\alpha {|}^2\beta -|{\beta }^2|\alpha =\alpha -\beta ,$ then
(Here, $\arg \left(z\right)$ denotes the principal argument with $-\pi <\arg \left(z\right)\le \pi$)

1. $\arg \left(\frac{\alpha }{\beta }\right)=\pi$
2. $\alpha \overline{\beta }=\beta \overline{\alpha }$
3. $\alpha \overline{\beta }=1$
4. $|\alpha |=|\beta |$