Properties of Argument

Q.

$cot\left[{\cos }^{-1}\left(\frac{7}{25}\right)\right]=?$

1. $\frac{25}{24}$

2. $\frac{25}{7}$

3. $\frac{24}{25}$

4. None of these

Q.

If the $5^{th}$ term of a G.P. is $\frac{1}{3}$ and $9^{th}$ term is $\frac{16}{243}$, then the $4^{th}$ term will be

1. 

2. 

3. 

4. 

Q. Let $z_1$ and $z_2$ be two complex numbers such that $\overline{z_1}+i\overline{z_2}=0,\arg \left(z_1{z}_2\right)=\pi$. Then which of the following is/are ture?

1. $\arg z_1\in (0,\frac{5\pi }{6})$
2. $\arg z_1\in (0,\frac{\pi }{2})$
3. $\arg z_2\in (\frac{\pi }{3},\frac{2\pi }{3})$
4. $\arg z_2\in (0,\frac{\pi }{2})$

Q.

If $\alpha \in \left(0,\frac{\pi }{2}\right)$, then $\sqrt{(x^2+x)}+\frac{\tan 2\alpha }{\sqrt{(x^2+x)}}$ is always greater than or equal to

1. $2\tan \alpha$

2. $1$

3. $2$

4. $se{c}^2\alpha$

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. Let z, w be complex numbers such that $z+iw=0$ and arg $zw=\pi$ then arg z equals

1. 3π / 4
2. π / 4
3. 5π / 4
4. π / 2

Q. If $\arg \left(z\right)<0,$ then $\arg (-z)-\arg \left(z\right)=$

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

Q. If arg (z) < 0, then arg (-z)-arg (z) equals

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

Q.

If z is a complex number $\overline{z^{-1}}\left(\overline{z}\right)$ = , then

1. 1

2. -1

3. 0

4. None of these

Q.

If the $4^{th},{10}^{th}and{16}^{th}$ terms of a G.P, are x, y and z respectively. Prove that x, y, z are in G.P

Q.

If arg($z_1$) = 11 $\frac{\pi }{18}$ and arg ($z_2$) = 19 $\frac{\pi }{18}$, then the principal argument of $z_1$$z_2$ is

1. (-5 π)/3

2. (7 π)/3

3. (-π)/3

4. None of these

Q. Let $z$ and $\omega$ be complex numbers such that $\overline{z}+i\overline{\omega }=0$ and $argz\omega =\pi$. Then $argz$ equals

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

Q. Let z, w be complex numbers such that $z+iw=0$ and arg $zw=\pi$ then arg z equals

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

Q.

Let z and $\omega$ be complex numbers such that $\overline{z}+i\overline{\omega }=0$ and arg $z\omega =\pi$. Then arg z equals

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

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

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

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

Q.

$\sqrt{3}+i=(a+ib)(c+id),then{\tan }^{-1}\frac{b}{a}+{\tan }^{-1}\frac{d}{c}$

has the value

1. 2n$\pi$+$\frac{\pi }{3}$, n$ϵ$I

2. n$\pi$+$\frac{\pi }{6}$, n$ϵ$I

3. n$\pi$-$\frac{\pi }{3}$, n$ϵ$I

4. 2n$\pi$-$\frac{\pi }{6}$, n$ϵ$I

Q.

Three distinct real numbers, a, b, c are in G.P. such that a + b+ c= x b, then

1. -1<x<2

2. -1<x<3

3. 0<x<1

4. x<-1 or x>3

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. If $a=\cos \alpha +i\sin \alpha ,b=\cos \beta +i\sin \beta ,$
$c=\cos \gamma +i\sin \gamma$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=1,$ then

1. $\cos (\alpha -\beta )+\cos (\beta -\gamma )+\cos (\gamma -\alpha )=0$
2. $\sin (\alpha -\beta )+\sin (\beta -\gamma )+\sin (\gamma -\alpha )=0$
3. $\cos (\alpha -\beta )+\cos (\beta -\gamma )+\cos (\gamma -\alpha )=1$
4. $\sin (\alpha -\beta )+\sin (\beta -\gamma )+\sin (\gamma -\alpha )=1$

Q. Let $z_1$ and $z_2$be nth roots of unity which subtend a right angle at the origin, then n must be of the form (where, k is an integer)

1. 4k+1
2. 4k+2
3. 4k+3
4. 4k

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 ${\theta }_1$ and ${\theta }_2$ are arguments of two non-zero complex numbers $z_1$ and $z_2$ respectively such that $3|z_1|=4|z_2|$. If $z=\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$, then

1. $|z|=\sqrt{\frac{5}{2}}$
2. $\text{Re}\left(z\right)=\frac{5}{2}\cos ({\theta }_1-{\theta }_2)$
3. $|z|=\sqrt{\frac{5}{4}}$
4. $\text{Re}\left(z\right)=\frac{5}{2}\cos ({\theta }_1+{\theta }_2)$

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

Q. If $z=(1+i)(1-i\sqrt{3})(-2-2i)\left(i\right)\left(3\right),$ then

1. $|z|=8$
2. $\text{arg}\left(z\right)=-\frac{\pi }{3}$
3. $|z|=24$
4. $\text{arg}\left(z\right)=\frac{\pi }{6}$

Q. Let $z$ be a complex number such that $\arg \left(\frac{z-2}{2z+3i}\right)=0$, then the locus of $z$ is

1. a straight line with slope $\frac{4}{3}$
2. a pair of straight lines passing through the origin
3. a straight line with slope $\frac{3}{4}$
4. a pair of straight lines passing through $(1,1)$

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

Q. Argument of the complex number $\frac{1}{(\sqrt{3}-i{)}^{25}}$ is

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

Q. If z and $\omega$ are two non-zero complex numbers such that $|z\omega |=1$ and arg $\left(z\right)-arg\left(\omega \right)=\frac{\pi }{2},$ then $\overline{z}\omega$ is equal to

1. 1
2. i
3. -1
4. -i

Q. If $z_1,z_2$ and $z_3,z_4$ are two pairs of conjugate complex numbers, then the value of $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ is

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