Geometrical Representation of Algebra of Complex Numbers

Q. Let $z_1,z_2$ and $z_3$ be three complex numbers such that $|z_1|=|z_2|=|z_3|=1$ and $\frac{z_1^2}{z_2{z}_3}+\frac{z_2^2}{z_3{z}_1}+\frac{z_3^2}{z_1{z}_2}+1=0.$ Then the sum of all possible values of $|z_1+z_2+z_3|$ is

Q. A region $S$ in complex plane is defined by $S=\{x+iy:-1\le x,y\le 1\}$. A complex number $z=x+iy$ is chosen uniformly at random from $S.$ If $P$ be the probability that the complex number $\frac{3}{4}(1+i)z$ is also in $S,$ then the value of $27P$ is

Q. If a complex number $z$ satisfies $|2z+10+10i|<5\sqrt{3}-5$, then the least principal argument of $z$ is

1. $-\frac{5\pi }{6}$
2. $-\frac{3\pi }{4}$
3. $-\frac{11\pi }{12}$
4. $-\frac{2\pi }{3}$

Q. If $z,iz$ and $z+iz$ are the vertices of a triangle whose area is 2 units, then the value of $|z|$ is

1. -2
2. 2
3. 4
4. 8

Q. The area (in sq. units) of the region $A=\left\{\right(x,y):(x-1\left)\right[x]\le y\le 2\sqrt{x},0\le x\le 2\},$ where $\left[t\right]$ denotes the greatest integer function, is:

1. $\frac{4}{3}\sqrt{2}-\frac{1}{2}$
2. $\frac{8}{3}\sqrt{2}-\frac{1}{2}$
3. $\frac{8}{3}\sqrt{2}-1$
4. $\frac{4}{3}\sqrt{2}+1$

Q. If $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$, then area of the triangle whose vertices are $z_1,z_2,z_3$ is

1. $\frac{\sqrt{3}}{4}\text{sq. units}$
2. $1\text{sq. unit}$
3. $\frac{3\sqrt{3}}{4}\text{sq. unit}$
4. $2\text{sq. units}$

Q. Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1–t)z_1+t{z}_2$ for some real number $t$ with $0<t<1$. If $\text{Arg}\left(w\right)$ denotes the principal argument of a non-zero complex number $w$, then

1. $\text{Arg}(z–z_1)=\text{Arg}(z_2–z_1)$
2. $|z–z_1|+|z–z_2|=|z_1–z_2|$
3. $\text{Arg}(z–z_1)=\text{Arg}(z–z_2)$
4. $\left|\begin{array}{cc}z-z_1& \overline{z}-\overline{z_1}\\ z_2-z_1& \overline{z_2}-\overline{z_1}\end{array}\right|=0$

Q. Locus of complex number $z$ if $z,i$ and $iz$ are collinear is

1. $x^2+y^2-x-y=0$
2. $2x^2+2y^2+x+y=0$
3. $x^2+y^2+2x+2y=0$
4. $2x^2+2y^2-x-y=0$

Q. The roots of the equation $t^3+3a{t}^2+3bt+c=0$ are $z_1,z_2,z_3$ which represent the vertices of an equilateral triangle, then

1. $a^2=3b$
2. $b^2=a$
3. $a^2=b$
4. $b^2=3a$

Q. If $z_1,z_2,z_3$ are complex numbers such that $|z_1|=|z_2|=|z_3|=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1,$then $|z_1+z_2+z_3|$is

1. Less than 1
2. Greater than 3
3. Equal to 3
   
4. Equal to 1

Q. The locus of the complex number $z$ satisfying $|z-1|=|z-i|$ on the argand plane, is

1. a line passing through origin and $(1,1)$
2. a circle of radius $1\text{units}$
3. a circle of radius $\frac{1}{2}\text{units}$
4. a line passing through origin and $(1,-1)$

Q. If $z$ is a complex number such that $0\le \arg z\le \frac{\pi }{2}$, then which of the following inequality is true?

1. $|z-\overline{z}|\le |z|(\arg z-\arg \overline{z})$
2. $|z-\overline{z}|\ge |z|(\arg z-\arg \overline{z})$
3. $|z-\overline{z}|<|z|(\arg z-\arg \overline{z})$
4. None of these

Q. Show that points $A(4,2)B(7,5)C(9,7)$ are collinear.

Q. If $a,b$ are the numbers between $0$ and $1$ such that the points $z_1=a+i,z_2=1+bi$ and $z_0=0$ form an equilateral triangle, then

1. $a=2-\sqrt{3}\text{and}b=2-\sqrt{3}$
2. $a=2-\sqrt{3}\text{and}b=2+\sqrt{3}$
3. $a=2+\sqrt{3}\text{and}b=2-\sqrt{3}$
4. $a=2+\sqrt{3}\text{and}b=2+\sqrt{3}$

Q.

If z lies on the circle $|z-1|=1$, then $\frac{z-2}{z}$ equals

1. -1

2. 2

3. None of these

4. 0

Q. Let $f\left(x\right)=lnmx(m>0)$ and $g\left(x\right)=px$. Then the equation $|f\left(x\right)|=g\left(x\right)$ has exactly three solutions for

1. $p=\frac{m}{e}$
2. $0<p<\frac{m}{e}$
3. $0<p<\frac{e}{m}$
4. $p<\frac{e}{m}$

Q. Let $\alpha ,\beta \in R$ and $(1-\cos \alpha {)}^2+{\sin }^2\beta \ne 0$. Suppose $S=\{z\in C:z=\frac{1}{(1-\cos \alpha )+ik\sin \beta },k\in R-\{0\left\}\right\},$ where $i=\sqrt{-1}.$ If $z=x+iy,z\in S$ and $(x,y)$ lies on a circle, then

1. the minimum radius of the circle is $\frac{1}{4}$ and corresponding centre is $(\frac{1}{4},0)$
2. the minimum radius of the circle is $\frac{1}{2}$ and corresponding centre is $(\frac{1}{2},0)$
3. $\alpha \ne 2n\pi$ and $\beta \in R$
4. $\alpha =2n\pi$ and $\beta \ne n\pi$

Q. If $z_1,z_2,z_3$ are three complex numbers such that $5z_1-13z_2+8z_3=0$, then the value of $\left|\begin{array}{ccc}{z}_1& \overline{z_1}& 1\\ z_2& \overline{z_2}& 1\\ z_3& \overline{z_3}& 1\end{array}\right|$ is

Q. If a complex number $z$ satisfies $|2z+10+10i|<5\sqrt{3}-5$, then the least principal argument of $z$ is

1. $-\frac{5\pi }{6}$
2. $-\frac{11\pi }{12}$
3. $-\frac{3\pi }{4}$
4. $-\frac{2\pi }{3}$

Q. The locus of the centre of the circle $xcos\alpha +ysin\alpha =a$ and $xsin\alpha +ycos\alpha =b$

1. $x^2-y^2=a^2+b^2$
2. $x^2+y^2=a^2{b}^2$
3. $x^2+y^2=a^2+b^2$
4. $x^2+y^2=a^2-b^2$

Q. If $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$, then area of the triangle whose vertices are $z_1,z_2,z_3$ is

1. $\frac{3\sqrt{3}}{4}\text{sq. unit}$
2. $\frac{\sqrt{3}}{4}\text{sq. units}$
3. $1\text{sq. unit}$
4. $2\text{sq. units}$

Q. If $z_1,z_2,z_3$ are three complex numbers such that $5z_1-13z_2+8z_3=0$, then the value of $\left|\begin{array}{ccc}{z}_1& \overline{z_1}& 1\\ z_2& \overline{z_2}& 1\\ z_3& \overline{z_3}& 1\end{array}\right|$ is

Q. Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a=z^2+z+1$ is real. Then $a$ cannot take the value

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

Q. If $z$ is a complex number such that $0\le \arg z\le \frac{\pi }{2}$, then which of the following inequality is true?

1. $|z-\overline{z}|\le |z|(\arg z-\arg \overline{z})$
2. $|z-\overline{z}|\ge |z|(\arg z-\arg \overline{z})$
3. $|z-\overline{z}|<|z|(\arg z-\arg \overline{z})$
4. None of these

Q. If the imaginary part of $\frac{z-4}{z-2i}$ is $0$ , then the locus of $z$ is

1. Ellipse
2. Circle
3. Straight line
4. Parabola

Q. If a is an integer lying in $(-5,30]$, then probability that the graph of $y=x^2+2(a+4)x-5a+64$ is strictly above the x-axis is

1. $\frac{7}{36}$
2. $\frac{4}{5}$
3. $\frac{2}{9}$
4. $\frac{1}{5}$

Q. Match the following

Column - I	Column - II
(A) If z is a complex number satisfying $|z^3+z^{-3}|\le 2m\text{then}|z+\frac{1}{z}|$ can't take the value	(p) 4
(B) If z is a complex number satisfying $|z^3+z^{-3}|\le 2,\text{then}|z+\frac{1}{z}|$ can take the value	(q) 1
(C) $\left|{[i^{19}+\frac{1}{i^{25}}]}^2\right|$ equals	(r) 2
(D) ${\left(\frac{1+i}{1-i}\right)}^{2008}$ equals	(s) 5
	(t) 3

2. A(p, s, t), B(q, r, t), C(p), D(q)

Q. Find the principal and general value of ${\log }_e(-1+i)$

1. $\frac{1}{2}\log 2+\frac{3}{4}\pi i,\frac{1}{2}{\log }_{e}2+(n+\frac{3}{4})\pi i$ ($n$ integer)
2. $\frac{1}{2}\log 2+\frac{1}{4}\pi i,\frac{1}{2}{\log }_{e}2+(2n+\frac{1}{4})\pi i$ ($n$ integer)
3. $\frac{1}{2}\log 2+\frac{1}{4}\pi i,\frac{1}{2}{\log }_{e}2+(n+\frac{1}{4})\pi i$ ($n$ integer)
4. $\frac{1}{2}\log 2+\frac{3}{4}\pi i,\frac{1}{2}{\log }_{e}2+(2n+\frac{3}{4})\pi i$ ($n$ integer)

Q. For any vector $\overrightarrow{a}$, the value of $(\overrightarrow{a}\times \hat{i}{)}^2+(\overrightarrow{a}\times \hat{j}{)}^2+(\overrightarrow{a}\times \hat{k}{)}^2$ is equal to

1. ${\overrightarrow{a}}^2$
2. $3{\overrightarrow{a}}^2$
3. $4{\overrightarrow{a}}^2$
4. $2{\overrightarrow{a}}^2$

Q. If$amp\frac{z-2}{2z+3i}=0$ and $z_0=3+4i$ then

1. $z_0\overline{z}+\overline{z_0}z=12$
2. $z_{0}z+\overline{z_0}\overline{z}=12$
3. $z_0\overline{z}+\overline{z_0}z=0$
4. none of these