Geometrical Representation of a Complex Number

Q. Let $A=\{x\in Z:3^{(x+1)(x^2-7x+12)}=1\}\text{and}$ $B=\{x\in Z:-5<2x-1\le 7\},$ where $Z$ is the set of integers. If the number of relations from $A$ to $B$ is $2^k$ then the value of $k$ is

Q. The domain of the function ${\cos }^{-1}(3x-2)$ is

1. $\left(\frac{1}{3},1\right]$
2. $\left(-1,\frac{1}{3}\right]$
3. $\left(-1,1\right]$
4. $\left(-\frac{1}{3},\frac{1}{3}\right]$

Q.

A foot of the normal from the point $(4,3)$ to a circle is $(2,1)$ and a diameter of the circle has the equation $2x-y=2$. Then the equation of the circle is

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

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

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

4. None of these

Q. If $z_1,z_2,z_3$ are complex numbers such that $\left(\frac{2}{z_1}\right)=\left(\frac{1}{z_2}\right)+\left(\frac{1}{z_3}\right)$ and $\arg \left(\frac{z_3}{z_2}\right)\ne n\pi ,n\in I$, then the points $z_1,z_2,z_3$ and $O$(origin) will always lie on

1. a circle
2. a straight line
3. An equilateral triangle with $O$ as centriod
4. a rectangle

Q. If $(3+x^{2020}+x^{2021}{)}^{2022}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{10}{x}^n$, then the value of $a_0-\frac{1}{2}{a}_1-\frac{1}{2}{a}_2+a_3-\frac{1}{2}{a}_4-\frac{1}{2}{a}_5+a_6\cdots$ is
(correct answer + 2, wrong answer - 0.50)

1. $2^{2021}$
2. $1$
3. $2^{2022}$
4. $0$

Q. If $a,b,c$ and $u,v,w$ are complex numbers representing the vertices of two triangles such that $c=(1-r)a+rb$ and $w=(1-r)u+rv$ where $r$ is a complex number, then the two triangles

1. have the same area
2. are similar
3. are congruent
4. none of these

Q. If $z=\frac{\sqrt{3}}{2}+\frac{i}{2},$ then $\arg (-z)$ is

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

Q. If $A\left(z_1\right)$ is a variable point in the Argand plane such that $z_1\overline{z_1}=5,0<\arg \left(z_1\right)<\pi$. Also $B\left(z_2\right)\&C\left(z_3\right)$ are two fixed points in the Argand plane satisfying $z^4-6z^2+25=0\&-\pi <arg\left(z\right)<0$.

1. Internal angle bisector of $\angle A$ of $△ABC$ will always pass through a fixed point.
2. Circum centre of $△ABC$ is the Origin.
3. Orthocentre $\left(z_4\right)$ of $△ABC$ satisfies $|z_4+2i|=5.(i=\sqrt{-1})$
4. Orthocentre $\left(z_4\right)$ of $△ABC$ satisfies $|z_4-2i|=\sqrt{5}$.

Q. If the points $A\left(z\right),B(-z)$ and $C(1-z)$ are the vertices of an equilateral triangle $ABC$, then

1. sum of possible $z$ is $\frac{1}{2}$
2. Sum of possible $z$ is $1$
3. Product of possible $z$ is $\frac{1}{4}$
4. Product of possible $z$ is $\frac{1}{2}$

Q. Let $P\left(z\right)=z^3+a{z}^2+bz+c,$ where $a,b,c\in R$. If there exists a complex number $w$ such that the three roots of $P\left(z\right)$ are $w+3i,w+9i$ and $2w-4$, where $i^2=-1$, then the value of $a+b+c$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

1. $136$
2. $-136$
3. $280$
4. $-280$

Q. The locus of the points representing complex number $z=x+iy$ for which $|z+5{|}^2-|z-5{|}^2=10$ is

1. $x+y=2$
2. $y=\frac{1}{2}$
3. $x=\frac{1}{2}$
4. $x+y=10$

Q. If $A\left(a\right),B\left(b\right),C\left(c\right)$ and $P\left(u\right),Q\left(v\right),R\left(w\right)$ are complex numbers representing the vertices of two triangles such that $△ABC$ and $△PQR$ are similar, then $\frac{a-c}{a-b}=$

1. $\frac{u-v}{u-w}$
2. $\frac{u-v}{v-w}$
3. $\frac{u-w}{u-v}$
4. $\frac{u-w}{w-v}$

Q. If the argument of complex number $\sin \theta +i(1-\cos \theta ),0<\theta <\pi$ is $\frac{\theta }{k},k\in R$, then value of $k$ is

Q. If $a,b,c$ and $u,v,w$ are complex numbers representing the vertices of two triangles such that $c=(1-r)a+rb$ and $w=(1-r)u+rv$ where $r$ is a complex number, then the two triangles

1. have the same area
2. are similar
3. are congruent
4. none of these

Q. Let $a,bϵR$ and $a^2+b^2\ne 0$. Suppose $S=\{zϵC:z=\frac{1}{a+ibt},tϵR,t\ne 0\}$, where $i=\sqrt{-1}$. If $z=x+iy$ and $zϵS$, 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 $a=0,b\ne 0$

Q. Let A, B, C are 3 points on the complex plane represented by complex numbers a, b, c respectively such that |a| = |b| = |c| =1, a+b+c=1 and abc=1, then

1. area of triangle ABC is 2 (Sq.unit)
2. $△ABC$ is an equilateral triangle.
3. $△ABC$ is right isosceles triangle.
4. orthocentre of $△ABC$ lies outside the triangle.

Q. If $\sin \theta =\left(\frac{z-1}{i}\right),$ where $z$ is non real, $\theta$ represents angle of a triangle, then

1. $Re\left(z\right)=1,Im\left(z\right)=2$
2. $Re\left(z\right)=1,-1\le Im\left(z\right)\le 1$
3. $Re\left(z\right)+Im\left(z\right)=0$
4. $Re\left(z\right)=0,Im\left(z\right)=1$

Q. A man walks a distance of 3 units from the origin towards the north-east $\left(N{45}^{o}E\right)$ direction. From there, he walks a distance of 4 units towards the north-west $\left(N{45}^{o}W\right)$ direction to reach a point P. Then the position of P in the Argand plane is

1. $3e^{\frac{i\pi }{4}}+4i$
2. $(3-4i)e^{\frac{i\pi }{4}}$
3. $(4+3i)e^{\frac{i\pi }{4}}$
4. $(3+4i)e^{\frac{i\pi }{4}}$

Q. If $A\left(z_1\right)$ is a variable point in the Argand plane such that $z_1\overline{z_1}=5,0<\arg \left(z_1\right)<\pi$. Also $B\left(z_2\right)\&C\left(z_3\right)$ are two fixed points in the Argand plane satisfying $z^4-6z^2+25=0\&-\pi <arg\left(z\right)<0$.

1. Internal angle bisector of $\angle A$ of $△ABC$ will always pass through a fixed point.
2. Circum centre of $△ABC$ is the Origin.
3. Orthocentre $\left(z_4\right)$ of $△ABC$ satisfies $|z_4+2i|=5.(i=\sqrt{-1})$
4. Orthocentre $\left(z_4\right)$ of $△ABC$ satisfies $|z_4-2i|=\sqrt{5}$.

Q. Let $S=S_1\cap S_2\cap S_3,$ where
$S_1=\{zϵC:|z|<4\},S_2=\{zϵC:Im\left[\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right]>0\}and{S}_3=\{zϵC:Re(z)>0\}.$
Area of $S=$

1. $\frac{10\pi }{3}$
2. $\frac{20\pi }{3}$
3. $\frac{16\pi }{3}$
4. $\frac{32\pi }{3}$