Rotation Concept

Q. Let $z_1=10+6i$ and $z_2=4+6i$. If $z$ is any complex number such that the argument of $(z-z_1)/(z-z_2)$ is $\pi /4$, then $|z-7-9i|$ is

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

Q. $ABCD$ is a rhombus. Its diagonals $AC$ and $BD$ intersect at the point $M$ and satisfy $BD=2AC$. If the points $D$ and $M$ represent the complex numbers $1+i$ and $2-i$, respectively, then $C$ represents the complex numbers

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

Q. Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1|=4|z_2|$. If $z=\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$ then :

1. $\text{Im}\left(z\right)=0$
2. $\text{Re}\left(z\right)=0$
3. $|z|=\frac{1}{2}\sqrt{\frac{17}{2}}$
4. $|z|=\sqrt{\frac{5}{2}}$
5. $\text{Re}\left(z\right)=\frac{5}{2}\cos ({\theta }_1-{\theta }_2)$

Q. A particle $P$ starts from the point $z_0=1+2i$ , where $i=\sqrt{-1}$ It moves first horizontally away from origin by $5$ units and then vertically away from origin by $3$ units to reach a point $z_1$. From $z_1$ the particle moves $\sqrt{2}$ units away from origin in the direction of $x=y$ and then it moves through an angle $\pi /2$ in anticlockwise direction on a circle with centre at origin to reach a point $z_2$. Then point $z_2$ is given by

1. $6+7i$
2. $-7+6i$
3. $7+6i$
4. $-6+7i$

Q. If one of the vertices of the square, inscribed in the circle $|z-1|=2$, is $2+\sqrt{3}i$, then other vertices of the square are

1. $i\sqrt{3}$
2. $(1-\sqrt{3})+i$
3. $(\sqrt{3}+1)-i$
4. $-i\sqrt{3}$

Q. If $\arg \left(\frac{z_1-\frac{z}{|z|}}{\frac{z}{|z|}}\right)=\frac{\pi }{2}$ and $|\frac{z}{|z|}-z_1|=3$ then $|z_1{|}^2$ equals to

Q. The altitude from the vertices $A,B$ and $C$ of the triangle $ABC$ meet its circumcircle at $D,E$ and $F$ respectively. The complex numbers representing the points $D,E\text{and}F$ are $z_1,z_2\text{and}{z}_3$ respectively. If $\frac{z_3-z_1}{z_2-z_1}$ is purely real, then the triangle $ABC$ is right angle at

1. $C$
2. $A$
3. $B$
4. None of these

Q. $A\left(z_1\right),B\left(z_2\right),C\left(z_3\right)$ are the vertices of the triangle $ABC$ (in anticlockwise order). If $\angle ABC=\frac{\pi }{4}$ and $AB=\sqrt{2}\left(BC\right)$, then

1. $z_2=(z_1-z_3)+i{z}_3$
2. $z_2=z_3+i(z_1-z_3)$
3. $z_2=z_1+z_3$
4. None of these

Q. A particle starts from a point $z_0=1+i$, where $i=\sqrt{i}$. It moves horizontally away from origin by $2$ units and then vertically away from origin by $3$ units to reach a point $z_1$. From $z_1$ particle moves $\sqrt{5}$ units in the direction of $2\hat{i}+\hat{j}$ and then it moves through an angle of ${\text{cosec}}^{-1}\sqrt{2}$ in anticlockwise direction of a circle with centre at origin to reach a point $z_2$. The $\arg z_2$ is given by

1. ${\sec }^{-1}2$
2. ${\cot }^{-1}0$
3. ${\sin }^{-1}\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)$
4. ${\cos }^{-1}(-\frac{1}{2})$

Q. Let $A\left(z_1\right)$ and $B\left(z_2\right)$represent two complex numbers on the complex plane. Suppose the complex slope of the line $l_1$ joining $A$ and $B$ is defined as $\frac{(z_1-z_2)}{(\overline{z_1}-\overline{z_2})}$. If the line $l_1$ with complex slope ${\omega }_1$ and $l_2$ with complex slope ${\omega }_2$ on the complex plane are perpendicular then

1. ${\omega }_1+{\omega }_2=-1$
2. ${\omega }_1+{\omega }_2=0$
3. ${\omega }_1+{\omega }_2=1$
4. ${\omega }_1\times {\omega }_2=-1$

Q. Let origin and the non- real roots of $2z^2+2z+\lambda =0$ form the three vertices of an equilateral triangle in the Argand plane then $3\lambda =$

Q. PQ and PR are two infinite rays.QAR is an arc . point lying in the shaded region excluding the boundry satisfies

1. $\mid z-1\mid >2;arg(z-1)<\frac{\pi }{4}$
2. $\mid z-1\mid >2;arg(z-1)<\frac{\pi }{2}$
3. $\mid z-1\mid >2;arg(z-1)<\frac{\pi }{4}$
4. $\mid z-1\mid >2;arg(z-1)<\frac{\pi }{2}$

Q. A rectangle of maximum area is inscribed in the circle $|z-3-4i|=1$. If one vertex of the rectangle is $4+4i,$ then another adjacent vertex of this rectangle can be

1. $2+4i$
2. $3+5i$
3. $3+3i$
4. $3-3i$

Q. Let $\overline{b}z+b\overline{z}=c,b\ne 0$ be a line in the complex plane. If a point $z_1$ is the reflection of a point $z_2$ through the line, then $c$ is

1. $\frac{{\overline{z}}_{1}b+z_2\overline{b}}{2}$
2. $2({\overline{z}}_{1}b+z_2\overline{b})$
3. ${\overline{z}}_{1}b+z_2\overline{b}$
4. $\frac{3({\overline{z}}_{1}b+z_2\overline{b})}{2}$

Q. If the line $L:3x-4y=0$ is rotated about the centre of the circle $(x-4{)}^2+(y-3{)}^2=25$ through an acute angle of $\theta$ in anticlockwise sense such that after rotation it becomes one of the members of the family of lines $x+\lambda y-3-\lambda =0,\lambda \in R$, then $\theta$ equals

1. ${\tan }^{-1}2$
2. ${\cot }^{-1}2$
3. ${\tan }^{-1}1$
4. ${\sec }^{-1}2$

Q. Complex numbers $z_1,z_2,z_3$ are the vertices $A,B,C$ respectively, of an isosceles right-angled triangle with right angle at $C$. Then which of the following is true?

1. $(z_1-z_2{)}^2=2(z_1-z_3\left)\right(z_3-z_2)$.
2. $(z_1-z_2{)}^2=3(z_1-z_3\left)\right(z_3-z_2)$.
3. $(z_1-z_2{)}^2=4(z_1-z_3\left)\right(z_3-z_2)$.
4. $(z_1-z_2{)}^2=(z_1-z_3\left)\right(z_3-z_2)$.

Q. The complex number associated with the vertices $A,B,C$ of $\Delta ABC$ are $e^{i\theta },\omega ,\overline{\omega }$, respectively [where $\omega ,\overline{\omega }$ are the complex cube roots of unity and $Re\left(e^{i\theta }\right)>Re\left(\omega \right)$, then the complex number of the point where angle bisector of $A$ meets the circumcircle of the triangle, is

1. $e^{i\theta }$
2. $\omega +\overline{\omega }$
3. $e^{-i\theta }$
4. $\omega \overline{\omega }$

Q. $A\left(z_1\right),B\left(z_2\right)\text{and}C\left(z_3\right)$ are the vertices of an isosceles triangle in anticlockwise direction with origin as in-centre. If $AB=AC$, then $z_2,z_1\text{and}k{z}_3$ will form $(\text{where}k=\frac{|z_1{|}^2}{|z_2||z_3|})$

1. $A.P.$
2. $G.P.$
3. $A.G.P.$
4. None of these

Q. Let $z$ and $z_0$ be two complex numbers. It is given that $|z|=1$ and the numbers $z,z_0,z{\overline{z}}_0,1$ and $0$ are represented in an Argand diagram by the points $P,P_0,Q,A$ and the origin, respectively, then the value of $\frac{|z-z_0|}{|z{\overline{z}}_0-1|}=$

Q. If $|z_2+i{z}_1|=|z_1|+|z_2|$ and $|z_1|=3$ and $|z_2|=4$, then area of $△ABC$, if affixes of $A,B$ and $C$ are $z_1,z_2$ and $\left[\frac{z_2-i{z}_1}{1-i}\right]$ respectively, is,

1. $0$ sq. unit
2. $\frac{5}{2}$ sq. units
3. $\frac{25}{2}$ sq. units
4. $\frac{25}{4}$ sq. units

Q. On the Argand plane $z_1,z_2\text{and}{z}_3$ are, respectively, the vertices of an isosceles triangle $ABC$ with $AC=BC$ and equal angles are $\theta$. If $z_4$ is the incentre of the triangle, then $\frac{(z_2-z_1)(z_3-z_1)}{(z_4-z_1{)}^2}=$

1. $1+\cos \theta$
2. $1+\sec \theta$
3. $\tan \theta$
4. $1$

Q. If a point in argand plane $A(2,3)$ rotated through origin about $\frac{\pi }{4}$ in anticlockwise, then new coordinates of the point will be

1. $(\frac{1}{\sqrt{2}},-\frac{5}{\sqrt{2}})$
2. $(\frac{1}{\sqrt{2}},\frac{5}{\sqrt{2}})$
3. $(-\frac{1}{\sqrt{2}},\frac{5}{\sqrt{2}})$
4. $(-\frac{1}{\sqrt{2}},-\frac{5}{\sqrt{2}})$

Q. If $P$ and $Q$ are represented by the complex numbers $z_1$ and $z_2$, such that $|\frac{1}{z_2}+\frac{1}{z_1}|=|\frac{1}{z_2}-\frac{1}{z_1}|$, then

1. $△OPQ$ (where $O$ is the origin) is equilateral
2. $△OPQ$ is right angled
3. The circumcentre of $△OPQ$ is $\frac{1}{2}(z_1+z_2)$
4. The circumcentre of $△OPQ$ is $\frac{1}{3}(z_1+z_2)$

Q. The points $z_1=3+\sqrt{3}i\text{and}{z}_2=2\sqrt{3}+6i$ are given on a complex plane. The complex number lying on the bisector of the angle formed by the vector $z_1$ and $z_2$ is

1. $z=5+5i$
2. $z=\frac{(3+2\sqrt{3})}{2}+\frac{\sqrt{3}+2}{2}i$
3. $z=\frac{(3+2\sqrt{3})}{2}-\frac{\sqrt{3}+2}{2}i$
4. $z=(3+3\sqrt{3})+2i$

Q. If one vertex of a square whose diagonals intersect at the orign is $3(\cos \theta +i\sin \theta )$, then other vertices are

1. $3(i\sin \theta -\cos \theta )$
2. $3(i\cos \theta -\sin \theta )$
3. $3(\sin \theta -i\cos \theta )$
4. $-3(\cos \theta +i\sin \theta )$

Q. $A\left(z_1\right),B\left(z_2\right),C\left(z_3\right)$ are the vertices of the triangle $ABC$ (in anticlockwise order). If $\angle ABC=\frac{\pi }{4}$ and $AB=\sqrt{2}\left(BC\right)$, then

1. $z_2=(z_1-z_3)+i{z}_3$
2. $z_2=z_3+i(z_1-z_3)$
3. $z_2=z_1+z_3$
4. None of these

Q. Let $z$ and $z_0$ be two complex numbers. It is given that $|z|=1$ and the numbers $z,z_0,z{\overline{z}}_0,1\text{and}0$ are represented in an Argand diagram by the points $P,P_0,Q,A$ and the origin, respectively, then the value of $\frac{|z-z_0|}{|z{\overline{z}}_0-1|}=$

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

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

Q. Let $A,B,C,D$ be four concyclic points in order in which $AD:AB=CD:CB$. If $A,B,C$ are represented by complex numbers $a,b,c$, then vertex $D$ can be represented as

1. $\frac{2ac+b(a-c)}{a+c+2b}$
2. $\frac{2ac+b(a+c)}{a+c+2b}$
3. $\frac{2ac+b(a+c)}{a+c-2b}$
4. $\frac{2ac-b(a+c)}{a+c-2b}$

Q. Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1|=4|z_2|$. If $z=\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$ then :

1. $\text{Im}\left(z\right)=0$
2. $\text{Re}\left(z\right)=0$
3. $|z|=\frac{1}{2}\sqrt{\frac{17}{2}}$
4. $|z|=\sqrt{\frac{5}{2}}$
5. $\text{Re}\left(z\right)=\frac{5}{2}\cos ({\theta }_1-{\theta }_2)$