Rotation Concept

Q.

The maximum area of the rectangle that can be inscribed in a circle of radius $r$ is

1. $\pi r^2$

2. $r^2$

3. $\frac{\pi r^2}{4}$

4. $2r^2$

Q.

If $z=2+\sqrt{3}i$, then find the value of $z.\overline{z}$.

Q. Let $z_1$ and $z_2$ be roots of the equation $z^2+pz+q=0$ where the coefficients $p$ and $q$ may be complex numbers. Let $A$ and $B$ represent $z_1$ and $z_2$ in the complex plane, If $\angle AOB=\theta \ne 0$ and $OA=OB$ where $O$ is the origin, then $p^2$ is

1. $q{\cos }^2\left(\theta /2\right)$
2. $2q{\cos }^2\left(\theta /2\right)$
3. $3q{\cos }^2\left(\theta /2\right)$
4. $4q{\cos }^2\left(\theta /2\right)$

Q. If a point in argand plane $A(2,3)$ rotated through origin by $\frac{\pi }{4}$ in anticlockwise direction, then the 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. $A\left(z_1\right),B\left(z_2\right)$ 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 $z_2,z_1$ 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. If a point in argand plane $A(2,3)$ rotated through origin by $\frac{\pi }{4}$ in anticlockwise direction, then the 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. 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 in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\frac{\pi }{2}$in anti-clockwise direction on a circle with centre at origin, to reach a point $z_2$. The point $z_2$is given by

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

Q. Consider an equilateral triangle having vertices at the points $A\left(\frac{2}{\sqrt{3}}{e}^{i\pi /2}\right),B\left(\frac{2}{\sqrt{3}}{e}^{-i\pi /6}\right)$ and $C\left(\frac{2}{\sqrt{3}}{e}^{-i5\pi /6}\right).$ Let $P$ be any point on its incircle. then $A{P}^2+B{P}^2+C{P}^2=$

Q. On the Argand plane $z_1,z_2$ 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. 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. ${\overline{z}}_{1}b+z_2\overline{b}$
3. $\frac{3({\overline{z}}_{1}b+z_2\overline{b})}{2}$
4. $2({\overline{z}}_{1}b+z_2\overline{b})$

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 $P$ 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 $\frac{\pi }{2}$ 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. 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 $z_1=10+6i$ 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. $3\sqrt{2}$
4. $4\sqrt{2}$

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

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

Q. If one vertex of a square whose diagonals intersect at the origin 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. If $z$ is a complex number such that $-\frac{\pi }{2}\le \arg \left(z\right)\le \frac{\pi }{2}$, then which of the following inequality is always true?

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

Q. The locus of mid-point of line segment intercepted between real and imaginary axes by the line $a\overline{z}+\overline{a}z+b=0;$ where $b$ is a real parameter and $a$ is a fixed complex number with non- zero real and imaginary parts, is

1. $az+\overline{a}z=0$
2. $a\overline{z}+\overline{a}z=0$
3. $az+\overline{az}=0$
4. $az-\overline{a}z=0$

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. If $z_1,z_2,z_3$ are the vertices of an equilateral $△ABC$ such that $|z_1-i|=|z_2-i|=|z_3-i|$, then $|z_1+z_2+z_3|=$

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

Q. A triangle is formed on the Argand plane by the complex numbers $z,iz$ and $z+iz$. If $|z|=2$, then area of triangle is sq. unit

Q. If the triangle formed by the complex coordinates $A\left(z\right),B(2+3i),C(4+5i)$ which satisfy the relations $|z-(2+3i)|=|z-(4+5i)|$ and $|z-(3+4i\left)\right|\le 4$, then

1. ar. $(△ABC{)}_{max}=4\sqrt{2}$ sq. unit
2. ar. $(△ABC{)}_{max}=4$ sq. unit
3. if area of $△ABC$ is maximum, then unequal angle is $<\frac{\pi }{4}$
4. if area of $△ABC$ is maximum, then unequal angle is $\ge \frac{\pi }{4}$

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

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

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

Q. If $z=\sqrt{2}-i\sqrt{2}$ is rotated through an angle 45° in the anti-clockwise direction about the origin, then the coordinates of its new position are
[Kerala (Engg.) 2005]

1. (2, 0)

2. $(\sqrt{2},\sqrt{2})$

3. $(\sqrt{2},-\sqrt{2})$

4. $(\sqrt{2},0$
5. (4, 0)

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. ${\cos }^{-1}(-\frac{1}{2})$