Rotation Concept

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. (4, 0)

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

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

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

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. Given a point $A\left(z\right)$ in the Argand plane. $B\left(z^{\prime }\right)$ and $C\left(z^{\prime \prime }\right)$ are points obtained on rotating $A\left(z\right)$ about the Origin through an angle $\alpha$ anticlockwise and clockwise, respectively, without changing modulus of $z$. Identify the correct statement(s).

1. $z^{\prime },z,z^{\prime \prime }$ form a G.P.
2. $z^{\prime },z,z^{\prime \prime }$ form an A.P.
3. $(z^{\prime }{)}^2+(z^{\prime \prime }{)}^2=2z^{2}cos2\alpha$
4. $z^{\prime }+z^{\prime \prime }=2zcos2\alpha$

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 $z_1$ 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 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. 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. $A\left(z_1\right),B\left(z_2\right)\text{and}C\left(z_3\right)$ 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 $z_1$ and $z_2$ are the roots of $3z^2+3z+b=0$. If $O\left(0\right),\left(z_1\right),\left(z_2\right)$ form an equilateral triangle, then $b=$

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