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

1. (2, 0)

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

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

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

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

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.

If $(asec\theta .btan\theta )$ and $(asec\Phi ,btan\Phi )$ are the ends of a focal chord of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose eccentricity is e, then $tan\frac{\theta }{2}\times tan\frac{\oslash }{2}$ equal to.

1. 

2. 

3. 

4. 

Q.

A ray of light passing through the point A and the reflected ray passes through the point (1, 2) reflects on the x-axis at point A and the reflected ray passes through the point (5, 3). Find the coordinates of A.

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. Suppose $z_1,z_2,z_3$ are the vertices of an equilateral triangle inscribed in the circle $|z|=2$. If $z_1=1+i\sqrt{3}$, then the other two vertices are

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

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. 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.

Find the angle between the vectors 1 + i and -1 + i

1. 60

2. 90

3. 45

4. 180

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. Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद

A solid sphere of mass m and radius R = 0.5 m is rolling without slipping on rough horizontal surface with velocity v₀ = 2 m/s and acceleration a₀ = 3 m/s² as shown.

द्रव्यमान m तथा त्रिज्या R = 0.5 m का एक ठोस गोला खुरदरे क्षैतिज पृष्ठ पर बिना फिसले दर्शाए गए वेग v₀ = 2 m/s तथा त्वरण a₀ = 3 m/s² से लुढ़क रहा है।



Q. Acceleration of top most point A is

प्रश्न - शीर्षतम बिन्दु A का त्वरण है

1. 8 m/s²
2. 6 m/s²
3. 10 m/s²
4. $6\sqrt{2}\text{m/}{\text{s}}^2$

Q. Z is rotated through an angle of anticlockwise to get $z_1$ and clockwise to get $z_2$, then

1. , z, are in GP

2. , z, are in AP

3. + = 2 z cos

4. + = 2 cos(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 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. 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. If a point in argand plane $A(3,2)$ rotated through $B(1,1)$ about $\frac{\pi }{4}$ to obtain $C$, then area of $△ABC$ is

1. $\frac{5}{4}$ sq. unit
2. $\frac{5}{2}$ sq. unit
3. $\frac{\sqrt{50}}{2}$ sq. unit
4. $\frac{\sqrt{50}}{4}$ sq. unit

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