Properties of Conjugate of a Complex Number

Q.

The locus of $z$ given by $\left|\frac{z-1}{z-i}\right|=1$ is

1. Circle

2. Ellipse

3. Straight line

4. None of these

Q. If $P\left(x\right)=a{x}^2+bx+c\text{and}Q\left(x\right)=-a{x}^2+dx+c$, where $ac\ne 0$ [a, b, c, d are all real], then P(x).Q(x) = 0 has

1. four real roots
2. no real root
3. at least two real roots
4. two real roots

Q.

Find the terminal point on the unit circle determined by $\frac{2\mathrm{\pi }}{3}$ radians $?$

Q. If $z$ is non zero complex number satisfying $\overline{z}=i{z}^2$, then $z$ can be

1. $\frac{\sqrt{3}}{2}-\frac{i}{2}$
2. $-\frac{\sqrt{3}}{2}-\frac{i}{2}$
3. $0$
4. $i$

Q.

How do you find the square root of a complex number?

Q.

For all real values of x, the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is

a) zero b)1 c) 3 d) $\frac{1}{3}$

Q.

The number of real roots of the equation ${\left(x+\frac{1}{x}\right)}^3+\left(x+\frac{1}{x}\right)=0$ is

1. $0$

2. $2$

3. $4$

4. $6$

Q. If $\overline{z}$ be the conjugate of the complex number z, then which of the following relations is false
[MP PET 1987]

1. $|z|=|\overline{z}|$
2. $z.\overline{z}|\overline{z}{|}^2$
3. $\overline{z_1}+\overline{z_2}=\overline{z_1}+\overline{z_2}$
4. $argz=arg\overline{z}$

Q.

${\cos }^2\left(76°\right)+{\cos }^2\left(16°\right)-\cos \left(76°\right)\cos \left(16°\right)=$

1. $-\frac{1}{4}$

2. $\frac{1}{2}$

3. $0$

4. $\frac{3}{4}$

Q.

Two men A and B start with velocities V at the same time from the junction of two roads inclined at ${45}^{\circ }$ to each other. If they travel by different roads, then find the rate at which they are being separated.

Q.

If (1 + i) z = (1 - i) $\overline{z}$, then show that z = -i$\overline{z}$.

Q. If $z$ is a complex number, then the conjugate of $z+2\overline{z}$ is

1. $2\overline{z}+z$
2. $\overline{z}+2z$
3. $\overline{z}-2z$
4. $2\overline{z}-z$

Q. If $z_1,z_2,\cdots z_n$ lie on the circle $|z|=2$ then the value of $|z_1+z_2+\cdots +z_n|-4|\frac{1}{z_1}+\frac{1}{z_2}+\cdots +\frac{1}{z_n}|=$

1. $n$
2. $-n$
3. $1$
4. $0$

Q. If the equation $a{x}^2+bx+c=0,$ where $a,b,c$ are the sides of a $△ABC,$ and the equation $x^2+\sqrt{2}x+1=0$ have a common root, then $\angle C=$

1. ${90}^{\circ }$
2. ${35}^{\circ }$
3. ${45}^{\circ }$
4. ${60}^{\circ }$

Q. The length of the perpendicular from (1, 0, 2) on the line ​$\frac{x+1}{3}=\frac{y+1}{-2}=\frac{z+1}{-1}$ is

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

Q. The characteristic roots of the two rowed orthogonal matrix $\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$ are $\lambda$ and $\overline{\lambda }$. ($\lambda$ lies in $I$ quadrant for $\theta \in (0,{90}^o))$, then ${\lambda }^5$ is equal to

1. $\cos 5\theta -i\sin 5\theta$
2. $\cos 5\theta +i\sin 5\theta$
3. ${\cos }^5\theta -i{\sin }^5\theta$
4. ${\cos }^5\theta +i{\sin }^5\theta$

Q.

The function $x\sqrt{1-x^2},(x>0)$ has

1. A local maxima

2. A local minima

3. Neither local maxima nor a local minima

4. None of the above

Q.

Let A (-1, 1), B (5, 2) and c (3, -1) form the vertices of a triangle. Find the length of the altitude from A to BC.

1. 

2. 

3. 

4. 

Q.

If one root of the equation $4x^2-6x+p=0$ is $q+2i$, where $p,q\in R$ then $p+q=$

1. $10$

2. $19$

3. $-24$

4. $-32$

Q. The equation $z^{10}+(13z-1{)}^{10}=0$ has $5$ pairs of complex roots $a_1,b_1,a_2,b_2,a_3,b_3,a_4,b_4,a_5,b_5$. If each pair $a_i,b_i$ are complex conjugates, then

1. $\sum _{i=1}^5(\frac{1}{a_i}+\frac{1}{b_i})=130$
2. $\sum _{i=1}^5(\frac{1}{a_i}+\frac{1}{b_i})=260$
3. $\sum _{i=1}^5\left(\frac{1}{a_i{b}_i}\right)=1700$
4. $\sum _{i=1}^5\left(\frac{1}{a_i{b}_i}\right)=850$

Q.

The negation of the Boolean expression $x\leftrightarrow y$ is equivalent to

1. $(x\wedge y)\wedge (~x\vee ~y)$

2. $\left(x\wedge ~y\right)\vee \left(y\wedge ~x\right)$

3. $(x\wedge ~y)\vee (~x\wedge ~y)$

4. $(~x\wedge y)\vee (~x\wedge ~y)$

Q.

If $\sqrt{\left(x+iy\right)}=\pm \left(a+ib\right)$, then $\sqrt{\left(-x-iy\right)}$ is equal to

1. $\pm \left(b+ia\right)$

2. $\pm \left(a+ib\right)$

3. $\pm \left(a-ib\right)$

4. $\pm \left(b-ia\right)$

5. None of these

Q.

Find the projection of the vector $\hat{i}-\hat{j}$ on the vector $\hat{i}+\hat{j}$

Q. Let $f\left(x\right)=\left|{\sin }^{-1}\right(\sin x\left)\right|-\left(\frac{\pi -x}{2}\right)$. Then number of solutions of equation $f\left(x\right)=0$ in $x\in [-\pi ,\pi ]$ is

1. $2$
2. $3$
3. $0$
4. $4$

Q.

If $w$ (not equal $1$) is a cube root of unity and ${\left(1+w^2\right)}^n={\left(1+w^4\right)}^n$ , then the least positive value of $n$ is.

1. $2$

2. $3$

3. $5$

4. $6$

Q.

$E_1:a+b+c=0$, if $1$ is root of $a{x}^2+bx+c=0$,

$E_2:b^2-a^2=2ac$, if $\sin \left(\theta \right)$ and $\cos \left(\theta \right)$ are the roots of $a{x}^2+bx+c=0$, which of the following is correct ?

1. $E_1$is correct, $E_2$ is correct.

2. $E_1$is correct, $E_2$ is incorrect.

3. $E_1$is incorrect, $E_2$ is correct.

4. $E_1$is incorrect, $E_2$ is incorrect.

Q. The conjugate of $(4+5i{)}^2$ is

1. $-9-40i$
2. $-9+40i$
3. $9-40i$
4. $9+40i$

Q. If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg \left(\frac{1+z}{1+\overline{z}}\right)$ equals:

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

Q. Let $z$ be a complex number satisfying $\overline{z}=\frac{16}{z-i}-i$. Then which of the following is/are true?

1. Centre of locus of $z$ is $(0,1)$
2. $|z-i|=16$
3. Maximum value of $|z|$ is $\sqrt{15}$
4. Maximum value of $|z|$ is $5$

Q. Find $\left|\right(1+i\left)\frac{(2+i)}{(3+i)}\right|$