Inequality of 2 Complex Numbers

Q.

If the two regression coefficients between $x$ and $y$ are $0.8$ and $0.2$, then the coefficient of correlation between them is

1. $0.4$

2. $0.6$

3. $0.3$

4. $0.5$

Q.

The range of the function$f\left(x\right)=\sqrt{(x-1)(3-x)}$ is

1. $[0,1]$

2. $(-1,1)$

3. $(-3,3)$

4. $(-3,1)$

Q. If $|z-2|=min\{|z-1|,|z-5|\}$, where $z$ is a complex number, then possible value(s) of $Re\left(z\right)$ is/are

1. $\frac{3}{2}$
2. $\frac{5}{2}$
3. $\frac{7}{2}$
4. $\frac{9}{2}$

Q. The perimeter of the locus represented by $\arg \left(\frac{z+i}{z-i}\right)=\frac{\pi }{4}$ is equal to

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

Q.

The region represented by $\left\{z=x+iy\in C:\left|z\right|-Re\left(z\right)\le 1\right\}$ is also given by the inequality

1. $y^2\le 2\left(x+\frac{1}{2}\right)$

2. $y^2\le \left(x+\frac{1}{2}\right)$

3. $y^2\le 2\left(x+1\right)$

4. $y^2\le \left(x+1\right)$

Q. Let $z_1$ and $z_2$ be two complex numbers represented by points on the circle $|z_1|=1$ and $|z_2|=2$ respectively. Let $P$ be the point whose coordinates are $(1,1).$ Then

1. the maximum value of $|2z_1+z_2|$ is $4.$
2. the minimum value of $|z_1-z_2|$ is $1.$
3. $|z_2+\frac{1}{z_1}|\le 3$
4. the minimum distance of $P$ from $z_1$ is $2-\sqrt{2}$ unit.

Q. If $\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i$, then values of $x$ and $y$ are

1. $y=\frac{43}{63},x=-\frac{16}{7}$
2. $y=-\frac{43}{63},x=\frac{16}{7}$
3. $x=1,y=0$
4. $x=3,y=-1$

Q. If $w=\frac{z}{z-\frac{1}{3}i}$ , and $\mid w\mid =1$ , then z lies on

1. An ellipse
2. A circle
3. A straight line
4. A parabola

Q. For a complex number z, the minimum value of $\mid z\mid +\mid z-2\mid$ is

1. 1
2. 2
3. 3
4. none of these

Q. The locus of complex number $z$ satisfying $\arg [z-(1+i\left)\right]=\{\begin{array}{cc}\frac{3\pi }{4}& ;|z|\le |z-2|\\ -\frac{\pi }{4}& ;|z|>|z-2|\end{array}$ is

1. a straight line passing through $(2,0)$.
2. a straight line passing through $(2,0),(1,1)$.
3. is a line segment.
4. is a set of two rays.

Q. Let $z$ be a complex number satisfying $|z-3|\le |z-2|,$ $|z-3|\le |z-6|,$ $|z-i|\le |z+i|$ and $|z-i|\le |z-5i|$. Then the area of region in sq. units in which $z$ lies is

1. $9$
2. $6$
3. $2$
4. $14$

Q. If at least one value of the complex number z = x + iy satisfy the condition $|z+\sqrt{2}|=a^2-3a+2$ and the inequality $|z+i\sqrt{2}|=a^2,$ then

1. None of these
2. a > 2
3. a = 12
4. a < 2

Q. If $z_1,z_2,z_3$ are the vertices of a triangle in argand plane such that $|z_1-z_2|=|z_1-z_3|,$ then $\arg (\frac{2z_1-z_2-z_3}{z_3-z_2})$ is

1. $\pm \frac{\pi }{3}$
2. $0$
3. $\pm \frac{\pi }{2}$
4. $\pm \frac{\pi }{6}$

Q. If $z=x+iy$ is a complex number which satisfy $\frac{|z-5i|}{|z+5i|}=1$, then $z$ lies on

1. the imaginary axis
2. the straight line, $y=5$
3. a circle passing through the origin
4. the real axis

Q. If three complex numbers $z_1,z_2$ and $z_3$ satisfy $\frac{z_1-z_3}{z_2-z_3}=\frac{1-i\sqrt{3}}{2}$, then the triangle formed by $z_1,z_2$ and $z_3$ is

1. scalene
2. equilateral
3. obtuse-angled isosceles
4. right-angled isosceles

Q. If $x,y$ and $z$ are real numbers such that $4x+2y+z=31$ and $2x+4y-z=19,$ then the value of $9x+7y+z$

1. equals $\frac{218}{3}$
2. equals $\frac{281}{3}$
3. equals $\frac{182}{3}$
4. cannot be computed from the given information

Q. Let $P(3,3)$ be a point on the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal to it at $P$ intersects the $x-$axis at $(9,0)$ and $e$ is its eccentricity, then the ordered pair $(a^2,e^2)$ is equal to

1. $(9,3)$
2. $(\frac{9}{2},2)$
3. $(\frac{9}{2},3)$
4. $(\frac{3}{2},2)$

Q. If $z_1$ and $z_2$ are two complex numbers, then the inequality $|z_1+z_2{|}^2\le (1+c)|z_1{|}^2+(1+c^{-1})|z_2{|}^2$ is true if

1. $c\in R$
2. $c\in (0,\infty )$
3. $c\in R-\left\{0\right\}$
4. $c\in (-\infty ,0)$

Q. Let $z=x+iy$ and $\omega =\frac{1-iz}{z-i}$. If $|\omega |=1$, then $z$ lies on

1. a unit circle
2. on the real axis
3. a parabola
4. the imaginary axis

Q. If $a,b,c$ are positive real numbers such that $\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},$ find the value of $\frac{\left(a+b\right)\left(b+c\right)\left(c+a\right)}{abc}.$

Q. If |z - 25i| $\le$ 15, then |max amp(z) - min.amp(z)| =

1. $\pi -2co{s}^{-1}\left(\frac{3}{5}\right)$
2. $\frac{\pi }{2}+co{s}^{-1}\left(\frac{3}{5}\right)$
3. $si{n}^{-1}\left(\frac{3}{5}\right)-co{s}^{-1}\left(\frac{3}{5}\right)$
4. $co{s}^{-1}\left(\frac{3}{5}\right)$

Q.

If z = (1 + i $\sqrt{3}$).Find the value of ${log}_e$z.

1. In2 + i

2. In2 + i

3. In3 + i

4. In3 + i

Q. If $z=x+iy$ is a complex number which satisfies $\frac{|z-5i|}{|z+5i|}=1$, then $z$ lies on

1. the real axis
2. the straight line, $y=5$
3. a circle passing through the origin
4. the imaginary axis

Q. If $|z-2|=min\{|z-1|,|z-5|\}$, where $z$ is a complex number, then possible value(s) of $Re\left(z\right)$ is/are

1. $\frac{3}{2}$
2. $\frac{5}{2}$
3. $\frac{7}{2}$
4. $\frac{9}{2}$

Q.

Let |z-3+2i| $\le$ 4 , then the absolute different between the maximum and minimum values of |z| is

1. 

2. 

3. 

4. 

Q. Let $c=(a+ib{)}^3-47i,$ where $a,b,c\in Z^{+}$ and $i=\sqrt{-1}$. Then the value of $a+b+c$ is

1. $52$
2. $49$
3. $57$
4. $63$

Q. The locus of z satisfying the inequality $lo{g}_{1/3}|z+1|>lo{g}_{1/3}|z-1|$ is

1. R (z) < 0
2. R (z) > 0
3. I (z) < 0
4. None of these

Q. Let $z_1$ and $z_2$ be two complex numbers such that $|z_1|=12$ and $|z_2-3-4i|=5.$ Then the minimum value of $|z_1-z_2|$ is

1. $0$
2. $2$
3. $7$
4. $17$

Q. The locus of complex number $z$ satisfying $\arg [z-(1+i\left)\right]=\{\begin{array}{cc}\frac{3\pi }{4}& ;|z|\le |z-2|\\ -\frac{\pi }{4}& ;|z|>|z-2|\end{array}$ is

1. a straight line passing through $(2,0)$.
2. a straight line passing through $(2,0),(1,1)$.
3. is a line segment.
4. is a set of two rays.

Q. If $2a+3b-5c=0$ then prove that the points $A\left(\overline{a}\right),B\left(\overline{b}\right)$ and $C\left(\overline{c}\right)$ are collinear. Hence, find the ratio in which the point $C$ divides the line segment $AB$.