Geometrical Representation of Conjugate of a Complex Number

Q. Let $s,t,r$ be non-zero complex numbers and $L$ be the set of solutions $z=x+iy(x,y\in R,i=\sqrt{-1})$ of the equation $sz+t\overline{z}+r=0$, where $\overline{z}=x-iy$. Then, which of the following statement(s) is (are) TRUE?

1. If $L$ has exactly one element, then $|s|\ne |t|$
2. If $|s|=|t|$, then $L$ has infinitely many elements
3. The number of elements in $L\cap \{z:|z-1+i|=5\}$ is atmost $2$
4. If $L$ has more than one element, then $L$ has infinitely many elements

Q. Let $C$ be the set of all complex numbers. Let $S_1=\{z\in C:|z-3-2i{|}^2=8\},$
$S_2=\{z\in C:Re(z)\ge 5\}$ and
$S_3=\{z\in C:|z-\overline{z}|\ge 8\}.$
Then the number of elements in $S_1\cap S_2\cap S_3$ is equal to

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

Q.

Find the values of other five trigonometric functions if , x lies in third quadrant.

Q.

Graphs of y=mx and graph of y=mx+c separately and tell in which quadrant do they lie.

Q. If $\overline{z}=\overline{z_0}+A(z-z_0)$, where $A$ and $z_0$ are fixed complex numbers, then the locus of $z$ is

1. hyperbola
2. parabola
3. circle
4. straight line

Q.

If $z=a+ab$ lies in third quadrant, then $\frac{\overline{z}}{z}$ also lies in the third quadrant if

1. a < b < 0

2. b < a < 0

3. b > a > 0

4. a > b > 0

Q. The function f (x) = |cos x| is
(a) everywhere continuous and differentiable
(b) everywhere continuous but not differentiable at (2n + 1) π/2, n ∈ Z
(c) neither continuous nor differentiable at (2n + 1) π/2, n ∈ Z
(d) none of these

Q. Find the area of the region bounded by x² = 16y, y = 1, y = 4 and the y-axis in the first quadrant.

Q. The function f (x) = |cos x| is
(a) differentiable at x = (2n + 1) π/2, n ∈ Z
(b) continuous but not differentiable at x = (2n + 1) π/2, n ∈ Z
(c) neither differentiable nor continuous at x = n ∈ Z
(d) none of these

Q. Let f (x) = |cos x|. Then,
(a) f (x) is everywhere differentiable
(b) f (x) is everywhere continuous but not differentiable at x = n π, n ∈ Z
(c) f (x) is everywhere continuous but not differentiable at $x=\left(2n+1\right)\frac{\pi }{2},n\in Z$.
(d) none of these

Q. If $z=x+iy$ lies in the third quadrant, then $\frac{\overline{z}}{z}$ also lies in the third quadrant if

1. $x>y>0$
2. $x<y<0$
3. $y<x<0$
4. $y>x>0$

Q. Let $A=\{x:2<|x|\le 5\text{and}x\in Z\}$ and $B$ be the set of values of $a$ for which the equation $||x-1|+a|=4$ can have real solutions. Then $n(A\cap B)$ is

1. $3$
2. $2$
3. $5$
4. $0$

Q.

If z = a + ib where a and b are less than zero. Conjugate of z lies on which quadrant in Argand plane.

1. I

2. II

3. III

4. IV

Q.

, x in quadrant II

Q.

Find for , x in quadrant II

Q. If $\overline{z}=\overline{z_0}+A(z-z_0)$, where $A$ and $z_0$ are fixed complex numbers, then the locus of $z$ is a

1. parabola
2. hyperbola
3. circle
4. straight line

Q.

, x in quadrant III

Q.

Find for , x in quadrant III

Q.

If z = a + ib where a and b are less than zero. Conjugate of z lies on which quadrant in Argand plane.

1. I

2. II

3. III

4. IV

Q.

Conjugate of z is the mirror image of z along _______ on the Argand plane.

1. Imaginary axis

2. Real axis

3. Origin

4. None of these