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The value of \(\sum_{n=1}^{13}\left( i^n+i^{n+1} \right)\), is equals to______
If \(\begin{vmatrix}
6i & 3i & 1 \\
4& 3i &1 \\
20 & 3 & i
\end{vmatrix}=x+iy \), then_____
The smallest positive integer \( n \) for which \( \left( \dfrac{1+i}{1i} \right)^n=1\) is______
The smallest positive integer for which is .
A complex number \( Z \) is said to be unimodular, if \( Z \neq 1 \). If \(Z_1 \text { and } Z_2 \) are complex number such that \(\dfrac{Z_12Z_2}{2Z_1Z_2} \) is unimodular and \( Z_2\) is not unimodular, then the points \( Z_1\) lies on a_____
is not unimodular i.e., and is unimodular.
Point lies on a circle of radius .
If \( Z \) is a complex number such that \( Z \geq 2 \), then the minimum value of \( Z+\dfrac{1}{2} \).
is the region on or outside circle whose center is and radius is . Minimum is distance of , which lies on circle from .
Minimum distance of from
Hence, minimum value of lies in the interval .
Let \( z \) be a complex number such that the imaginary part of \( z \) is nonzero and \( a=z^2+z+1\) is real. Then, \( a \) cannot take the value______
For do not have real roots , < <
So, other 3 options are showing imaginary roots.
Let \( z=x+iy \), be a complex number where, \( x \text { and } y \) are integers. Then, the area of the rectangle whose vertices are the root of the equation \( z \overline{z}^3+\overline{z}z^3=350\).
Hence
Area of rectangle
If \( z=1 \) and \( z \neq \pm 1\), then all the values of \(\dfrac{z}{1z^2}\) lie on ______
Hence , lies on the imaginary axis i.e. yaxis.
If \( w=\alpha+i \beta\), where \(\beta \neq 0\) and \( z \neq 1 \) satisfies the condition that \( \left( \dfrac{w\overline{w}z}{1z} \right)\) is purely real, then the set of values of \( z \) is_____
be purely real.
If \( z=1 \) and \( \dfrac{z1}{z+1}( \text { where } z \neq 1)\) then \( Re(w) \) is______
and
For all complex numbers \( z_1, z_2 \) satisfying \( z_1=12\) and \( z_234i=5 \), the minimum value of \( z_1z_2\) is_____
The minimum value of is .
If \( z_1, z_2 \text { and } z_3 \) are complex numbers such that \(z_1=z_2=z_3=\left \dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3} \right=1 \) then \( z_1+z_2+z_3 \) is_____
Similarly,
Again,
For positive integers \( n_1, n_2\) the value of expression \( (1+i)^n+(1+i^3)^{n_1}+(1+i^5)^{n_2}+(1+i^7)^{n_2}\) here \(i=\sqrt{1} \) is a real number , iff_____
This is a real number irrespective of the values > 0 and > 0.
The complex numbers \( \sin x + i \cos 2x \) and \(\cos x i \sin 2x \) are conjugate to each other, for_____
Since,
which is not possible at the same time.
Hence, no solution exists.
The points \( z_1, z_2, z_3 \text { and } z_4\) in the complex plane are the vertices of parallelogram taken in order , iff_____
Midpoint of midpoint of .
If \( z=x+iy \) and \( w=\left( \dfrac{1iz}{zi} \right)\) then \( w=1\) implies that in the complex plane.
It is a perpendicular bisector of and , i.e., xaxis. Thus, lies on the real axis.
The inequality \(z4 \) < \(z2 \) represents the region given by_____
<
Since , > represents the region on right side of the perpendicular bisector of and .
>
> and
If \(z= \left( \dfrac{\sqrt{3}}{2}+\dfrac{i}{2} \right)^5+\left( \dfrac{\sqrt{3}}{2}\dfrac{i}{2} \right)^5 \) then ______
Now
and
< and
The complex numbers \(z=x+iy \) which satisfy the equation \( \left \dfrac{z5i}{z+5i} \right=1\), lies on_______
Given,
Perpendicualr bisector of and is xaxis.
Let \(z_1 \text { and }z_2 \) be complex numbers such that \(z_1 \neq z_2 \text { and } z_1 =z_2 \). If \( z_1\) has positive real part and \(z_2 \) has negative imaginary part, then \( \dfrac{z_1+z_2}{z_1z_2}\) may be______
Now,
As,
Which is purely imaginary.
If \( z=a+ib\) and \( z_2=c+id\) are complex numbers such that \(z_1=z_2=1 \) and \( Re(z_1\overline{z_2})=0\), then the pair of complex numbers \( w_1=a+ic\) and \( w_2=b+id\) satisfies, _____
Also,
Given,
Let \(A, B ,C \) are three sets of complex number as defined below:
\( A=\left\{ z: Im~ z \geq 1 \right\}, \quad B=\left\{ z: z2i=3 \right\}, \quad C=\left\{ z: Re(1i)z=\sqrt{2} \right\}\),
then find \(\text { min } z \in S \) \( 13iz\) is equal to_____
perpendicular distance of point from the line
Let \( z \) be any point in \( A\cap B\cap C\) and let \( w \) be any point satisfying \( w2i \leq 3\) then \( zw+3\) lies between_______
Since, <
<
< <
< <
Also,
< <
\( S=S_1\cap S_2\cap S_3\), where \(S_1= z \in C: z \) 0 \right\}[/latex] and \(S_3=\left\{ z \in C : Re z > 0 \right\}\), where \( z \) be any point in \( A\cap B \cap C\), then \(z+1i^2+z5i^2 \) lies between______
< <
The set of points \( z \) satisfying \( ziz=z+iz\) is contained in or equal to_____
is equidistance from the points and whose perpendicular bisector is