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The set of points \( z \) satisfying \( z+4+z4=0\) is contained in or equal to____
Sum of distance of from and is a constant . Hence, locus of is an ellipse with semimajor axis and focus at , .
If \( w=2\), then the set of points \( z=w+\dfrac{1}{w}\) is contained in or equal to______
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If \(w=1\), then the set of points \(z=w+\dfrac{1}{w} \) is contained in or equal to______
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If \(\alpha, \beta, \gamma \) are the cube roots of \( P\), \( P\) < 0, then for any \(x, y \) and \( z \) then \( \dfrac{x \alpha +y \beta+z \gamma}{x \beta+y\gamma+z\alpha}\) is equal to_______
For any two complex numbers \(z_1,z_2 \) and any real number \( a \text { and } b,\quad az_1bz_2^2+bz_1+az_2^2=\)_____
If \( iz^3+z^2z+i=0\), then _______
If , then
If , then
If , then
Find the real values of \( x \text { and } y\) for which of the following equation is satisfied \(\dfrac{(1+i)x2i}{3+i}+\dfrac{(23i)y+i}{3i}=i \)
If \( x+iy=\sqrt{\dfrac{a+ib}{c+id}}\), then \( (x^2+y^2)^2 \) is_____
If \( z \) is any complex number satisfying \(z32i \leq 2\), then the minimum value of \(2z6+5i \) is______
Given ,
To find the minimum value of or , using triangle inequality i.e.,
Minimum value
If \( z \) is a complex number of unit modulus and argument \(\theta \), then \(\arg \left( \dfrac{1+z}{1+\overline{z}} \right) \) is equal to______
Given,
, But
If \( \arg (z)\) < \(0 \), then \(\arg (z)\arg (z) \) is equal to______
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Let \( z \) and \( w \) be two complex numbers such that \(z \leq 1, ~w \leq 1 \) and \(z+iw =z\overline{iw}=2 \) then \( z \) equals____
lies on the perpendicular bisector of the line joining and . Since, is the mirror image of in the xaxis, the locus of is the xaxis.
Let and
Now
may take values and .
Let \( z \) and \( w \) be two nonzero complex number such that \( z=w \) and \(\arg (z)+\arg (w) =\pi \) then \( z \) equals_____
and
Let , then
If \( z_1\) and \(z_2 \) are two nonzero complex numbers such that \(z_1+z_2=z_1+z_2\), then \( \arg (z_1)\arg (z_2)\) is equal to_____
Given,
If \( a,b,c\) and \( u, v, w\) are the complex numbers representing the vertices of two triangles such that \( c=(1r)a+rb \) and \( w=(1r)u+rv \) where \( r \) is a complex number, then the two triangles_____
Since, and are the vertices of two triangles.
Also
Consider
Applying
, i.e., two triangles are similar.
If \(z\neq 0 \) is a complex number, and \(Re (z)=0 \), then______
Let
then or
If \( z=a+ib\neq0 \) be a complex number and \( \arg (z)=\dfrac{\pi}{4}\), then_____
Given,
i.e.,
Let \( z_1=10+6i\) and \(z_2=4+6i \). If \( z \) is any complex number such that the argument of \(\dfrac{(zz_1)}{(zz_2)} \) is \(\dfrac{\pi}{4} \), then ______
represents locus of is a circle whose center is and < , clearly
Center=(7, 9) and radius
Equation of circle is
A value of \(\theta \) for which \( \dfrac{2+3 i \sin \theta}{12i \sin \theta}\) is purely imaginary, is________
is purely imaginary, then .
As
Let \( z=\cos \theta+i \sin \theta\), then the value of \( \sum_{m=1}^{15} ~Im \left(z^{2m1} \right)\) at \(\theta=2^o \) is_______
If \( w \) be a cube root of unity and \( (1+w^2)^n=(1+w^4)^n\), then the least positive value of \( n \) is ______
Let \(w=\dfrac{1}{2}+i \dfrac{\sqrt{3}}{2}\), then the value of the determinant \(\begin{vmatrix}
1 & 1 & 1 \\
1 &1w^2 &w^2 \\
1& w^2 &w
\end{vmatrix} \) is______
Let \( z_1 \text { and } z_2\) be \(n^{\text{th}} \) roots of unity which subtend a right angle at the origin, then \( n \) must be of the form ( where \( k \) is an integer).
Since,
If \( i=\sqrt{1}\), then \( 4+5 \left( \dfrac{1}{2}+\dfrac{i \sqrt{3}}{2} \right)^{334}+3\left( \dfrac{1}{2}+\dfrac{i\sqrt{3}}{2} \right)^{365}\) is equal to_____
Let
If \( w \) is an imaginary cube root of unity, then \((1+ww^2)^7 \) is_____