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If \( w \) is a cube root of unity and \((1+w)^7=A+Bw \), the \( A \) and \( B \) are respectively_____
The value of \( \sum_{k=1}^{6}\left( \sin \dfrac{2\pi k}{7}i \cos \dfrac{2 \pi k}{7} \right)\) is______
Let \(Z_k=\cos \left( \dfrac{3 k \pi}{10} \right)+i \sin \left( \dfrac{2k \pi}{10} \right); k=1,2,…9 \). For each \(Z_k \), there exists a \(Z_j \) such that \( Z_k \cdot Z_j\) is equal to______
Given,
is root of unity.
Taking, as , we have
Let \( w \) be the complex number \( \cos \dfrac{2\pi}{3}+i \sin \dfrac{2 \pi}{3}\). Then the number of distinct complex number \( z \) satisfying \( \begin{vmatrix}
z+1 & w &w^2 \\
w & z+w &1 \\
w^2 & 1 & z+w
\end{vmatrix}=0\) is equal to_____
Let
Now,
, the number of satisfying the given equation is .
The value of the expression \( 1(2w)(2w^2)+2(3w)(3w^2)+…+(n1)(nw)(nw^2)\) where, \( w \) is an imaginary cube root of unity is______
The cube roots of unity when represented on the Argand plane form the vertices of an or a _______
Since, cube roots of unity are given by
is an equilateral triangle.
If \( x=a+b, \quad y=a \alpha+b \beta,\quad z=a\beta+b \alpha\), where \(\alpha, \beta \) are complex cube roots of unity, then \( xyz\) is equal to_____
Let
For \(n \in N \), find the value of \( (1i)^n\left( 1\dfrac{1}{i} \right)^n\).
Solve: \(\sum_{n=1}^{13}\left( i^n+i^{n+1} \right) \), where \( n \in N \).
If \( \dfrac{(1+i)^2}{2i}=x+iy\), then find the value of \( x+y \).
If \(\left( \dfrac{1i}{1+i} \right)^{100}=a+ib \), then find \( (a, b)\).
Hence
If \(a=\cos \theta+i \sin \theta \), find the value of \( \dfrac{1+a}{1a}\).
If \( (1+i)z=(1i)\overline {z}\), then \( z \) equals to_______
If \(z=x+iy \), where \( z \overline{z}+2(z+\overline{z})+b=0\), \(b \in R \), represents_____
, which is the equation of a circle.
If \( \dfrac{z1}{z+1}\) is a purely imaginary number, then find the value of \( z (z \neq 1)\)
Since is purely imaginary number then
If \(z_1 \text { and } z_2 \) are two complex numbers such that \( z_1=z_2\) and \(\arg (z_1) +\arg (z_2)=\pi \), then _____
Let
If if
Now,
If \( z_1=1 \text { and } z_2=\dfrac{z_11}{z_1+1} (z_1 \neq 1)\) , then \( Re (z_2)\) is_____
If \( z_1, z_2 \text { and } z_3,z_4\) are two pairs of conjugate number, then find \(\arg \left( \dfrac{z_1}{z_4} \right)+\arg \left( \dfrac{z_2}{z_3} \right) \)
Let
Let
The value of \(\sqrt{25}\times \sqrt{9} \) is______
Multiplicative inverse of \( 1+i \) is______
Multiplicative inverse of
\( \arg (z)+\arg (\overline{z})\), where \( (\overline{z} \neq 0) \) is_____
Solve:\(\dfrac{(1i)^3}{1i^3} \)
The sum of the series \( i+i^2+i^3+…\) up to \(1000\) terms____
up to terms
For any complex \( z \), the minimum value of \( z+z1 \) is_______
Let
If , then value of
If \( z \) is a complex number such that \( z \neq 0\) and \( Re (z)=0\), then_____
Let i.e.,
, which is real.