Practice Questions 2-Complex Numbers & Quadratic Equations-Class XI

The set of points \( z  \) satisfying \( |z+4|+|z-4|=0\) is contained in or equal to____

If \( |w|=2\), then the set of points \( z=w+\dfrac{1}{w}\) is contained in or equal to______

If \(|w|=1\), then the set of points \(z=w+\dfrac{1}{w} \) is contained in or equal to______

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 \( a \text { and } b,\quad |az_1-bz_2|^2+|bz_1+az_2|^2=\)_____

If \( iz^3+z^2-z+i=0\), then _______

Find the real values of \(  x \text { and } y\) for which of the following equation is satisfied \(\dfrac{(1+i)x-2i}{3+i}+\dfrac{(2-3i)y+i}{3-i}=i \)

If \( x+iy=\sqrt{\dfrac{a+ib}{c+id}}\), then \( (x^2+y^2)^2 \) is_____

If \( z \) is any complex number satisfying \(|z-3-2i| \leq 2\), then the minimum value of \(|2z-6+5i| \) is______

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______

If \( \arg (z)\) < \(0 \), then \(\arg (-z)-\arg (z) \) is equal to______

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____

Let \(  z \)  and \( w \) be two non-zero complex number such that \( |z|=|w| \) and \(\arg (z)+\arg (w) =\pi \) then \( z \) equals_____

If \( z_1\) and \(z_2 \) are two non-zero complex numbers such that \(|z_1+z_2|=|z_1|+|z_2|\), then \( \arg (z_1)-\arg (z_2)\) is equal to_____

If \( a,b,c\) and \( u, v, w\) are the complex numbers representing the vertices of two triangles such that \( c=(1-r)a+rb \) and \( w=(1-r)u+rv \) where \( r \) is a complex number, then the two triangles_____

If \(z\neq 0 \) is a complex number, and \(Re (z)=0 \), then______

If \( z=a+ib\neq0 \) be a complex number and \( \arg (z)=\dfrac{\pi}{4}\), then_____

Let \( z_1=10+6i\) and \(z_2=4+6i \). If \( z \) is any complex number such that the argument of \(\dfrac{(z-z_1)}{(z-z_2)} \) is \(\dfrac{\pi}{4} \), then ______

A value of \(\theta \) for which \( \dfrac{2+3 i \sin \theta}{1-2i \sin \theta}\) is purely imaginary, is________

Let \( z=\cos \theta+i \sin \theta\), then the value of \( \sum_{m=1}^{15} ~Im \left(z^{2m-1} \right)\) 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 \(\begin{vmatrix}
1 & 1 & 1 \\
1 &-1-w^2 &w^2 \\
1& w^2 &w
\end{vmatrix} \) 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).

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_____

If \( w \) is an imaginary cube root of unity, then \((1+w-w^2)^7 \) is_____