91Ó°ÊÓ

The complex number which satisfies the equation \(z+\sqrt{2}|z+1|+i=0\) is (A) \(2-i\) (B) \(-2-i\) (C) \(2+i\) (D) \(-2+i\)

\(\tan \left[i \log \frac{a-i b}{a+i b}\right]\) is equal to (A) \(\frac{2 a b}{a^{2}+b^{2}}\) (B) \(\frac{a^{2}-b^{2}}{2 a b}\) (C) \(\frac{2 a b}{2}\) (D) \(a b\)

If \(z=a+i b\) where \(a>0, b>0\), then (A) \(|z| \geq \frac{1}{\sqrt{2}}(a-b)\) (B) \(|z| \geq \frac{1}{\sqrt{2}}(a+b)\) (C) \(|z|<\frac{1}{\sqrt{2}}(a+b)\) (D) None of these

If \(\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{2 n} x^{2 n}\), then \(a_{0}+a_{3}+a_{6}+\ldots=\) (A) \(3^{n}\) (B) \(3^{n-1}\) (C) \(3^{n-2}\) (D) None of these

The closest distance of the origin from a curve given as \(a \bar{z}+\bar{a} z+a \bar{a}=0\) ( \(a\) is a complex number) is (A) 1 (B) \(\frac{|a|}{2}\) (B) \(\frac{\operatorname{Re} a}{|a|}\) (D) \(\frac{\operatorname{Im} a}{|a|}\)

The integral solution of the equation \((1-i)^{n}=2^{n}\) is (A) \(n=0\) (B) \(n=1\) (C) \(n=-1\) (D) None of these

The locus of the complex number \(z\) in an argand plane satisfying the equation $$ \operatorname{Arg}(z+i)-\operatorname{Arg}(z-i)=\frac{\pi}{2} \text { is } $$ (A) boundary of a circle (B) interior of a circle (C) exterior of a circle (D) None of these

If for the complex numbers \(z_{1}\) and \(z_{2},\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right|\), then \(\operatorname{amp} z_{1} \sim a m p z_{2}=\) (A) \(\pi\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{4}\) (D) None of these

If for the complex numbers \(z_{1}\) and \(z_{2},\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right|\), then \(\operatorname{amp} z_{1} \sim a m p z_{2}=\) (A) \(\pi\) (B) \(\frac{\pi}{2}\) (C) \(\frac{\pi}{4}\) (D) None of these

Let \(z_{1}\) and \(z_{2}\) be two complex numbers such that \(\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}}=1\), then (A) \(z_{1}, z_{2}\) are collinear (B) \(z_{1}, z_{2}\) and the origin from a right angled triangle (C) \(z_{1}, z_{2}\) and the origin form an equilateral triangle (D) None of these