Chapter 3

The complex numbers \(\sin x+i \cos 2 x\) and \(\cos x-\) \(i \sin 2 x\) are conjugate to each other, for (A) \(x=n \pi\) (B) \(x=0\) (C) \(x=\left(n+\frac{1}{2}\right) \pi\) (D) no value of \(x\)

If \(z_{1}\) and \(z_{2}\) are two non-zero complex numbers such that \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\), then \(\arg z_{1}-\arg z_{2}\) is equal to (A) \(-\pi\) (B) \(-\frac{\pi}{2}\) (C) \(\pi\) (D) \(\frac{\pi}{2}\)

The number of solutions of the equation \(z^{2}+|z|^{2}=0\), where \(z \in C\) is (A) one (B) two (C) three (D) infinitely many

The number of solutions of the equation \(z^{2}+|z|^{2}=0\), where \(z \in C\) is (A) one (B) two (C) three (D) infinitely many

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

\(z_{1}, z_{2}\) are two non-real complex numbers such that \(\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}}=1\). Then \(z_{1}, z_{2}\) and the origin(A) are collinear (B) form right angled triangle (C) form right angle isosceles triangle (D) form an equilateral triangle

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

vIf \((\sqrt{3}+i)^{100}=2^{99}(a+i b)\), then \(b=\) (A) \(\sqrt{3}\) (B) \(\sqrt{2}\) (C) 1 (D) None of these

The real value of \(\alpha\) for which the expression \(\frac{1-i \sin \alpha}{1+2 i \sin \alpha}\) is purely real is (A) \((2 n+1) \frac{\pi}{2}\) (B) \((n+1) \frac{\pi}{2}\) (C) \(n \pi\) (D) None of these

The locus of \(z\) which satisfies the inequality \(\log _{0.3}|z-1|>\log _{0.3}|z-i|\) is given by, (A) \(x+y>0\) (B) \(x-y<0\) (C) \(x+y<0\) (D) \(x-y>0\)

If centre of a regular hexagon is at origin and on of the vertices on argand diagram is \(1+2 i\) then it perimeter is (A) \(2 \sqrt{5}\) (B) \(6 \sqrt{2}\) (C) \(4 \sqrt{5}\) (D) \(6 \sqrt{5}\)

If \(z_{1}, z_{2}, z_{3}\) are three complex numbers, then \(z_{1} \mathrm{Im}\) \(\left(\bar{z}_{2} z_{3}\right)+z_{2} \operatorname{lm}\left(\bar{z}_{3} z_{1}\right)+z_{3} \operatorname{Im}\left(\bar{z}_{1} z_{2}\right)\) is equal to (A) 1 (B) \(-1\) (C) 0 (D) None of these

If \(\frac{2 z_{1}}{3 z_{2}}\) is purely imaginary number, then \(\left|\frac{z_{1}-z_{2}}{z_{1}+z_{2}}\right|^{4}\) is equal to (A) \(\frac{3}{2}\) (B) 1 (C) \(\frac{2}{3}\) (D) \(\frac{4}{9}\)

If \(x^{6}=(4-3 i)^{5}\), then the product of all of its roots is (where \(\left.\theta=-\tan ^{-1}(3 / 4)\right)\) (A) \(5^{5}(\cos 5 \theta+i \sin 5 \theta)\) (B) \(-5^{5}(\cos 5 \theta+i \sin 5 \theta)\) (C) \(5^{5}(\cos 5 \theta-i \sin 5 \theta)\) (D) \(-5^{5}(\cos 5 \theta-i \sin 5 \theta)\)

\(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\) is possible if (A) \(z_{2}=\overline{\bar{z}}_{1}\) (B) \(z_{2}=\frac{1}{z_{1}}\) (C) \(\arg z_{1}=\arg z_{2}\) (D) \(\left|z_{1}\right|=\left|z_{2}\right|\)

If \(z=x+i y, x, y\) real, then \(|x|+|y| \leq k|z|\), where \(k\) is equal to (A) 1 (B) \(\sqrt{2}\) (C) \(\sqrt{3}\) (D) None of these

Let \(z_{1}=a+i b, z_{2}=p+i q\) be two unimodular complex numbers such that \(\operatorname{Im}\left(z_{1} \bar{z}_{2}\right)=1\). If \(\omega_{1}=a+i p, \omega_{2}=b+i q\), then (A) \(\operatorname{Re}\left(\omega_{1} \omega_{2}\right)=1\) (B) \(\operatorname{Im}\left(\omega_{1} \omega_{2}\right)=1\) (C) \(\operatorname{Re}\left(\omega_{1} \omega_{2}\right)=0\) (D) \(\operatorname{Im}\left(\omega_{1} \overline{\omega_{2}}\right)=1\)

If \(\sqrt[3]{a+i b}=x+i y\), then \(\frac{a}{x}+\frac{b}{y}=\) (A) \(4\left(x^{2}+y^{2}\right)\) (B) \(4\left(x^{2}-y^{2}\right)\) (C) \(2\left(x^{2}-y^{2}\right)\) (D) None of these

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

The complex numbers \(z_{1}, z_{2}\) and \(z_{3}\) satisfying \(\frac{z_{1}-z_{3}}{z_{2}-z_{3}}=\) \(\frac{1-\sqrt{3} i}{2}\) are the vertices of a triangle which is (A) of area zero (B) right angled isosceles (C) equilateral (D) obtuse angled isosceles

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

If \(1, a_{1}, a_{2}, \ldots, a_{n-1}\) are the \(n n\)th roots of unity, then \(\left(1-a_{1}\right)\left(1-a_{2}\right)\left(1-a_{3}\right) \ldots\left(1-a_{n-1}\right)=\) (A) \(n+1\) (B) \(n\) (C) \(n-1\) (D) None of these

\(|z-1|+|z+3| \leq 8\), then the range of values of \(|z-4|\) is (A) \((0,8)\) (B) \([0,8]\) (C) \([1,9]\) (D) \([5,9]\)

The roots of the equation \(z^{4}+1=0\) are (A) \((\pm 1 \pm i)\) (B) \((\pm 2 \pm 2 i)\) (C) \(\frac{1}{\sqrt{3}}(\pm 1 \pm i)\) (D) None of these

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 greatest value of the moduli of complex numbres \(z\) satisfying the equation \(\left|z-\frac{4}{z}\right|=2\) is (A) \(\sqrt{5}\) (C) \(\sqrt{5}+1\) (B) \(\sqrt{5}-1\) (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 5

\(\frac{z^{2}}{z-1}\) is always real, then (A) \(z\) lies only on a circle (B) \(z\) lies only on the real axis (C) \(z\) lies either on the real axis or on a circle (D) None of these

and \(z_{2}\) are two complex numbers such that \(\frac{z_{1}-2 z_{2}}{2-z_{1} \bar{z}_{2}}\) unimodular whereas \(z_{2}\) is not unimodular. Then \(\left|z_{1}\right|=\) A) 1 (B) 2 (C) 3 (D) 4

The locus of the complex number \(z\) in an argand plane satisfying the inequality \(\log _{1 / 2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1\left(\right.\) where \(\left.|z-1| \neq \frac{2}{3}\right)\) is (A) a circle (B) interior of a circle (C) exterior of a circle (D) None of these

The equation \(z^{3}+i z-1=0\) has (A) three real roots (B) one real root (C) no real roots (D) no real or complex roots

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

If \(S(n)=i^{n}+i^{-n}\), where \(i=\sqrt{-1}\) and \(n\) is a positive integer, then the total number of distinct values of \(S(n)\) is (A) 1 (B) 2 (C) 3 (D) 4

If \(z_{1} \neq-z_{2}\) and \(\left|z_{1}+z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|\), then (A) at least one of \(z_{1}, z_{2}\) is unimodular (B) \(z_{1} \times z_{2}\) is unimodular (C) both \(z_{1}, z_{2}\) are unimodular (D) None of these

If \(z=x+i y\) satisfies amp \((z-1)=\operatorname{amp}(z+3 i)\) then the value of \((x-1): y\) is equal to (A) \(2: 1\) (B) \(-1: 3\) (C) \(1: 3\) (D) None of these

If \(z_{1}, z_{2}, z_{3}, z_{4}\) are the four complex numbers represented by the vertices of a quadrilateral taken in order such that \(z_{1}-z_{4}=z_{2}-z_{3}\) and amp \(\frac{z_{4}-z_{1}}{z_{2}-z_{1}}=\frac{\pi}{2}\) then the quadrilateral is a (A) square (B) rhombus (C) rectangle (D) a cyclic quadrilateral

Let \(z\) be a complex number with modulus 2 and argument \(\frac{2 \pi}{3}\), then \(z\) is equal to (A) \(-1+i \sqrt{3}\) (B) \(1-i \sqrt{3}\) (C) \(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\) (D) None of these

vIf \(\log _{\sqrt{3}}\left(\frac{|z|^{2}-|z|+1}{2+|z|}\right)<2\), then the locus of \(z\) is (A) \(|z|<5\) (B) \(|z|=5\) (C) \(|z|>5\) (D) None of these

If \(|z|=1\), then the value of \(\left(\frac{z-1}{z+1}\right)\) is (A) 0 (B) purely real (C) purely imaginary (D) complex number

If \(z_{1}\) and \(z_{2}\) are complex numbers, such that \(z_{1}+z_{2}\) is a real number, then (A) \(z_{1}=-\bar{z}_{2}\) (B) \(z_{2}=\bar{z}_{1}\) (C) \(z_{1}\) and \(z_{2}\) are any two complex numbers (D) \(z_{1}=\bar{z}_{1}, z_{2}=\bar{z}_{2}\)

The locus of the points representing the complex numbers which satisfy \(|z|-2=0,|z-i|-|z+5 i|=0\) is: (A) a circle with centre at origin (B) a straight line passing through origin (C) the single point \((0,-2)\) (D) None of these

The locus of the points representing the complex numbers which satisfy \(|z|-2=0,|z-i|-|z+5 i|=0\) is: (A) a circle with centre at origin (B) a straight line passing through origin (C) the single point \((0,-2)\) (D) None of these

If \(P, P^{\prime}\) represent the complex number \(z_{1}\) and its additive inverse respectively then the complex equation of the circle with \(P P^{\prime}\) as a diameter is (A) \(\frac{z}{z_{1}}=\overline{\left(\frac{z_{1}}{z}\right)}\) (B) \(z \bar{z}+z_{1} \bar{z}_{1}=0\) (C) \(z \bar{z}_{1}+\bar{z} z_{1}\) (D) None of these

If \(a, b, c, p, q, r\) are three non-zero complex numbers such that \(\frac{p}{a}+\frac{q}{b}+\frac{r}{c}=1+i\) and \(\frac{a}{p}+\frac{b}{q}+\frac{c}{r}=0\), then value of \(\frac{p^{2}}{a^{2}}+\frac{q^{2}}{b^{2}}+\frac{r^{2}}{c^{2}}\) is (A) 0 (B) \(-1\) (C) \(2 i\) (D) \(-2 i\)

If \(z_{1}, z_{2}\) are two complex numbers such that \(\left|\frac{z_{1}-z_{2}}{z_{1}+z_{2}}\right|=1\) and \(t z_{1}=k z_{2}\) where \(k \in \mathbb{R}\), then the angle between \(\left(z_{1}-z_{2}\right)\) and \(\left(z_{1}+z_{2}\right)\) is(A) \(\tan ^{-1}\left(\frac{2 k}{k^{2}+1}\right)\) (B) \(\tan ^{-1}\left(\frac{2 k}{1-k^{2}}\right)\) (C) \(-2 \tan ^{-1}(k)\) (D) \(2 \tan ^{-1}(k)\)

\(1+x^{2}=\sqrt{3} x\), then \(\sum_{n=1}^{24}\left(x^{n}-\frac{1}{x^{n}}\right)^{2}\) is equal to (A) 48 (B) \(-48\) \(\begin{array}{ll}(\mathrm{C})+48\left(\omega-\omega^{2}\right) & \text { (D) } 1+48\end{array}\)

For any two complex numbers \(z_{1}\) and \(z_{2}\) with \(\left|z_{1}\right| \neq\left|z_{2}\right|\) \(\left|\sqrt{2} z_{1}+i \sqrt{3} \bar{z}_{2}\right|^{2}+\left|\sqrt{3} \bar{z}_{1}+i \sqrt{2} z_{2}\right|^{2}\) is (A) less than \(5\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\) (B) greater than \(10\left|\mathrm{z}_{1} z_{2}\right|\) (C) equal to \(2\left|z_{1}\right|^{2}+3\left|z_{2}\right|^{2}\) (D) 7ero

If the complex numbers \(z_{1}, z_{2}, z_{3}\) are in \(\mathrm{AP}\), then they lie on a (A) circle (B) parabola (C) line (D) ellipse

If the roots of \((z-1)^{25}=2 \omega^{2}(z+1)^{25}\) where \(\omega\) is a complex cube root of unity are plotted in the argand plane, they lie on (A) a straight line (B) a circle (C) an ellipse (D) None of these

Let \(A_{0} A_{1} A_{2} A_{3} A_{4} A_{5}\) be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments \(A_{0} A_{1}, A_{0} A_{2}\) and \(A_{0} A_{4}\) is (A) \(\frac{3}{4}\) (B) \(3 \sqrt{3}\) (C) 3 (D) \(\frac{3 \sqrt{3}}{2}\)