Chapter 2

Write the following complex numbers in the polar form (i) \(-1-i\) (ii) \(\frac{2+6 \sqrt{3} i}{5+\sqrt{3} i}\)

For any complex number \(z\), which of the following is not true? (a) \(\operatorname{Re}(z)=\frac{z+\bar{z}}{2}\) (b) \(\operatorname{Im}(z) \frac{\mathrm{z}-\bar{z}}{2 i}\) (c) \(|z|^{2}=z \bar{z}\) (d) \(-|\operatorname{Re}(z)| \leq|z| \leq|\operatorname{Re}(z)|\)

Find the modulus and argument of each of the following complex numbers (i) \(1+i \sqrt{3}\) (ii) \(-4\) (iii) \(\frac{1}{1+i}\) (iv) \(1+i\)

If \(|z-i|<|z+i|\), then (a) \(\operatorname{Re}(z)>0\) (b) \(\operatorname{Re}(z)<0\) (c) \(\operatorname{Im}(z)>0\) (d) \(\operatorname{Im}(z)<0\)

If points \(p\) represents the complex number \(z=x+i y\) on the argand plane, then find the locus of the point \(p\) such that \(\arg (z)=0\).

Express \((1-\cos \theta+i \sin \theta)\) in polar form.

\((\cos \theta+i \sin \theta)^{2}\) is equal to (a) \(\cos 2 \theta+i \sin 2 \theta\) (b) \(\sin 2 \theta+i \cos 2 \theta\) (c) \(\cos 2 \theta-i \sin 2 \theta\) (d) none of these

If \(z_{1}\) and \(z_{2}\) be two complex numbers such that \(\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right|\), then prove that arg \(\left(z_{1}\right)-\arg \left(z_{2}\right)=\pi / 2 .\)

Prove that \(\left|z_{1}+z_{2}\right|^{2}+\left|z_{1}-z_{2}\right|^{2}=2\left[\left|z_{1}\right|^{2}+\right.\) \(\left.\left|z_{2}\right|^{2}\right]\)

\(\left(\frac{1}{\sqrt{3}+i}\right)^{24}=\) (a) \(2^{24}\) (b) \(-2^{24}\) (c) \(\frac{1}{2^{24}}\) (d) none of these

If \(z_{1}, z_{2}\) are two non \(-\) zero complex numbers such that \(\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|\). Then, prove that arg \(z_{1}-\arg z_{2}=0\).

\(\left(\frac{\sqrt{3}+i}{2}\right)^{69}\) is equal to (a) 1 (b) \(-1\) (c) \(-i\) (d) \(i\)

If \(z=\frac{\sqrt{3}+i}{2}\) then prove that \(Z^{69}=-i\).

\(\left(\frac{1-i}{1+i}\right)^{100}=a+i b\), then (a) \(a=2, b=-1\) (b) \(a=1, b=0\) (c) \(a=0, b=1\) (d) \(a=-1, b=2\)

Find the square roots of the following (i) \(7-24 i\) (ii) \(-8 i\)

Find the square roots of the following (i) \(5+12 i\) (ii) \(-15-8 i\)

If \(z_{1}\) and \(z_{2}\) are any two complex numbers then \(\left|z_{1}+z_{2}\right|^{2}+\left|z_{1}-z_{2}\right|^{2}\) is equal to (a) \(2\left|z_{1}\right|^{2}\left|z_{2}\right|^{2}\) (b) \(2\left|z_{1}\right|^{2}+2\left|z_{2}\right|^{2}\) (c) \(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\) (d) \(2\left|z_{1}\right|\left|z_{2}\right|\)

If \(\omega\) is one cube root of unity, then prove that \(\left(1-\omega+\omega^{2}\right)^{5}+\left(1+\omega-\omega^{2}\right)^{5}=32\)

If \(1, \omega, \omega^{2}\) are the cube roots of unity, then prove that \((1-\omega)^{3}-\left(1+\omega^{2}\right)^{3}=0 .\)

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 \(\arg \left(z_{1}\right)-\arg \left(z_{2}\right)\) is equal to (a) \(-\pi\) (b) \(-\pi / 2\) (c) \(\pi / 2\) (d) 0

Find the cube roots of \(27 .\)

If \(z\) and \(\omega\) are two non-zero complex numbers such that \(|\mathrm{z}|=|\omega|\) and \(\arg (z)+\arg\) \((\omega)=\pi\), then \(z\) is equal to (a) \(\omega\) (b) \(-\omega\) (c) \(\bar{\omega}\) (d) \(-\bar{\omega}\)

If \(\omega_{1}\) and \(\omega_{2}\) are complex cube roots of unity, then prove that \(\omega_{1}^{4}+\omega_{2}^{4}=-\frac{1}{\omega_{1} \omega_{2}}\).

\(-1+\sqrt{-3}=r e^{\theta}\), then \(\theta=\) (a) \(\frac{2 \pi}{3}\) (b) \(-\frac{2 \pi}{3}\) (c) \(\frac{\pi}{3}\) (d) \(-\frac{\pi}{3}\)

If \(z \neq 0\) is a complex number such that \(\operatorname{Arg}(z)=\pi / 4\), then (a) \(\operatorname{Im}\left(z^{2}\right)=0\) (b) \(\operatorname{Re}\left(z^{2}\right)=0\) (c) \(\operatorname{Re}(z)=i \mathrm{~m}\left(z^{2}\right)\) (d) none of these

The value of \((i)^{i}\) is (a) \(\omega\) (b) \(\omega^{2}\) (c) \(e^{-\pi / 2}\) (d) \(2 \sqrt{2}\)

Argument of the complex number \(\left(\frac{-1-3 i}{2+i}\right)\) is (a) \(45^{\circ}\) (b) \(135^{\circ}\) (c) \(225^{\circ}\) (d) \(240^{\circ}\)

What is Arg (bi) where \(b>0 ?\) (a) 0 (b) \(\frac{\pi}{2}\) (c) \(\pi\) (d) \(\frac{3 \pi}{2}\)

Let \(c\) be the set of complex numbers and \(z_{1}, z_{2}\) are in \(C .\) (i). Argument \(z_{1}=\) argument \(z_{2} \Rightarrow z_{1}=z_{2}\) (ii). \(\left|z_{1}\right|=\left|z_{2}\right| \Rightarrow z_{1}=z_{2}\) Which of the statements given above is/are correct? (a) 1 only (b) 2 only (c) both 1 and 2 (d) neither 1 nor 2

If \(y=\cos \theta+i \sin \theta\), then the value of \(y+\frac{1}{y}\) is (a) \(2 \cos \theta\) (b) \(2 \sin \theta\) (c) \(2 \operatorname{cosec} \theta\) (d) \(2 \tan \theta\)

The imaginary part of \(i^{i}\) is (a) 0 (b) 1 (c) 2 (d) \(-1\)

If \(z_{1}\) and \(z_{2}\) are any two complex numbers, then which of the following is not true. (a) \(\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|\) (b) \(\left|z_{1}-z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|\) (c) \(\left|z_{1}-z_{2}\right| \geq \| z_{1}|-| z_{2}||\) (d) \(\left|z_{1}\right|-\left|z_{2}\right| \geq\left|z_{1}-z_{2}\right|\)

If \(z^{2}=-i\), then \(z=\) (a) \(\frac{1}{\sqrt{2}}(1+i)\) (b) \(\frac{1}{\sqrt{2}}(1+i)\) (c) \(\pm \frac{1}{\sqrt{2}}(1-i)\) (d) none of these

If \(\omega\) is a non real cube root of unity, then \(\frac{a+b \omega+c \omega^{2}}{a \omega+c+b \omega^{2}}\) (a) 1 (b) \(\omega^{2}\) (c) \(\omega\) (d) none of these

If \(\alpha\) and \(\beta\) are imaginary cube roots of unity, then the value of \(\alpha^{4}+\beta^{28}+\frac{1}{\alpha \beta}\), is (a) 1 (b) \(-1\) (c) 0 (d) none of these

If \(\omega\) is an imaginary cube root of unity, \(\left(1+\omega-\omega^{2}\right)^{7}\) equals (a) \(128 \omega\) (b) \(-128 \omega\) (c) \(128 \omega^{2}\) (d) \(-128 \omega^{2}\)

If \(z\) is any complex number such that \(|z+4|\) \(\leq 3\), then the greatest value of \(|z+1|\) is (a) 6 (b) 4 (c) 5 (d) 3

If \(n\) is a positive integer not a multiple of 3 , then \(1+\omega^{n}+\omega^{2 n}=\) (a) 3 (b) 1 (c) 0 (d) none of these

If \(1, \omega, \omega^{2}\) are the three cube roots of unity, then \(\left(3+\omega^{2}+\omega^{4}\right)^{6}=\) (a) 64 (b) 729 (c) 2 (d) 0

If \(\omega\) is a cube root of unity, then \((1+\omega-\) \(\left.\omega^{2}\right)\left(1-\omega+\omega^{2}\right)=\) (a) 1 (b) 0 (c) 2 (d) 4

One of the cube roots of unity is (a) \(\frac{-1+i \sqrt{3}}{2}\) (b) \(\frac{1+i \sqrt{3}}{2}\) (c) \(\frac{1-i \sqrt{3}}{2}\) (d) \(\frac{\sqrt{3}-i}{2}\)

If \(\omega\) is the cube root of unity, then \((3+5 \omega\) \(\left.+3 \omega^{2}\right)^{2}+\left(3+3 \omega+5 \omega^{2}\right)^{2}=\) (a) 4 (b) 0 (c) \(-4\) (d) none of these

If \(\omega\) is cube root of unity then the value of \((1-\omega)\left(1-\omega^{2}\right)\left(1-\omega^{4}\right)\left(1-\omega^{8}\right)=\) (a) 0 (b) 1 (c) \(-1\) (d) 9

\(\left(\frac{-1+i \sqrt{3}}{2}\right)^{20}+\left(\frac{-1-i \sqrt{3}}{2}\right)^{20}=\) (a) \(20 \sqrt{3} i\) (b) 1 (c) \(\frac{1}{2^{19}}\) (d) \(-1\)

If \(\left|Z-\frac{4}{z}\right|=2\), then the maximum value of \(|z|\) is equal to (a) \(\sqrt{3}+1\) (b) \(\sqrt{5}+1\) (c) 2 (d) \(2+\sqrt{2}\)

What is the value of \(\left(\frac{-1+i \sqrt{3}}{2}\right)^{900}+\left(\frac{-1+i \sqrt{3}}{2}\right)^{301} ?\) (a) \(\frac{-1+i \sqrt{3}}{2}\) (b) \(\frac{1-i \sqrt{3}}{2}\) (c) \(\frac{-1-i \sqrt{3}}{2}\) (b) \(\frac{1+i \sqrt{3}}{2}\)

If \(z^{2}+z+1=0\), where \(z\) is a complex number, then the value of \(\left(z+\frac{1}{z}\right)^{2}+\) \(\left(z^{2}+\frac{1}{z^{2}}\right)^{2}+\left(z^{3}+\frac{1}{z^{3}}\right)^{2}+\ldots+\left(z^{6}+\frac{1}{z^{6}}\right)^{2}\) is (a) 6 (b) 12 (c) 18 (d) 24

The number of solutions of the equation \(z^{2}+\bar{z}=0\) is (a) 1 (b) 2 (c) 3 (d) 4

If \(x\) is a positive integer, then \((1+i \sqrt{3})^{n}\) \(+(1-i \sqrt{3}) \mathrm{n}\) is equal to (a) \(2^{n-1} \cos \frac{n \pi}{3}\) (a) \(2^{n} \cos \frac{n \pi}{3}\) (c) \(2^{n+1} \cos \frac{n \pi}{3}\) (d) none of these