Rd Sharma Xi 2018 Solutions for Class 11 Science Math Chapter 13 Complex Numbers are provided here with simple step-by-step explanations. These solutions for Complex Numbers are extremely popular among Class 11 Science students for Math Complex Numbers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2018 Book of Class 11 Science Math Chapter 13 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2018 Solutions. All Rd Sharma Xi 2018 Solutions for class Class 11 Science Math are prepared by experts and are 100% accurate.

#### Question 1:

Evaluate the following:
(i) i457
(ii) i528
(iii) $\frac{1}{{i}^{58}}$
(iv) ${i}^{37}+\frac{1}{{i}^{67}}$
(v) ${\left({i}^{41}+\frac{1}{{i}^{257}}\right)}^{9}$
(vi) $\left({i}^{77}+{i}^{70}+{i}^{87}+{i}^{414}{\right)}^{3}$
(vii)  ${i}^{30}+{i}^{40}+{i}^{60}$
(viii) ${i}^{49}+{i}^{68}+{i}^{89}+{i}^{110}$

#### Question 1:

Express the following complex numbers in the standard form a + i b:
(i) $\left(1+i\right)\left(1+2i\right)$
(ii) $\frac{3+2i}{-2+i}$
(iii) $\frac{1}{\left(2+i{\right)}^{2}}$
(iv) $\frac{1-i}{1+i}$
(v) $\frac{\left(2+i{\right)}^{3}}{2+3i}$
(vi) $\frac{\left(1+i\right)\left(1+\sqrt{3}i\right)}{1-i}$
(vii) $\frac{2+3i}{4+5i}$
(viii) $\frac{\left(1-i{\right)}^{3}}{1-{i}^{3}}$
(ix) $\left(1+2i{\right)}^{-3}$
(x) $\frac{3-4i}{\left(4-2i\right)\left(1+i\right)}$
(xi) $\left(\frac{1}{1-4i}-\frac{2}{1+i}\right)\left(\frac{1-4i}{5+i}\right)$
(xii) $\frac{5+\sqrt{2}i}{1-2\sqrt{i}}$

#### Question 2:

Find the real values of x and y, if
(i) $\left(x+iy\right)\left(2-3i\right)=4+i$
(ii) $\left(3x-2iy\right)\left(2+i{\right)}^{2}=10\left(1+i\right)$
(iii) $\frac{\left(1+i\right)x-2i}{3+i}+\frac{\left(2-3i\right)y+i}{3-i}$
(iv) $\left(1+i\right)\left(x+iy\right)=2-5i$

#### Question 3:

Find the conjugates of the following complex numbers:
(i) 4 − 5 i
(ii) $\frac{1}{3+5i}$
(iii) $\frac{1}{1+i}$
(iv) $\frac{\left(3-i{\right)}^{2}}{2+i}$
(v) $\frac{\left(1+i\right)\left(2+i\right)}{3+i}$
(vi) $\frac{\left(3-2i\right)\left(2+3i\right)}{\left(1+2i\right)\left(2-i\right)}$

#### Question 4:

Find the multiplicative inverse of the following complex numbers:
(i) 1 − i
(ii) $\left(1+i\sqrt{3}{\right)}^{2}$
(iii) 4 − 3i
(iv) $\sqrt{5}+3i$

If

#### Question 6:

If ${z}_{1}=2-i,{z}_{2}=-2+i,$ find
(i) Re $\left(\frac{{z}_{1}{z}_{2}}{{z}_{1}}\right)$
(ii) Im $\left(\frac{1}{{z}_{1}{\overline{)z}}_{1}}\right)$

#### Question 7:

Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$

#### Question 8:

If $x+iy=\frac{a+ib}{a-ib}$, prove that x2 + y2 = 1

#### Question 9:

Find the least positive integral value of n for which ${\left(\frac{1+i}{1-i}\right)}^{n}$ is real.

#### Question 10:

Find the real values of θ for which the complex number  is purely real.

#### Question 11:

Find the smallest positive integer value of m for which $\frac{\left(1+i{\right)}^{n}}{\left(1-i{\right)}^{n-2}}$ is a real number.

#### Question 12:

If ${\left(\frac{1+i}{1-i}\right)}^{3}-{\left(\frac{1-i}{1+i}\right)}^{3}=x+iy$, find (x, y).

Also,

It is given that,

Thus, (xy) = (0, −2).

#### Question 13:

If $\frac{{\left(1+i\right)}^{2}}{2-i}=x+iy$, find x + y.

It is given that,

Thus, x + y = $\frac{2}{5}$.

#### Question 14:

If ${\left(\frac{1-i}{1+i}\right)}^{100}=a+ib$, find (ab).

It is given that,

Thus, (ab) = (1, 0).

#### Question 15:

If $a=\mathrm{cos}\theta +i\mathrm{sin}\theta$, find the value of $\frac{1+a}{1-a}$.

Thus, $\frac{1+a}{1-a}=2i\mathrm{cot}\frac{\theta }{2}$.

#### Question 16:

Evaluate the following:
(i)
(ii)
(iii)
(iv)
(v)

#### Question 17:

For a positive integer n, find the value of $\left(1-i{\right)}^{n}{\left(1-\frac{1}{i}\right)}^{n}$.

Thus, the value of $\left(1-i{\right)}^{n}{\left(1-\frac{1}{i}\right)}^{n}$ is 2n.

#### Question 18:

If $\left(1+i\right)z=\left(1-i\right)\overline{z}$, then show that $z=-i\overline{z}$.

Hence,  $z=-i\overline{z}$.

#### Question 19:

Solve the system of equations

Let $z=x+iy$.
Then ,

and $\left|z\right|=\sqrt{{x}^{2}+{y}^{2}}$

According to the question,

Thus, .

#### Question 20:

If $\frac{z-1}{z+1}$ is purely imaginary number ($z\ne -1$), find the value of $\left|z\right|$.

Let $z=x+iy$.
Then,

If $\frac{z-1}{z+1}$ is purely imaginary number, then
$\mathrm{Re}\left(\frac{z-1}{z+1}\right)=0\phantom{\rule{0ex}{0ex}}⇒{x}^{2}+{y}^{2}-1=0\phantom{\rule{0ex}{0ex}}⇒{x}^{2}+{y}^{2}=1\phantom{\rule{0ex}{0ex}}⇒{\left|z\right|}^{2}=1\phantom{\rule{0ex}{0ex}}⇒\left|z\right|=1$

Thus, the value of $\left|z\right|$ is 1.

#### Question 21:

If z1 is a complex number other than −1 such that $\left|{z}_{1}\right|=1$ and ${z}_{2}=\frac{{z}_{1}-1}{{z}_{1}+1}$, then show that the real parts of z2 is zero.

Let $z=x+iy$.
Then,

Now,

Thus, the real parts of z2 is zero.

#### Question 22:

If $\left|z+1\right|=z+2\left(1+i\right)$, find z.

Let $z=x+iy$.
Then,
$z+1=\left(x+1\right)+iy\phantom{\rule{0ex}{0ex}}⇒\left|z+1\right|=\sqrt{{\left(x+1\right)}^{2}+{y}^{2}}$

$\therefore z=x+iy=\frac{1}{2}-2i$

Thus, $z=\frac{1}{2}-2i$

#### Question 23:

Solve the equation $\left|z\right|=z+1+2i$.

Let $z=x+iy$.
Then,
$\left|z\right|=\sqrt{{x}^{2}+{y}^{2}}$

$\therefore z=x+iy=\frac{3}{2}-2i$

​Thus, $z=\frac{3}{2}-2i$

#### Question 24:

What is the smallest positive integer n for which ${\left(1+i\right)}^{2n}={\left(1-i\right)}^{2n}$?

Thus, the smallest positive integer n for which ${\left(1+i\right)}^{2n}={\left(1-i\right)}^{2n}$ is 2.

#### Question 25:

If z1, z2, z3 are complex numbers such that $\left|{z}_{1}\right|=\left|{z}_{2}\right|=\left|{z}_{3}\right|=\left|\frac{1}{{z}_{1}}+\frac{1}{{z}_{2}}+\frac{1}{{z}_{3}}\right|=1$, then find the value of $\left|{z}_{1}+{z}_{2}+{z}_{3}\right|$.

Thus, the value of $\left|{z}_{1}+{z}_{2}+{z}_{3}\right|$ is 1.

#### Question 26:

Find the number of solutions of ${z}^{2}+{\left|z\right|}^{2}=0$

Let $z=x+iy$.
Then,
$\left|z\right|=\sqrt{{x}^{2}+{y}^{2}}$

For

​Thus, there are infinitely many solutions of the form .

#### Question 1:

Find the square root of the following complex numbers:
(i) −5 + 12i
(ii) −7 − 24i
(iii) 1 − i
(iv) −8 − 6i
(v) 8 −15i
(vi) $-11-60\sqrt{-1}$
(vii)  $1+4\sqrt{-3}$
(viii) 4i
(ix) −i

#### Question 2:

Show that 1 + i10 + i20 + i30 is a real number.

#### Question 3:

Find the values of the following expressions:
(i) i49 + i68 + i89 + i110
(ii) i30 + i80 + i120
(iii) i + i2 + i3 + i4
(iv) i5 + i10 + i15
(v) $\frac{{i}^{592}+{i}^{590}+{i}^{588}+{i}^{586}+{i}^{584}}{{i}^{582}+{i}^{580}+{i}^{578}+{i}^{576}+{i}^{574}}$
(vi) 1+ i2 + i4 + i6 + i8 + ... + i20
(vii) (1 + i)6 + (1 − i)3

(vii) (1 + i)6 + (1 − i)3
= [(1 + i)2]3 + (1 − i)3
= [12 + i2 + 2i]3 + (13 − i3 + 3i− 3i)
= [1 − 1 + 2i]3 + (1 + i − 3 − 3i)           [∵ i2 = −1, i= −i]
= (2i)3 + (−2 − 2i)
= 8i3 − 2 − 2i
= −8i − 2 − 2i                                        [∵ i= −i]
= −10i − 2

#### Question 1:

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
(i) 1 + i
(ii) $\sqrt{3}+i$
(iii) 1 − i
(iv) $\frac{1-i}{1+i}$
(v) $\frac{1}{1+i}$
(vi) $\frac{1+2i}{1-3i}$
(vii)
(viii) $\frac{-16}{1+i\sqrt{3}}$

#### Question 2:

Write (i25)3 in polar form.

Let $z=0-i$.
Then, $\left|z\right|=\sqrt{{0}^{2}+{\left(-1\right)}^{2}}=1$.

Let θ be the argument of z and α be the acute angle given by $\mathrm{tan}\alpha =\frac{\left|\mathrm{Im}\left(z\right)\right|}{\left|\mathrm{Re}\left(z\right)\right|}$.
Then,
$\mathrm{tan}\alpha =\frac{1}{0}=\infty \phantom{\rule{0ex}{0ex}}⇒\alpha =\frac{\mathrm{\pi }}{2}$

Clearly, z lies in fourth quadrant. So, arg(z) = $-\alpha =-\frac{\mathrm{\pi }}{2}$.

∴ the polar form of z is $\left|z\right|\left(\mathrm{cos}\theta +i\mathrm{sin}\theta \right)=\mathrm{cos}\left(-\frac{\mathrm{\pi }}{2}\right)+i\mathrm{sin}\left(-\frac{\mathrm{\pi }}{2}\right)$.

Thus, the polar form of (i25)is $\mathrm{cos}\left(\frac{\mathrm{\pi }}{2}\right)-i\mathrm{sin}\left(\frac{\mathrm{\pi }}{2}\right)$.

#### Question 3:

Express the following complex in the form r(cos θ + i sin θ):
(i) 1 + i tan α
(ii) tan α − i
(iii) 1 − sin α + i cos α
(iv) $\frac{1-i}{\mathrm{cos}\frac{\mathrm{\pi }}{3}+i\mathrm{sin}\frac{\mathrm{\pi }}{3}}$

#### Question 4:

If z1 and z2 are two complex numbers such that $\left|{z}_{1}\right|=\left|{z}_{2}\right|$ and arg(z1) + arg(z2) = $\mathrm{\pi }$, then show that ${z}_{1}=-\overline{{z}_{2}}$.

Let θbe the arg(z1) and θbe the arg(z2).

It is given that $\left|{z}_{1}\right|=\left|{z}_{2}\right|$ and arg(z1) + arg(z2) = $\mathrm{\pi }$.

Since, z1 is a complex number.

Hence,  ${z}_{1}=-\overline{{z}_{2}}$.

#### Question 5:

If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, prove that $\mathrm{arg}\left(\frac{{z}_{1}}{{z}_{4}}\right)+\mathrm{arg}\left(\frac{{z}_{2}}{{z}_{3}}\right)=0$.

Given that z1, z2 and z3, z4 are two pairs of conjugate complex numbers.

Then,

and

Hence,  $\mathrm{arg}\left(\frac{{z}_{1}}{{z}_{4}}\right)+\mathrm{arg}\left(\frac{{z}_{2}}{{z}_{3}}\right)=0$.

#### Question 6:

Express $\mathrm{sin}\frac{\mathrm{\pi }}{5}+i\left(1-\mathrm{cos}\frac{\mathrm{\pi }}{5}\right)$ in polar form.

#### Question 1:

Write the values of the square root of i.

#### Question 2:

Write the values of the square root of −i.

#### Question 3:

If x + iy = $\sqrt{\frac{a+ib}{c+id}}$, then write the value of (x2 + y2)2.

#### Question 4:

If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of $\left|z\right|$.

#### Question 5:

If n is any positive integer, write the value of $\frac{{i}^{4n+1}-{i}^{4n-1}}{2}$.

#### Question 6:

Write the value of $\frac{{i}^{592}+{i}^{590}+{i}^{588}+{i}^{586}+{i}^{584}}{{i}^{582}+{i}^{580}+{i}^{578}+{i}^{576}+{i}^{574}}$.

#### Question 7:

Write 1 − i in polar form.

#### Question 8:

Write −1 + $\sqrt{3}$ in polar form

#### Question 9:

Write the argument of −i.

#### Question 10:

Write the least positive integral value of n for which ${\left(\frac{1+i}{1-i}\right)}^{n}$ is real.

#### Question 11:

Find the principal argument of ${\left(1+i\sqrt{3}\right)}^{2}$.

#### Question 12:

Find z, if

We know that,

Thus, $z=-2\sqrt{3}+2i$.

#### Question 13:

If $\left|z-5i\right|=\left|z+5i\right|$, then find the locus of z.

Hence, the locus of z is real axis.

#### Question 14:

If $\frac{{\left({a}^{2}+1\right)}^{2}}{2a-i}=x+iy$, find the value of ${x}^{2}+{y}^{2}$.

Hence, ${x}^{2}+{y}^{2}=\frac{{\left({a}^{2}+1\right)}^{4}}{4{a}^{2}+1}$.

#### Question 15:

Write the value of $\sqrt{-25}×\sqrt{-9}$.

Hence, $\sqrt{-25}×\sqrt{-9}=-15$.

#### Question 16:

Write the sum of the series $i+{i}^{2}+{i}^{3}+....$upto 1000 terms.

We know that,
$i+{i}^{2}+{i}^{3}+{i}^{4}=i-1-i+1=0$

$\therefore i+{i}^{2}+{i}^{3}+....+{i}^{1000}\phantom{\rule{0ex}{0ex}}=\left(i+{i}^{2}+{i}^{3}+{i}^{4}\right)+\left({i}^{5}+{i}^{6}+{i}^{7}+{i}^{8}\right)+...+\left({i}^{997}+{i}^{998}+{i}^{999}+{i}^{1000}\right)\phantom{\rule{0ex}{0ex}}=\left(i+{i}^{2}+{i}^{3}+{i}^{4}\right)+\left({i}^{4}i+{i}^{4}{i}^{2}+{i}^{4}{i}^{3}+{i}^{4}{i}^{4}\right)+...+\left[{\left({i}^{4}\right)}^{249}i+{\left({i}^{4}\right)}^{249}{i}^{2}+{\left({i}^{4}\right)}^{249}{i}^{3}+{\left({i}^{4}\right)}^{249}{i}^{4}\right]\phantom{\rule{0ex}{0ex}}=\left(i+{i}^{2}+{i}^{3}+{i}^{4}\right)+\left(i+{i}^{2}+{i}^{3}+{i}^{4}\right)+...+\left(i+{i}^{2}+{i}^{3}+{i}^{4}\right)\phantom{\rule{0ex}{0ex}}=0$

Thus, the sum of the series $i+{i}^{2}+{i}^{3}+....$upto 1000 terms is 0.

#### Question 17:

Write the value of $\mathrm{arg}\left(z\right)+\mathrm{arg}\left(\overline{z}\right)$.

Let z be a complex number with argument θ.
Then,
$z=r{e}^{i\theta }\phantom{\rule{0ex}{0ex}}⇒\overline{z}=\overline{r{e}^{i\theta }}=r{e}^{-i\theta }$
⇒ argument of $\overline{z}$ is −θ.

Thus, $\mathrm{arg}\left(z\right)+\mathrm{arg}\left(\overline{z}\right)=0$.

#### Question 18:

If $\left|z+4\right|\le 3$, then find the greatest and least values of $\left|z+1\right|$.

Hence, the greatest and least values of $\left|z+1\right|$ is 6 and 0.

#### Question 19:

For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of ${\left|a{z}_{1}-b{z}_{2}\right|}^{2}+{\left|a{z}_{2}+b{z}_{1}\right|}^{2}$.

Hence, ${\left|a{z}_{1}-b{z}_{2}\right|}^{2}+{\left|a{z}_{2}+b{z}_{1}\right|}^{2}=\left({a}^{2}+{b}^{2}\right)\left({\left|{z}_{1}\right|}^{2}+{\left|{z}_{2}\right|}^{2}\right)$.

#### Question 20:

Write the conjugate of $\frac{2-i}{{\left(1-2i\right)}^{2}}$.

∴ Conjugate of $\frac{2-i}{{\left(1-2i\right)}^{2}}=\left(\overline{-\frac{2}{25}+\frac{11}{25}i}\right)=-\frac{2}{25}-\frac{11}{25}i$

Hence, Conjugate of $\frac{2-i}{{\left(1-2i\right)}^{2}}$ is $-\frac{2}{25}-\frac{11}{25}i$.

#### Question 21:

If n ∈ $\mathrm{ℕ}$, then find the value of ${i}^{n}+{i}^{n+1}+{i}^{n+2}+{i}^{n+3}$.

${i}^{n}+{i}^{n+1}+{i}^{n+2}+{i}^{n+3}\phantom{\rule{0ex}{0ex}}={i}^{n}+{i}^{n}.i+{i}^{n}.{i}^{2}+{i}^{n}.{i}^{3}\phantom{\rule{0ex}{0ex}}={i}^{n}+{i}^{n}.i+{i}^{n}.\left(-1\right)+{i}^{n}.\left(-i\right)\phantom{\rule{0ex}{0ex}}={i}^{n}+{i}^{n}.i-{i}^{n}-{i}^{n}.i\phantom{\rule{0ex}{0ex}}=0$

Thus, the value of ${i}^{n}+{i}^{n+1}+{i}^{n+2}+{i}^{n+3}$ is 0.

#### Question 22:

Find the real value of a for which $3{i}^{3}-2a{i}^{2}+\left(1-a\right)i+5$ is real.

$3{i}^{3}-2a{i}^{2}+\left(1-a\right)i+5\phantom{\rule{0ex}{0ex}}=-3i+2a+\left(1-a\right)i+5\phantom{\rule{0ex}{0ex}}=\left(2a+5\right)+i\left(1-a-3\right)\phantom{\rule{0ex}{0ex}}=\left(2a+5\right)+i\left(-2-a\right)$

Since, $3{i}^{3}-2a{i}^{2}+\left(1-a\right)i+5$ is real.

Hence, the real value of for which $3{i}^{3}-2a{i}^{2}+\left(1-a\right)i+5$ is real is −2.

#### Question 23:

If , find z.

We know that,

Hence, $z=\sqrt{2}\left(1+i\right)$.

#### Question 24:

Write the argument of $\left(1+i\sqrt{3}\right)\left(1+i\right)\left(\mathrm{cos}\theta +i\mathrm{sin}\theta \right)$.

Disclaimer: There is a misprinting in the question. It should be $\left(1+i\sqrt{3}\right)$ instead of $\left(1+\sqrt{3}\right)$.

Let the argument of $\left(1+i\sqrt{3}\right)$ be α. Then,
$\mathrm{tan}\alpha =\frac{\sqrt{3}}{1}=\mathrm{tan}\frac{\mathrm{\pi }}{3}\phantom{\rule{0ex}{0ex}}⇒\alpha =\frac{\mathrm{\pi }}{3}$

Let the argument of $\left(1+i\right)$ be β. Then,
$\mathrm{tan\beta }=\frac{1}{1}=\mathrm{tan}\frac{\mathrm{\pi }}{4}\phantom{\rule{0ex}{0ex}}⇒\mathrm{\beta }=\frac{\mathrm{\pi }}{4}$

Let the argument of $\left(\mathrm{cos\theta }+i\mathrm{sin\theta }\right)$ be γ. Then,
$\mathrm{tan\gamma }=\frac{\mathrm{sin\theta }}{\mathrm{cos\theta }}=\mathrm{tan\theta }\phantom{\rule{0ex}{0ex}}⇒\mathrm{\gamma }=\mathrm{\theta }$

∴ The argument of $\left(1+i\sqrt{3}\right)\left(1+i\right)\left(\mathrm{cos\theta }+i\mathrm{sin\theta }\right)=\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=\frac{\mathrm{\pi }}{3}+\frac{\mathrm{\pi }}{4}+\mathrm{\theta }=\frac{7\mathrm{\pi }}{12}+\mathrm{\theta }$

Hence, the argument of .

#### Question 1:

The value of $\left(1+i\right)\left(1+{i}^{2}\right)\left(1+{i}^{3}\right)\left(1+{i}^{4}\right)$ is
(a) 2
(b) 0
(c) 1
(d) i

(b) 0
(1+ i) (1 + i2) (1 + i3) (1 + i4)
= (1+ i) (1 $-$ 1) (1 $-$ i) (1 + 1)      ($\because$i2 = $-$1,  i3 = $-$i and i4  = 1)
= (1 + i) (0) (1 $-$ i) (2)
= 0

#### Question 2:

If  is a real number and 0 < θ < 2π, then θ =
(a) π
(b) $\frac{\mathrm{\pi }}{2}$
(c) $\frac{\mathrm{\pi }}{3}$
(d) $\frac{\mathrm{\pi }}{6}$

(a) π

Given:

is a real number

On rationalising, we get,

For the above term to be real, the imaginary part has to be zero.

$\therefore \frac{8\mathrm{sin}\theta }{1+4{\mathrm{sin}}^{2}\theta }=0\phantom{\rule{0ex}{0ex}}⇒8\mathrm{sin}\theta =0$

For this to be zero,
sin $\theta$= 0
$⇒$ $\theta$ = 0,
But $0<\theta <2\pi$
Hence, $\theta =\pi$

#### Question 3:

If is equal to
(a) $\sqrt{{a}^{2}+{b}^{2}}$
(b) $\sqrt{{a}^{2}-{b}^{2}}$
(c) ${a}^{2}+{b}^{2}$
(d) ${a}^{2}-{b}^{2}$
(e) $a+b$

(c) a2 + b2

(1 + i)(1 + 2i)(1 + 3i) ......(1 + ni) = a + ib

Taking modulus on both the sides, we get:

Squaring on both the sides, we get:

2

#### Question 4:

If $\sqrt{a+ib}=x+iy,$ then possible value of $\sqrt{a-ib}$ is
(a) ${x}^{2}+{y}^{2}$
(b) $\sqrt{{x}^{2}+{y}^{2}}$
(c) x + iy
(d) xiy
(e) $\sqrt{{x}^{2}-{y}^{2}}$

(d) x $-$ iy

If , then
(a)
(b)
(c)
(d)

(d)

#### Question 6:

The polar form of (i25)3 is
(a)
(b) cos π + i sin π
(c) cos π − i sin π
(d)

(d)
(i25)3 = (i)75
= (i)4$×$18+ 3
= (i)3

= $-$i            ($\because$ i4  = 1)

Modulus, r =

$\therefore$ Polar form = r (cos $\theta$ + i sin $\theta$)
= cos$\left(\frac{-\mathrm{\pi }}{2}\right)$+i sin$\left(\frac{-\mathrm{\pi }}{2}\right)$
= cos$\frac{\pi }{2}$ $-$ i sin $\frac{\pi }{2}$

#### Question 7:

If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
(a) 1
(b) −1
(c) i
(d) 0

(d) 0

#### Question 8:

If $z=\frac{-2}{1+i\sqrt{3}}$, then the value of arg (z) is
(a) π
(b) $\frac{\mathrm{\pi }}{3}$
(c) $\frac{2\mathrm{\pi }}{3}$
(d) $\frac{\mathrm{\pi }}{4}$

(c) $\frac{2\pi }{3}$
z =

Rationalising z, we get,

#### Question 9:

If a = cos θ + i sin θ, then $\frac{1+a}{1-a}=$
(a) $\mathrm{cot}\frac{\mathrm{\theta }}{2}$
(b) cot θ
(c)
(d)

(c)

#### Question 10:

If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n2)=
(a) aib
(b) a2b2
(c) a2 + b2
(d) none of these

(c) a2 + b2

(1 + i)(1 + 2i)(1 + 3i) ......(1 + ni) = a + ib

Taking modulus on both the sides, we get,

Squaring on both the sides, we get:

2×5×10×.....(1 + n2)  = a2 + b2

#### Question 11:

If  is equal to
(a) $\frac{\left({a}^{2}+1{\right)}^{4}}{4{a}^{2}+1}$
(b) $\frac{\left(a+1{\right)}^{2}}{4{a}^{2}+1}$
(c) $\frac{\left({a}^{2}-1{\right)}^{2}}{\left(4{a}^{2}-1{\right)}^{2}}$
(d) none of these

(a)$\frac{{\left({a}^{2}+1\right)}^{4}}{4{a}^{2}+1}$

Taking modulus on both the sides, we get:

#### Question 12:

The principal value of the amplitude of (1 + i) is
(a) $\frac{\mathrm{\pi }}{4}$
(b) $\frac{\mathrm{\pi }}{12}$
(c) $\frac{3\mathrm{\pi }}{4}$
(d) π

(a)$\frac{\pi }{4}$

Let z = (1+i)

Therefore, arg (z) = $\frac{\pi }{4}$

#### Question 13:

The least positive integer n such that ${\left(\frac{2i}{1+i}\right)}^{n}$ is a positive integer, is
(a) 16
(b) 8
(c) 4
(d) 2

#### Question 14:

If z is a non-zero complex number, then  is equal to
(a) $\left|\frac{\overline{)z}}{z}\right|$
(b)
(c)
(d) none of these

(a) $\left|\frac{\overline{)z}}{z}\right|$

#### Question 15:

If a = 1 + i, then a2 equals
(a) 1 − i
(b) 2i
(c) (1 + i) (1 − i)
(d) i − 1.

(b) 2i

a = 1 + i
On squaring both the sides, we get,
a2 = (1 + i)2
$⇒$a2  = 1 + i2  + 2i
$⇒$a2  = 1$-$1 + 2i          ($\because$ i2 = $-$1)
$⇒$a2  = 2i

#### Question 16:

If (x + iy)1/3 = a + ib, then $\frac{x}{a}+\frac{y}{b}=$
(a) 0
(b) 1
(c) −1
(d) none of these

(d) none of these

#### Question 17:

$\left(\sqrt{-2}\right)\left(\sqrt{-3}\right)$ is equal to
(a) $\sqrt{6}$
(b) $-\sqrt{6}$
(c) $i\sqrt{6}$
(d) none of these.

(b) $-\sqrt{6}$

#### Question 18:

The argument of $\frac{1-i\sqrt{3}}{1+i\sqrt{3}}$ is
(a) 60°
(b) 120°
(c) 210°
(d) 240°

(d) 240°

#### Question 19:

If $z=\left(\frac{1+i}{1-i}\right)$, then z4 equals
(a) 1
(b) −1
(c) 0
(d) none of these

(a) 1

Rationalising the denominator:

$⇒z=\frac{1+{i}^{2}+2i}{1-{i}^{2}}\phantom{\rule{0ex}{0ex}}$

$⇒z=\frac{2i}{2}\phantom{\rule{0ex}{0ex}}⇒z=i$

#### Question 20:

If $z=\frac{1+2i}{1-\left(1-i{\right)}^{2}}$, then arg (z) equal
(a) 0
(b) $\frac{\mathrm{\pi }}{2}$
(c) π
(d) none of these.

(a) 0

#### Question 21:

(a) $\frac{1}{13}$
(b) $\frac{1}{5}$
(c) $\frac{1}{12}$
(d) none of these

(a) $\frac{1}{13}$

$⇒\left|z\right|=\frac{1}{13}$

#### Question 22:

(a) 1
(b) $1/\sqrt{26}$
(c) $5/\sqrt{26}$
(d) none of these

(b) $\frac{1}{\sqrt{26}}$

$⇒z=\frac{1}{\sqrt{26}}$

#### Question 23:

(a)
(b)
(c) $2\left|\mathrm{sin}\frac{\mathrm{\theta }}{2}\right|$
(d) $2\left|\mathrm{cos}\frac{\mathrm{\theta }}{2}\right|$

(c)

#### Question 24:

If $x+iy=\left(1+i\right)\left(1+2i\right)\left(1+3i\right)$, then x2 + y2 =
(a) 0
(b) 1
(c) 100
(d) none of these

(c) 100

#### Question 25:

If , then Re (z) =
(a) 0
(b) $\frac{1}{2}$
(c) $\mathrm{cot}\frac{\mathrm{\theta }}{2}$
(d) $\frac{1}{2}\mathrm{cot}\frac{\mathrm{\theta }}{2}$

(b) $\frac{1}{2}$

#### Question 26:

If $x+iy=\frac{3+5i}{7-6i},$ then y =
(a) 9/85
(b) −9/85
(c) 53/85
(d) none of these

(c) $\frac{53}{85}$

#### Question 27:

If $\frac{1-ix}{1+ix}=a+ib$, then ${a}^{2}+{b}^{2}$=
(a) 1
(b) −1
(c) 0
(d) none of these

(a) 1

#### Question 28:

If θ is the amplitude of $\frac{a+ib}{a-ib}$, than tan θ =
(a) $\frac{2a}{{a}^{2}+{b}^{2}}$
(b) $\frac{2ab}{{a}^{2}-{b}^{2}}$
(c) $\frac{{a}^{2}-{b}^{2}}{{a}^{2}+{b}^{2}}$
(d) none of these

(b) $\frac{2ab}{{a}^{2}-{b}^{2}}$

#### Question 29:

If $z=\frac{1+7i}{\left(2-i{\right)}^{2}}$, then
(a)
(b)
(c) amp (z) = $\frac{\mathrm{\pi }}{4}$
(d) amp (z) = $\frac{3\mathrm{\pi }}{4}$

(d) amp (z) = $\frac{3\mathrm{\pi }}{4}$

#### Question 30:

The amplitude of $\frac{1}{i}$ is equal to
(a) 0
(b) $\frac{\mathrm{\pi }}{2}$
(c) $-\frac{\mathrm{\pi }}{2}$
(d) π

(c) $-\frac{\pi }{2}$

#### Question 31:

The argument of $\frac{1-i}{1+i}$ is
(a) $-\frac{\mathrm{\pi }}{2}$
(b) $\frac{\mathrm{\pi }}{2}$
(c) $\frac{3\mathrm{\pi }}{2}$
(d) $\frac{5\mathrm{\pi }}{2}$

(a) $-\frac{\pi }{2}$

#### Question 32:

The amplitude of $\frac{1+i\sqrt{3}}{\sqrt{3}+i}$ is
(a) $\frac{\mathrm{\pi }}{3}$
(b) $-\frac{\mathrm{\pi }}{3}$
(c) $\frac{\mathrm{\pi }}{6}$
(d) $-\frac{\mathrm{\pi }}{6}$

(c) $\frac{\pi }{6}$

#### Question 33:

The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
(a) $\frac{1}{2}\left(1+i\right)$
(b) $\frac{1}{2}\left(1-i\right)$
(c) 1
(d) $\frac{1}{2}$

(a) $\frac{1}{2}\left(1+i\right)$

#### Question 34:

$\frac{1+2i+3{i}^{2}}{1-2i+3{i}^{2}}$ equals
(a) i
(b) −1
(c) −i
(d) 4

(c) $-$i

#### Question 35:

The value of $\frac{{i}^{592}+{i}^{590}+{i}^{588}+{i}^{586}+{i}^{584}}{{i}^{582}+{i}^{580}+{i}^{578}+{i}^{576}+{i}^{574}}-1$ is
(a) −1
(b) −2
(c) −3
(d) −4

(b) $-$2

#### Question 36:

The value of $\left(1+i{\right)}^{4}+\left(1-i{\right)}^{4}$ is
(a) 8
(b) 4
(c) −8
(d) −4

(c) $-$8

#### Question 37:

If $z=a+ib$ lies in third quadrant, then $\frac{\overline{z}}{z}$ also lies in third quadrant if

(a) $a>b>0$
(b) $a
(c) $b
(d) $b>a>0$

Since, $z=a+ib$ lies in third quadrant.

Now,

Since, $\frac{\overline{z}}{z}$ also lies in third quadrant.

From (1) and (2),
$b

Hence, the correct option is (c).

#### Question 38:

If $f\left(z\right)=\frac{7-z}{1-{z}^{2}}$, where $z=1+2i$, then $\left|f\left(z\right)\right|$ is

(a) $\frac{\left|z\right|}{2}$
(b) $\left|z\right|$
(c) $2\left|z\right|$
(d) none of these

Since $z=1+2i$,

Hence, the correct answer is option (a).

#### Question 39:

A real value of x satisfies the equation

(a) 1
(b) −1
(c) 2
(d) −2

Hence, the correct option is (a).

#### Question 40:

The complex number z which satisfies the condition $\left|\frac{i+z}{i-z}\right|=1$ lies on

(a) circle x2 + y2 = 1
(b) the x−axis
(c) the y−axis
(d) the line x + y = 1

Hence, the correct option is (b).

#### Question 41:

If z is a complex number, then

(a) ${\left|z\right|}^{2}>{\left|z\right|}^{2}$
(b) ${\left|z\right|}^{2}={\left|z\right|}^{2}$
(c) ${\left|z\right|}^{2}<{\left|z\right|}^{2}$
(d) ${\left|z\right|}^{2}\ge {\left|z\right|}^{2}$

It is obvious that, for any complex number z,
${\left|z\right|}^{2}={\left|z\right|}^{2}$

Hence, the correct option is (b).

#### Question 42:

Which of the following is correct for any two complex numbers z1 and z2?

(a) $\left|{z}_{1}{z}_{2}\right|=\left|{z}_{1}\right|\left|{z}_{2}\right|$
(b)
(c) $\left|{z}_{1}+{z}_{2}\right|=\left|{z}_{1}\right|+\left|{z}_{2}\right|$
(d) $\left|{z}_{1}+{z}_{2}\right|\ge \left|{z}_{1}\right|+\left|{z}_{2}\right|$

Since, it is known that
$\left|{z}_{1}{z}_{2}\right|=\left|{z}_{1}\right|\left|{z}_{2}\right|\phantom{\rule{0ex}{0ex}}$,
$\mathrm{arg}\left({z}_{1}{z}_{2}\right)=\mathrm{arg}\left({z}_{1}\right)+\mathrm{arg}\left({z}_{2}\right)$ and
$\left|{z}_{1}+{z}_{2}\right|\le \left|{z}_{1}\right|+\left|{z}_{2}\right|$

Hence, the correct option is (a).

#### Question 43:

If the complex number $z=x+iy$ satisfies the condition $\left|z+1\right|=1$, then z lies on

(a) x−axis
(b) circle with centre (−1, 0) and radius 1
(c) y−axis
(d) none of these