Rd Sharma Xi 2020 _volume 1 Solutions for Class 12 Commerce Math Chapter 13 Complex Numbers are provided here with simple step-by-step explanations. These solutions for Complex Numbers are extremely popular among Class 12 Commerce 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 2020 _volume 1 Book of Class 12 Commerce Math Chapter 13 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2020 _volume 1 Solutions. All Rd Sharma Xi 2020 _volume 1 Solutions for class Class 12 Commerce 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:

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

Hence, the correct option is (b).

#### Question 44:

sin x + i cos 2x and cos xi sin 2x are conjugate to each other for

(a) x = nπ

(b) $x=\left(n+\frac{1}{2}\right)\frac{\mathrm{\pi }}{2}$

(c) x = 0

(d) No value of x

Given sin x + i cos 2x and cos xi sin 2x are conjugate to each other
i.e sin x + i cos 2x = cos x i sin 2x
i.e sin xi cos 2x = cos x i sin 2x
on comparing real and imaginary part,
sin x = cos x and cos 2x = sin 2x
i.e. sin x = cos x and 2cos2 x – 1 = 2 sin x cos x
i.e 2cos2 x – 1 = 2 cos x cos         (∴ sin x = cos x)
i.e 2cos2 x – 1 = 2cos2 x
i.e – 1 = 0
which is a false statement.
Hence no value of x exist
Therefore, the correct answer is option D.

#### Question 45:

The real value of α for which the expression is purely real, is

(a)

(b)

(c)

(d) none of these where nN.

Given is purely real

Which is given to purely real

Hence, the correct answer is option C.

#### Question 46:

The value of  is equivalent to

(a) |z + 3|2

(b) |z – 3|

(c) z2 + 3

(d) none of these

Since |z + 3|2

Hence, the correct answer is option A.

#### Question 47:

If ${\left(\frac{1+i}{1-i}\right)}^{n}=1,$ then n =
(a) 2m + 1
(b) 4m
(c) 2m
(d) 4m + 1 where mN

Given :- ${\left(\frac{1+i}{1-i}\right)}^{n}=1$
$\begin{array}{rcl}\mathrm{Since},\frac{1+i}{1-i}& =& \frac{1+i}{1-i}×\frac{1+i}{1+i}\\ & =& \frac{{\left(1+i\right)}^{2}}{1-{i}^{2}}\\ & =& \frac{1+{i}^{2}+2i}{1+1}\end{array}$

Hence, the correct answer is option B.

#### Question 48:

The vector represented by the complex number 2 – i is rotated about the origin through an angle $\frac{\mathrm{\pi }}{2}$ in the clockwise direction, the new position of point is

(a) 1 + 2i

(b) –1 –2i

(c) 2 + i

(d) –1 + 2i

Given 2 – i is rotated by $\frac{\mathrm{\pi }}{2}$ angle in the clockwise direction about the origin

Let Z' denote the new position and Z denote the previous p

Hence, the correct answer is option B.

#### Question 49:

The real value of θ for which the expression is a real number, is

(a) $n\mathrm{\pi }+\frac{\mathrm{\pi }}{4}$

(b) $n\pi +{\left(-1\right)}^{n}\frac{\pi }{4}$

(c) $2n\mathrm{\pi }±\frac{\mathrm{\pi }}{2}$

(d) none of theses

Given is a real number

Since is a real number

Hence, the correct answer is option C.

#### Question 50:

|z1 + z2| = |z1| + |z2| is possible if

(a) ${z}_{2}={\overline{z}}_{1}$

(b) ${z}_{2}=\frac{1}{{z}_{1}}$

(c) arg (z1) = arg (z2)

(d) |z1| = |z2|

Since given |z1 + z2| = |z1| + |z2|
i.e |z1 + z2|2 = (|z1| + |z2|)2

Hence, the correct answer is option C.

#### Question 51:

The equation |z + 1 – i| = |z – 1 + i| represents a
(a) straight line
(b) circle
(c) parabola
(d) hyperbola

|z + 1 – i| = |z – 1 + i|
Let z = x + iy ; where x denote real part and y denote imaginary part of z

Hence, the correct answer is option A.

#### Question 52:

The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is

(a) |z|2

(b) $|\overline{z}{|}^{2}$

(c) $\frac{1}{2}{\left|z\right|}^{2}$

(d) none of these

For any complex number z, –iz represents complex number obtained by rotating z clockwise by $\frac{\mathrm{\pi }}{2}$ angle.
Hence, z, –iz and z + iz represents a right angled triangle with sides z, –iz and hypotenus z + iz
∴ Area of triangle formed is

Hence, the correct answer is option C.

#### Question 1:

The principal value of the argument of the complex number 1 –i is ____________.

Since z = 1 – i = 1 + i(–1)

Since z lies in 4th quadrant.
∴argument is given by $\frac{-\mathrm{\pi }}{y}$

#### Question 2:

The polar form of (i25)3 is ____________.

(i25)3
= (i24. i)3
Since i24 = 1
= (i)3 = i2. i = – i
i.e (i25)3 = – i
Here modulus $r=\sqrt{{0}^{2}+{1}^{2}}=1$ and argument $\theta ={\mathrm{tan}}^{-1}\left|\frac{-1}{0}\right|=\infty =-\frac{\mathrm{\pi }}{2}$

Polar form of –i is

#### Question 3:

The value of   is ____________.

#### Question 4:

The complex number $\frac{{\left(1-i\right)}^{3}}{1-{i}^{3}}$ in polar form is ____________.

#### Question 5:

The sum of the series i + i2i3 +_____ upto 1000 terms is ____________.

i + i2i3 +_____ 1000 terms
i.e i + i2i3 + i4 _____ + i1000
=
i + i2i + 1 + i5 _____ + i1000
= i – 1 – i + 1 + i5 + _______ + i1000
= 0 + i5 + i6 + ________ + i100
Similarly, sum of next form terms is also zero.   (∵ 1000 = 4(250)
i.e multiple of 4
Hence, i + i2i3 +_____ i1000 = 0

#### Question 6:

The multiplicative inverse of (1 + i) is ____________.

For z = 1 + i
Let us suppose multiplicative inverse of 1 + i is a + ib
then (1 + i) (a + ib) = 1
i.e a + ib + ai + i2b = 1
i.e a + ib + ia b = 1
i.e a b + i(a + b) = 1 + i0
On comparing, real and imaginary part, we get
a – b = 1 and a + b = 0
i.e a = 1 and a = – b
i.e a + a = 1

i.e multiplicative inverse of 1 + i is $\frac{1}{2}-\frac{i}{2}$

#### Question 7:

If |z| = 4 and  then z = ____________.

If |z| = 4 and

#### Question 8:

If z1 and z2 are two complex numbers such that z1 + z2 is a real number, then z2 = ____________.

Given for two complex numbers, z1 and z2, we have z1+ z2 is real number

#### Question 9:

For any non-zero complex number z, arg (z) + arg $\left(\overline{z}\right)$ = ____________.

For complex number z
Say z = x + iy = re where r = modulus of z, θ = argument of $\overline{z}=x-iy$
$⇒\overline{z}=r{e}^{-i\theta }$
Let us arg z = θ
Since arg $\overline{z}$ = – arg z
⇒ arg z + arg $\overline{z}$ = θ + (–θ)
i.e arg z + arg $\left(\overline{z}\right)$ = 0

#### Question 10:

If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are _______ and ____________.

Given |z + 4| ≤ 3
here |z + 1| = |z + 1 + 3 – 3| = |z + 4 + (–3)|
Since |a + b| ≤ |a| + |b| ≤ |z + 4| + |–3| = |z + 4| + 3
≤ 3 + 3    (given)
hence maximum value of |z + 1| is 6
|z + 1| = |z + 4 –3|
Since |a – b| ≥ ||a| – |b|| ≥ –|a| + |b|
⇒ |z + 1| ≥ – |z + 4| + 3
Since |z + 4| ≤ 3
⇒ –|z + 4| ≥ –3
i.e |z + 1| ≥ –|z + 4| + 3 ≥ –3 + 3 = 0
Hence, minimum value of |z + 1| is 0.

#### Question 11:

The modulus and argument of $\mathrm{sin}\frac{\mathrm{\pi }}{5}+i\left(1-\mathrm{cos}\frac{\mathrm{\pi }}{5}\right)$ are _______ and _______ respectively.

Since complex number is $\mathrm{sin}\frac{\mathrm{\pi }}{5}+i\left(1-\mathrm{cos}\frac{\mathrm{\pi }}{5}\right)$

and argument is

∴ argument is given by $\frac{\mathrm{\pi }}{10}$
∴ modulus of and argument is $\frac{\mathrm{\pi }}{10}$.

#### Question 12:

If $\left|\frac{z-2}{z+2}\right|=\frac{\mathrm{\pi }}{6},$ then the locus of z is ____________.

Given $\left|\frac{z-2}{z+2}\right|=\frac{\mathrm{\pi }}{6}$

which defines locus of a  circle.

#### Question 13:

If |z + 2i| = |z – 2i|, then the locus of z is ____________.

Given |z + 2i| = |z – 2i|
for z = |x + iy|
|x + iy + 2i| = |x + iy – 2i|
Squaring both sides, |x + i(y + 2)|2 = |x + i (y – 2)|2
i.e x2 + (y + 2)|2 = x2 + (y – 2)|2
i.e x2 + y2 + 4 + 4y = x2 + y2 + 4 – 4y
i.e 8y = 0
i.e y = 0
∴ locus is perpendicular bisector of the segment joining (0, –2) and (0, 2)

#### Question 14:

If |z + 2| = |z – 2|, then the locus of z is ____________.

|z + 2| = |z – 2| for z = x + iy
i.e |x + iy + 2| = |x + iy – 2|
i.e |(x + 2) + iy| = |(x – 2) + iy|
Square both sides,
|(x + 2)| + iy|2 = |(x – 2)| + iy|2
i.e (x + 2)2 + y2 = (x – 2)2 + y2
i.e x2 + 4 + 4x + y2 = x2 + 4 – 4x + y2
i.e fx = 0
i.e x = 0
Hence, locus is perpendicular bisector of the segment joining (–2, 0) and (2, 0).

#### Question 15:

If z = –1 +$\sqrt{–3}$, then arg (z) = ____________.

,
Since z lies in IV quadrant.
⇒ argument of z is π $-\frac{\mathrm{\pi }}{3}=\frac{2\mathrm{\pi }}{3}$.

#### Question 16:

If x < 0 is a real number, then arg (x) = ____________.

If x < 0
i.e z = x + i 0 and x is negative
$⇒\theta ={\mathrm{tan}}^{-1}\left|\frac{0}{x}\right|={\mathrm{tan}}^{-1}0=0$
Hence, z lies in II quadrant chg z = π.

#### Question 17:

The real value of 'a' for which 3i3 – 2ai2 + (1 – a) i + 5 is real is ____________.

3i3 – 2ai2 + (1 – a) i + 5
i.e 3i2 i – 2a(–1) + (1 – a) i + 5
i.e 3i + 2a + (1 – a) i + 5
i.e 2a + 5 + i(1 – a – 3)
i.e 2a + 5 + i (–a – 2)
Since the above expression is given to be real
⇒ –a – 2 = 0
a = – 2

#### Question 18:

If |z| = 2 and arg (z) = $\frac{\mathrm{\pi }}{4}$,  then z = ____________.

for |z| = 2 = r arg z = $\frac{\mathrm{\pi }}{4}$

z = r (cos(arg z) + i sin (arg z))

#### Question 19:

The value of ${\left(-\sqrt{-1}\right)}^{4n-3},$ where nN, is ____________.

#### Question 20:

The locus of z satisfying is ____________.

Given and for z = x + iy

#### Question 21:

The conjugate of the complex number $\frac{1-i}{1+i}$ is ____________.

#### Question 22:

If (2 + i) (2 + 2i) (2 + 3i) ...... (2 + ni) = x + iy, then 5.8.13...(4 + n2) = ____________.

Given:- (2 + i) (2 + 2i) (2 + 3i) ...... (2 + ni) = x + iy
Taking modulus both sides, we get
|(2 + i) (2 + 2i) (2 + 3i) ...... (2 + ni) = |x + iy|
By squaring both sides, we get

#### Question 23:

If the point representing a complex number lies in the third quadrant, then the point representing its conjugate lies in the ____________.

If z lies in III quadrant.

i.e z = – xiy ; x, y ≥ 0

i.e $\overline{z}$ lies in II quadrant.

#### Question 24:

The multiplication of a non-zero complex number by i rotates it through ____________ in the anti-clockwise direction.

Let z = x + iy then iz = ix + i2y
i.e iz = ixy
i.e iz = – y + ix

i.e iz is rotated by 90°.

#### Question 25:

The complex number cosθ + i sinθ __________ be zero for any θ.

z = cosθ + i sinθ can never be zero for any θ because z = 0
⇒ cosθ = 0 and sin θ = 0
Since no such value of  θ exists
⇒ cosθ + i sinθ  can never be zero.

#### Question 26:

The argument of the complex number is ____________.

Since z1 lies in II quadrant

Since z2 lies in I quadrant

#### Question 27:

If a complex number coincides with its conjugate, then it lies on ____________.

Let z = x + iy and $\overline{z}=\overline{x+iy}$
$\overline{z}=x-iy$
Since z = $\overline{z}$       (given)
x + iy = x iy
iy = – iy
⇒ 2iy = 0
i.e y = 0
Then z lies an x-axis.

#### Question 28:

The points representing the complex number z for which |z + 1| < |z – 1| lie on the left side of ____________.

Given |z + 1| < |z – 1|

Let z = x + iy where x, y R

Squaring both sides, we get

Hence, |z + 1| < |z – 1| lies on left side of y-axis

#### Question 29:

If three complex numbers z1, z2 and z3 are in A.P., then points representing them lie on ____________.

Since z1, z2 and z3 are in A.P
Hence 2z2 = z1 + z3

i.e z2 is the mid-point of line joining z1 and z3
z1, zand z3 lie on a straight.

#### Question 30:

The principal argument of i–1097 is ____________.

Let z = i–1097

Hence, principle argument is $-\frac{\mathrm{\pi }}{2}$.

#### Question 31:

The value of $\frac{{i}^{4n+1}-{i}^{4n-1}}{2}$ is ____________.

#### Question 32:

If ${z}_{1}=\sqrt{3}+i\sqrt{3}$ and ${z}_{2}=\sqrt{3}+i,$ then the point representing $\frac{{z}_{1}}{{z}_{2}}$ lies in ____________.

#### Question 33:

If 0 < arg (z) < π, then arg (z) – arg (–z) = ____________.

For 0 < arg z < π
Let z = r(cosθ, i sinθ)

i.e arg z = θ
Then –z = – r(cosθ + i sinθ)
= – r (+ cosθ + i (+ sinθ))

= (–1) re
= eiπ re
= rei(θ + π)
i.e arg (–z) = θ + π
⇒ arg z – arg(–z) = θ – θ – π
= – π

#### Question 34:

For any two complex numbers z1, z2 and any real numbers a, b, |az1bz2|2 + |bz1 + az2|2 = ____________.

For complex z1 and z2 and real numbers a and b

#### Question 35:

Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg (z1) – arg (z2) = ____________.

Given for complex number z1 and z2

i.e angle between z1 and z2 is 0
i.e arg (z1) – arg z2 = 0

#### Question 36:

Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1z2|, then arg (z1) – arg (z2) = ____________.

Given |z1 + z2| = | z1 – z2|
On squaring both sides, |z1 + z2|2 = | z1 – z2|2            (1)
i.e for

Using identities

#### Question 37:

If |z1| = |z2| and arg $\left(\frac{{z}_{1}}{{z}_{2}}\right)=\mathrm{\pi },$ then z1 + z2 = ____________.

Let |z1| = |z2| = r
Let arg z1 = θ1 and arg z2 = θ2

#### 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 .

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