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#### 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) ${i}^{49}+{i}^{68}+{i}^{89}+{i}^{110}$

#### Answer:

#### 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).

#### Answer:

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.

#### Answer:

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).

#### Answer:

It is given that,

Thus, (ab) = (1, 0).

#### Question 15:

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

#### Answer:

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 $\left(1-i{\right)}^{n}{\left(1-\frac{1}{i}\right)}^{n}$.

#### Answer:

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 $z=-i\overline{z}$.

#### Answer:

Hence,  $z=-i\overline{z}$.

#### Question 19:

Solve the system of equations

#### Answer:

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 $\left|z\right|$.

#### Answer:

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.

#### Answer:

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.

#### Answer:

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 $\left|z\right|=z+1+2i$.

#### Answer:

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 ${\left(1+i\right)}^{2n}={\left(1-i\right)}^{2n}$?

#### Answer:

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 $\left|{z}_{1}+{z}_{2}+{z}_{3}\right|$.

#### Answer:

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

#### Question 26:

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

#### Answer:

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

#### Answer:

(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.

#### Answer:

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 ${z}_{1}=-\overline{{z}_{2}}$.

#### Answer:

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

#### Answer:

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

#### Answer:

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

#### Answer:

(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) $a+b$

#### Answer:

(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) $\sqrt{{x}^{2}-{y}^{2}}$

#### Answer:

(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)

#### Answer:

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

#### Answer:

(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

#### Answer:

(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

#### Answer:

(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

#### Answer:

(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

#### Answer:

(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.

#### Answer:

(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

#### Answer:

(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.

#### Answer:

(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

#### Answer:

(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

#### Answer:

(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

#### Answer:

(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) $\frac{1}{2}\mathrm{cot}\frac{\mathrm{\theta }}{2}$

#### Answer:

(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

#### Answer:

(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

#### Answer:

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

#### Answer:

(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

#### Answer:

(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) $\frac{5\mathrm{\pi }}{2}$

#### Answer:

(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) $-\frac{\mathrm{\pi }}{6}$

#### Answer:

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

#### Answer:

(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

#### Answer:

(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

#### Answer:

(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

#### Answer:

(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) $b>a>0$

#### Answer:

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

#### Answer:

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

#### Answer:

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

#### Answer:

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) ${\left|z\right|}^{2}\ge {\left|z\right|}^{2}$

#### Answer:

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) $\left|{z}_{1}+{z}_{2}\right|\ge \left|{z}_{1}\right|+\left|{z}_{2}\right|$

#### Answer:

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

#### Answer:

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

#### Answer:

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.

#### Answer:

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

#### Answer:

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

#### Answer:

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

#### Answer:

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

#### Answer:

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|

#### Answer:

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

#### Answer:

|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

#### Answer:

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 ____________.

#### Answer:

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 ____________.

#### Answer:

(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 ____________.

#### Answer:

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 ____________.

#### Answer:

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 = ____________.

#### Answer:

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 $\left(\overline{z}\right)$ = ____________.

#### Answer:

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 ____________.

#### Answer:

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.

#### Answer:

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 ____________.

#### Answer:

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 ____________.

#### Answer:

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 ____________.

#### Answer:

|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) = ____________.

#### Answer:

,
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) = ____________.

#### Answer:

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 ____________.

#### Answer:

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 = ____________.

#### Answer:

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 ____________.

#### Answer:

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) = ____________.

#### Answer:

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 ____________.

#### Answer:

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.

#### Answer:

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.

#### Answer:

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 ____________.

#### Answer:

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 ____________.

#### Answer:

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 ____________.

#### Answer:

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 ____________.

#### Answer:

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 ____________.

#### Answer:

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) = ____________.

#### Answer:

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 = ____________.

#### Answer:

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) = ____________.

#### Answer:

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) = ____________.

#### Answer:

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 = ____________.

#### Answer:

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 ${\left(1+i\sqrt{3}\right)}^{2}$.

Find z, if

#### Answer:

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.

#### Answer:

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 ${x}^{2}+{y}^{2}$.

#### Answer:

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

#### Question 15:

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

#### Answer:

Hence, $\sqrt{-25}×\sqrt{-9}=-15$.

#### Question 16:

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

#### Answer:

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 $\mathrm{arg}\left(z\right)+\mathrm{arg}\left(\overline{z}\right)$.

#### Answer:

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 $\left|z+1\right|$.

#### Answer:

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 ${\left|a{z}_{1}-b{z}_{2}\right|}^{2}+{\left|a{z}_{2}+b{z}_{1}\right|}^{2}$.

#### Answer:

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 $\frac{2-i}{{\left(1-2i\right)}^{2}}$.

#### Answer:

∴ 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 ${i}^{n}+{i}^{n+1}+{i}^{n+2}+{i}^{n+3}$.

#### Answer:

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

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.

#### Answer:

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

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.

If , find z.

#### Answer:

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 $\left(1+\sqrt{3}\right)$.

#### Answer:

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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