RD Sharma solutions for Class 11 Mathematics Textbook chapter 13 - Complex Numbers [Latest edition]

Chapter 13: Complex Numbers

Exercise 13.1Exercise 13.2Exercise 13.3Exercise 13.4Others
Exercise 13.1 [Pages 3 - 13]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers Exercise 13.1 [Pages 3 - 13]

Exercise 13.1 | Q 1.1 | Page 3

Evaluate the following:

i457

Exercise 13.1 | Q 1.2 | Page 3

Evaluate the following:

(ii) i528

Exercise 13.1 | Q 1.3 | Page 3

Evaluate the following:

$\frac{1}{i^{58}}$

Exercise 13.1 | Q 1.4 | Page 3

Evaluate the following:

$i^{37} + \frac{1}{i^{67}}$

Exercise 13.1 | Q 1.5 | Page 3

Evaluate the following:

$\left( i^{41} + \frac{1}{i^{257}} \right)^9$

Exercise 13.1 | Q 1.6 | Page 3

Evaluate the following:

$( i^{77} + i^{70} + i^{87} + i^{414} )^3$

Exercise 13.1 | Q 1.7 | Page 13

Evaluate the following:

$i^{30} + i^{40} + i^{60}$

Exercise 13.1 | Q 1.8 | Page 3

Evaluate the following:

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

Exercise 13.1 | Q 2 | Page 4

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

Exercise 13.1 | Q 3.1 | Page 4

Find the value of the following expression:

i49 + i68 + i89 + i110

Exercise 13.1 | Q 3.2 | Page 4

Find the value of the following expression:

i30 + i80 + i120

Exercise 13.1 | Q 3.3 | Page 4

Find the value of the following expression:

i + i2 + i3 + i4

Exercise 13.1 | Q 3.4 | Page 4

Find the value of the following expression:

i5 + i10 + i15

Exercise 13.1 | Q 3.5 | Page 4

Find the value of the following expression:

$\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}$

Exercise 13.1 | Q 3.6 | Page 4

Find the value of the following expression:

1+ i2 + i4 + i6 + i8 + ... + i20

Exercise 13.1 | Q 3.7 | Page 4

Find the value of the following expression:

(1 + i)6 + (1 − i)3

Exercise 13.2 [Pages 31 - 33]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers Exercise 13.2 [Pages 31 - 33]

Exercise 13.2 | Q 1.01 | Page 31

Express the following complex number in the standard form a + i b:

$(1 + i)(1 + 2i)$

Exercise 13.2 | Q 1.02 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{3 + 2i}{- 2 + i}$

Exercise 13.2 | Q 1.03 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{1}{(2 + i )^2}$

Exercise 13.2 | Q 1.04 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{1 - i}{1 + i}$

Exercise 13.2 | Q 1.05 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{(2 + i )^3}{2 + 3i}$

Exercise 13.2 | Q 1.06 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}$ .

Exercise 13.2 | Q 1.07 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{2 + 3i}{4 + 5i}$

Exercise 13.2 | Q 1.08 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{(1 - i )^3}{1 - i^3}$

Exercise 13.2 | Q 1.09 | Page 31

Express the following complex number in the standard form a + i b:

$(1 + 2i )^{- 3}$

Exercise 13.2 | Q 1.1 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{3 - 4i}{(4 - 2i)(1 + i)}$

Exercise 13.2 | Q 1.11 | Page 31

Express the following complex number in the standard form a + i b:

$\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{1 - 4i}{5 + i} \right)$

Exercise 13.2 | Q 1.12 | Page 31

Express the following complex number in the standard form a + i b:

$\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}$

Exercise 13.2 | Q 2.1 | Page 31

Find the real value of x and y, if

$(x + iy)(2 - 3i) = 4 + i$

Exercise 13.2 | Q 2.2 | Page 31

Find the real value of x and y, if

$(3x - 2iy)(2 + i )^2 = 10(1 + i)$

Exercise 13.2 | Q 2.3 | Page 31

Find the real value of x and y, if

$\frac{(1 + i)x - 2i}{3 + i} + \frac{(2 - 3i)y + i}{3 - i}$

Exercise 13.2 | Q 2.4 | Page 31

Find the real value of x and y, if

$(1 + i)(x + iy) = 2 - 5i$

Exercise 13.2 | Q 3.1 | Page 31

Find the conjugate of the following complex number:

4 − 5 i

Exercise 13.2 | Q 3.2 | Page 31

Find the conjugate of the following complex number:

$\frac{1}{3 + 5i}$

Exercise 13.2 | Q 3.3 | Page 31

Find the conjugate of the following complex number:

$\frac{1}{1 + i}$

Exercise 13.2 | Q 3.4 | Page 31

Find the conjugate of the following complex number:

$\frac{(3 - i )^2}{2 + i}$

Exercise 13.2 | Q 3.5 | Page 31

Find the conjugate of the following complex number:

$\frac{(1 + i)(2 + i)}{3 + i}$

Exercise 13.2 | Q 3.6 | Page 31

Find the conjugate of the following complex number:

$\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}$

Exercise 13.2 | Q 4.1 | Page 32

Find the multiplicative inverse of the following complex number:

1 − i

Exercise 13.2 | Q 4.2 | Page 32

Find the multiplicative inverse of the following complex number:

$(1 + i\sqrt{3} )^2$

Exercise 13.2 | Q 4.3 | Page 32

Find the multiplicative inverse of the following complex number:

4 − 3i

Exercise 13.2 | Q 4.4 | Page 32

Find the multiplicative inverse of the following complex number:

$\sqrt{5} + 3i$

Exercise 13.2 | Q 5 | Page 32

If $z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|$

Exercise 13.2 | Q 6.1 | Page 32

If $z_1 = 2 - i, z_2 = - 2 + i,$ find

Re $\left( \frac{z_1 z_2}{z_1} \right)$

Exercise 13.2 | Q 6.2 | Page 32

If $z_1 = 2 - i, z_2 = - 2 + i,$ find

Im (1/(z_1overlinez_1))

Exercise 13.2 | Q 7 | Page 32

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

Exercise 13.2 | Q 8 | Page 32

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

Exercise 13.2 | Q 9 | Page 32

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

Exercise 13.2 | Q 10 | Page 32

Find the real values of θ for which the complex number $\frac{1 + i cos\theta}{1 - 2i cos\theta}$  is purely real.

Exercise 13.2 | Q 11 | Page 32

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

Exercise 13.2 | Q 12 | Page 32

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

Exercise 13.2 | Q 13 | Page 32

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

Exercise 13.2 | Q 14 | Page 32

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

Exercise 13.2 | Q 15 | Page 32

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

Exercise 13.2 | Q 16.1 | Page 32

Evaluate the following:

$2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}$

Exercise 13.2 | Q 16.2 | Page 32

Evaluate the following:

$x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i$

Exercise 13.2 | Q 16.3 | Page 32

Evaluate the following:

$x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}$

Exercise 13.2 | Q 16.4 | Page 32

Evaluate the following:

$x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}$

Exercise 13.2 | Q 16.5 | Page 32

Evaluate the following:

$2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i$

Exercise 13.2 | Q 17 | Page 32

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

Exercise 13.2 | Q 18 | Page 33

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

Exercise 13.2 | Q 19 | Page 33

Solve the system of equations $\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2$.

Exercise 13.2 | Q 20 | Page 33

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

Exercise 13.2 | Q 21 | Page 33

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.

Exercise 13.2 | Q 22 | Page 33

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

Exercise 13.2 | Q 23 | Page 33

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

Exercise 13.2 | Q 24 | Page 33

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

Exercise 13.2 | Q 25 | Page 33

If z1z2z3 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 $\left| z_1 + z_2 + z_3 \right|$ .

Exercise 13.2 | Q 26 | Page 33

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

Exercise 13.3 [Page 39]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers Exercise 13.3 [Page 39]

Exercise 13.3 | Q 1.1 | Page 39

Find the square root of the following complex number:

−5 + 12i

Exercise 13.3 | Q 1.2 | Page 39

Find the square root of the following complex number:

−7 − 24i

Exercise 13.3 | Q 1.3 | Page 39

Find the square root of the following complex number:

1 − i

Exercise 13.3 | Q 1.4 | Page 39

Find the square root of the following complex number:

−8 − 6i

Exercise 13.3 | Q 1.5 | Page 39

Find the square root of the following complex number:

8 −15i

Exercise 13.3 | Q 1.6 | Page 39

Find the square root of the following complex number:

$- 11 - 60\sqrt{- 1}$

Exercise 13.3 | Q 1.7 | Page 39

Find the square root of the following complex number:

$1 + 4\sqrt{- 3}$

Exercise 13.3 | Q 1.8 | Page 39

Find the square root of the following complex number:

4i

Exercise 13.3 | Q 1.9 | Page 39

Find the square root of the following complex number:

i

Exercise 13.4 [Pages 57 - 58]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers Exercise 13.4 [Pages 57 - 58]

Exercise 13.4 | Q 1.1 | Page 57

Find the modulus and argument of the following complex number and hence express in the polar form:

1 + i

Exercise 13.4 | Q 1.2 | Page 57

Find the modulus and argument of the following complex number and hence express in the polar form:

$\sqrt{3} + i$

Exercise 13.4 | Q 1.3 | Page 57

Find the modulus and argument of the following complex number and hence express in the polar form:

1 − i

Exercise 13.4 | Q 1.4 | Page 57

Find the modulus and argument of the following complex number and hence express in the polar form:

$\frac{1 - i}{1 + i}$

Exercise 13.4 | Q 1.5 | Page 57

Find the modulus and argument of the following complex number and hence express in the polar form:

$\frac{1}{1 + i}$

Exercise 13.4 | Q 1.6 | Page 57

Find the modulus and argument of the following complex number and hence express in the polar form:

$\frac{1 + 2i}{1 - 3i}$

Exercise 13.4 | Q 1.7 | Page 57

Find the modulus and argument of the following complex number and hence express in the polar form:

sin 120° - i cos 120°

Exercise 13.4 | Q 1.8 | Page 57

Find the modulus and argument of the following complex number and hence express in the polar form:

$\frac{- 16}{1 + i\sqrt{3}}$

Exercise 13.4 | Q 2 | Page 57

Write (i25)3 in polar form.

Exercise 13.4 | Q 3.1 | Page 57

Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α

Exercise 13.4 | Q 3.2 | Page 57

Express the following complex in the form r(cos θ + i sin θ):

tan α − i

Exercise 13.4 | Q 3.3 | Page 57

Express the following complex in the form r(cos θ + i sin θ):

1 − sin α + i cos α

Exercise 13.4 | Q 3.4 | Page 57

Express the following complex in the form r(cos θ + i sin θ):

$\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}$

Exercise 13.4 | Q 4 | Page 57

If z1 and z2 are two complex numbers such that $\left| z_1 \right| = \left| z_2 \right|$ and arg(z1) + arg(z2) = $\pi$ then show that $z_1 = - \bar{{z_2}}$.

Exercise 13.4 | Q 5 | Page 57

If z1z2 and z3z4 are two pairs of conjugate complex numbers, prove that $\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0$.

Exercise 13.4 | Q 6 | Page 58

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

[Pages 62 - 63]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers [Pages 62 - 63]

Q 1 | Page 62

Write the values of the square root of i.

Q 2 | Page 62

Write the values of the square root of −i.

Q 3 | Page 62

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

Q 4 | Page 62

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

Q 5 | Page 62

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

Q 6 | Page 62

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

Q 7 | Page 62

Write 1 − i in polar form.

Q 8 | Page 62

Write −1 + $\sqrt{3}$ in polar form .

Q 9 | Page 62

Write the argument of −i.

Q 10 | Page 62

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

Q 11 | Page 62

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

Q 12 | Page 62

Find z, if $\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .$

Q 13 | Page 62

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

Q 14 | Page 63

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

Q 15 | Page 63

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

Q 16 | Page 63

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

Q 17 | Page 63

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

Q 18 | Page 63

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

Q 19 | Page 63

For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of $\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2$.

Q 20 | Page 63

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

Q 21 | Page 63

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

Q 22 | Page 63

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

Q 23 | Page 63

If $\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}$,find z.

Q 24 | Page 63

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

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

[Pages 63 - 66]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers [Pages 63 - 66]

Q 1 | Page 63

The value of $(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )$ is.

• 2

• 0

• 1

• i

Q 2 | Page 63

If (3+2i sintheta)/(1-2 i sin theta)is a real number and 0 < θ < 2π, then θ =

• π

• pi/2

• pi/3

• pi/6

Q 3 | Page 63

If (1+i)(1 + 2i)(1+3i)..... (1+ ni) = a+ib,then 2 ×5 ×10 ×...... ×(1+n2) is equal to.

• sqrt(a^2 +b^2)

• sqrt(a^2 +b^2)

• sqrt(a^2 - b^2)

• a^2 +b^2

• a^2 -b^2

• a+b

Q 4 | Page 63

If$\sqrt{a + ib} = x + iy,$ then possible value of $\sqrt{a - ib}$ is

• $x^2 + y^2$

• $\sqrt{x^2 + y^2}$

• x + iy

• x − iy

• $\sqrt{x^2 - y^2}$

Q 5 | Page 64

If$z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}$, then

• $\left| z \right| = 1, \text { arg }(z) = \frac{\pi}{4}$

• $\left| z \right| = 1, \text { arg }(z) = \frac{\pi}{6}$

• $\left| z \right| = \frac{\sqrt{3}}{2},\text { arg }(z) = \frac{5\pi}{24}$

• $\left| z \right| = \frac{\sqrt{3}}{2}, \text { arg }(z) = \tan^{- 1} \frac{1}{\sqrt{2}}$

Q 6 | Page 64

The polar form of (i25)3 is

• $\cos\frac{\pi}{2} + i \sin\frac{\pi}{2}$

• cos π + i sin π

•  cos π − i sin π

• $\cos\frac{\pi}{2} - i \sin\frac{\pi}{2}$

Q 7 | Page 64

If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to

• 1

• −1

• i

• 0

Q 8 | Page 64

If $z = \frac{- 2}{1 + i\sqrt{3}}$,then the value of arg (z) is

• π

• $\frac{\pi}{3}$

• $\frac{2\pi}{3}$

• $\frac{\pi}{4}$

Q 9 | Page 64

If a = cos θ + i sin θ, then $\frac{1 + a}{1 - a} =$

• $\cot\frac{\theta}{2}$

• cot θ

• $i \cot\frac{\theta}{2}$

• $i \tan\frac{\theta}{2}$

Q 10 | Page 64

If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n2)=

• a − ib

• a2 − b2

• a2 + b2

• none of these

Q 11 | Page 64

If $\frac{( a^2 + 1 )^2}{2a - i} = x + iy, \text { then } x^2 + y^2$ is equal to

• $\frac{( a^2 + 1 )^4}{4 a^2 + 1}$

• $\frac{(a + 1 )^2}{4 a^2 + 1}$

• $\frac{( a^2 - 1 )^2}{(4 a^2 - 1 )^2}$

• none of these

Q 12 | Page 64

The principal value of the amplitude of (1 + i) is

• $\frac{\pi}{4}$

• $\frac{\pi}{12}$

• $\frac{3\pi}{4}$

• π

Q 13 | Page 64

The least positive integer n such that $\left( \frac{2i}{1 + i} \right)^n$ is a positive integer, is.

•  16

• 8

• 4

• 2

Q 14 | Page 64

If z is a non-zero complex number, then $\left| \frac{\left| z \right|^2}{zz} \right|$ is equal to

• |overlinez/z|

• $\left| z \right|$

• |overlinez|

• none of these

Q 15 | Page 64

If a = 1 + i, then a2 equals

• 1 − i

•  2i

•  (1 + i) (1 − i)

• i − 1.

Q 16 | Page 64

If (x + iy)1/3 = a + ib, then $\frac{x}{a} + \frac{y}{b} =$

• 0

• 1

• −1

• none of these

Q 17 | Page 64

$(\sqrt{- 2})(\sqrt{- 3})$ is equal to

• $\sqrt{6}$

• $- \sqrt{6}$

• $i\sqrt{6}$

• none of these.

Q 18 | Page 65

The argument of $\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}$ is

•  60°

• 120°

• 210°

• 240°

Q 19 | Page 65

If $z = \left( \frac{1 + i}{1 - i} \right)$ then z4 equals

•  1

• −1

• 0

• none of these

Q 20 | Page 65

If $z = \frac{1 + 2i}{1 - (1 - i )^2}$, then arg (z) equal

• 0

• $\frac{\pi}{2}$

• π

• none of these.

Q 21 | Page 65

$\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =$

• $\frac{1}{13}$

• $\frac{1}{5}$

• $\frac{1}{12}$

• none of these

Q 22 | Page 65

$\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =$

• 1

• $1/\sqrt{26}$

• $5/\sqrt{26}$

• none of these

Q 23 | Page 65

$\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =$

• $2 \sin\frac{\theta}{2}$

• $2 \cos\frac{\theta}{2}$

• $2\left| \sin\frac{\theta}{2} \right|$

• $2\left| \cos\frac{\theta}{2} \right|$

Q 24 | Page 65

If $x + iy = (1 + i)(1 + 2i)(1 + 3i)$,then x2 + y2 =

• 0

• 1

• 100

• none of these

Q 25 | Page 65

If $z = \frac{1}{1 - cos\theta - i sin\theta}$ then Re (z) =

• 0

• $\frac{1}{2}$

• $\cot\frac{\theta}{2}$

• $\frac{1}{2}\cot\frac{\theta}{2}$

Q 26 | Page 65

If $x + iy = \frac{3 + 5i}{7 - 6i},$  then y =

• 9/85

•  −9/85

•  53/85

• none of these

Q 27 | Page 65

If $\frac{1 - ix}{1 + ix} = a + ib$ then $a^2 + b^2$

• 1

• -1

• 0

• none of these

Q 28 | Page 65

If θ is the amplitude of $\frac{a + ib}{a - ib}$ , than tan θ =

• $\frac{2a}{a^2 + b^2}$

• $\frac{2ab}{a^2 - b^2}$

• $\frac{a^2 - b^2}{a^2 + b^2}$

• none of these

Q 29 | Page 65

If $z = \frac{1 + 7i}{(2 - i )^2}$ , then

• $\left| z \right| = 2$

• $\left| z \right| = \frac{1}{2}$

• amp (z) = $\frac{\pi}{4}$

•  amp (z) = $\frac{3\pi}{4}$

Q 30 | Page 65

The amplitude of $\frac{1}{i}$ is equal to

• 0

• $\frac{\pi}{2}$

• $- \frac{\pi}{2}$

•  π

Q 31 | Page 66

The argument of $\frac{1 - i}{1 + i}$ is

• $- \frac{\pi}{2}$

• $\frac{\pi}{2}$

• $\frac{3\pi}{2}$

• $\frac{5\pi}{2}$

Q 32 | Page 66

The amplitude of $\frac{1 + i\sqrt{3}}{\sqrt{3} + i}$ is

• $\frac{\pi}{3}$

• $- \frac{\pi}{3}$

• $\frac{\pi}{6}$

• $- \frac{\pi}{6}$

Q 33 | Page 66

The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is

• $\frac{1}{2}(1 + i)$

• $\frac{1}{2}(1 - i)$

• 1

• $\frac{1}{2}$

Q 34 | Page 66

$\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}$ equals

• i

• -1

• $-$i

• 4

Q 35 | Page 66

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

• -1

• -2

• -3

• -4

Q 36 | Page 66

The value of $(1 + i )^4 + (1 - i )^4$ is

• 8

• 4

• -8

• -4

Q 37 | Page 66

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

• $a > b > 0$

• $a < b < 0$

• $b < a < 0$

• $b > a > 0$

Q 38 | Page 66

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

• $\frac{\left| z \right|}{2}$

• $\left| z \right|$

• $2\left| z \right|$

• none of these

Q 39 | Page 66

A real value of x satisfies the equation  $\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =$

• 1

• -1

• 2

• -2

Q 40 | Page 66

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

• circle x2 + y2 = 1

• the x−axis

• the y−axis

• the line x + y = 1

Q 41 | Page 66

If z is a complex numberthen

• $\left| z \right|^2 > \left| z \right|^2$

• $\left| z \right|^2 = \left| z \right|^2$

• $\left| z \right|^2 < \left| z \right|^2$

• $\left| z \right|^2 \geq \left| z \right|^2$

Q 42 | Page 66

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

• $\left| z_1 z_2 \right| = \left| z_1 \right|\left| z_2 \right|$

• $\arg\left( z_1 z_2 \right) = \arg\left( z_1 \right) \arg\left( z_2 \right)$

• $\left| z_1 + z_2 \right| = \left| z_1 \right| + \left| z_2 \right|$

• $\left| z_1 + z_2 \right| \geq \left| z_1 \right| + \left| z_2 \right|$

Q 43 | Page 66

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

• x−axis

• circle with centre (−1, 0) and radius 1

• y−axis

• none of these

Chapter 13: Complex Numbers

Exercise 13.1Exercise 13.2Exercise 13.3Exercise 13.4Others

RD Sharma solutions for Class 11 Mathematics Textbook chapter 13 - Complex Numbers

RD Sharma solutions for Class 11 Mathematics Textbook chapter 13 (Complex Numbers) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics Textbook chapter 13 Complex Numbers are Argand Plane and Polar Representation, Quadratic Equations, Algebra of Complex Numbers - Equality, Algebraic Properties of Complex Numbers, Need for Complex Numbers, Square Root of a Complex Number, Algebra of Complex Numbers, The Modulus and the Conjugate of a Complex Number, Concept of Complex Numbers.

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