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

Mathematics Class 11

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Chapters

RD Sharma Mathematics Class 11

Mathematics Class 11

Chapter 13: Complex Numbers

Ex. 13.10Ex. 13.20Ex. 13.30Ex. 13.40Others

Chapter 13: Complex Numbers Exercise 13.10 solutions [Pages 3 - 13]

Ex. 13.10 | Q 1.1 | Page 3

Evaluate the following:

i457

Ex. 13.10 | Q 1.2 | Page 3

Evaluate the following:

(ii) i528

Ex. 13.10 | Q 1.3 | Page 3

Evaluate the following:

 \[\frac{1}{i^{58}}\]

Ex. 13.10 | Q 1.4 | Page 3

Evaluate the following:

\[i^{37} + \frac{1}{i^{67}}\]

Ex. 13.10 | Q 1.5 | Page 3

Evaluate the following:

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

Ex. 13.10 | Q 1.6 | Page 3

Evaluate the following:

\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]

Ex. 13.10 | Q 1.7 | Page 13

Evaluate the following:

 \[i^{30} + i^{40} + i^{60}\]

Ex. 13.10 | Q 1.8 | Page 3

Evaluate the following:

\[i^{49} + i^{68} + i^{89} + i^{110}\]

Ex. 13.10 | Q 2 | Page 4

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

Ex. 13.10 | Q 3.1 | Page 4

Find the value of the following expression:

i49 + i68 + i89 + i110

Ex. 13.10 | Q 3.2 | Page 4

Find the value of the following expression:

i30 + i80 + i120

Ex. 13.10 | Q 3.3 | Page 4

Find the value of the following expression:

i + i2 + i3 + i4

Ex. 13.10 | Q 3.4 | Page 4

Find the value of the following expression:

i5 + i10 + i15

Ex. 13.10 | Q 3.5 | Page 4

Find the value of the following expression:

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

Ex. 13.10 | Q 3.6 | Page 4

Find the value of the following expression:

1+ i2 + i4 + i6 + i8 + ... + i20

Ex. 13.10 | Q 3.7 | Page 4

Find the value of the following expression:

(1 + i)6 + (1 − i)3

Chapter 13: Complex Numbers Exercise 13.20 solutions [Pages 31 - 33]

Ex. 13.20 | Q 1.01 | Page 31

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

\[(1 + i)(1 + 2i)\]

Ex. 13.20 | Q 1.02 | Page 31

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

\[\frac{3 + 2i}{- 2 + i}\]

Ex. 13.20 | Q 1.03 | Page 31

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

\[\frac{1}{(2 + i )^2}\]

Ex. 13.20 | Q 1.04 | Page 31

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

\[\frac{1 - i}{1 + i}\]

Ex. 13.20 | Q 1.05 | Page 31

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

\[\frac{(2 + i )^3}{2 + 3i}\]

Ex. 13.20 | Q 1.06 | Page 31

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

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

Ex. 13.20 | Q 1.07 | Page 31

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

\[\frac{2 + 3i}{4 + 5i}\]

Ex. 13.20 | Q 1.08 | Page 31

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

\[\frac{(1 - i )^3}{1 - i^3}\]

Ex. 13.20 | Q 1.09 | Page 31

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

\[(1 + 2i )^{- 3}\]

Ex. 13.20 | Q 1.1 | Page 31

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

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

Ex. 13.20 | Q 1.11 | Page 31

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

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

 

Ex. 13.20 | Q 1.12 | Page 31

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

\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]

Ex. 13.20 | Q 2.1 | Page 31

Find the real value of x and y, if

\[(x + iy)(2 - 3i) = 4 + i\]

Ex. 13.20 | Q 2.2 | Page 31

Find the real value of x and y, if

\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]

Ex. 13.20 | Q 2.3 | Page 31

Find the real value of x and y, if

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

Ex. 13.20 | Q 2.4 | Page 31

Find the real value of x and y, if

\[(1 + i)(x + iy) = 2 - 5i\]

Ex. 13.20 | Q 3.1 | Page 31

Find the conjugate of the following complex number:

4 − 5 i

Ex. 13.20 | Q 3.2 | Page 31

Find the conjugate of the following complex number:

\[\frac{1}{3 + 5i}\]

Ex. 13.20 | Q 3.3 | Page 31

Find the conjugate of the following complex number:

\[\frac{1}{1 + i}\]

Ex. 13.20 | Q 3.4 | Page 31

Find the conjugate of the following complex number:

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

Ex. 13.20 | Q 3.5 | Page 31

Find the conjugate of the following complex number:

\[\frac{(1 + i)(2 + i)}{3 + i}\]

Ex. 13.20 | Q 3.6 | Page 31

Find the conjugate of the following complex number:

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

Ex. 13.20 | Q 4.1 | Page 32

Find the multiplicative inverse of the following complex number:

1 − i

Ex. 13.20 | Q 4.2 | Page 32

Find the multiplicative inverse of the following complex number:

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

Ex. 13.20 | Q 4.3 | Page 32

Find the multiplicative inverse of the following complex number:

 4 − 3i

Ex. 13.20 | Q 4.4 | Page 32

Find the multiplicative inverse of the following complex number:

\[\sqrt{5} + 3i\]

Ex. 13.20 | Q 5 | Page 32

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

Ex. 13.20 | Q 6.1 | Page 32

If \[z_1 = 2 - i, z_2 = - 2 + i,\] find 

Re \[\left( \frac{z_1 z_2}{z_1} \right)\]

Ex. 13.20 | Q 6.2 | Page 32

If \[z_1 = 2 - i, z_2 = - 2 + i,\] find 

Im `(1/(z_1overlinez_1))`

Ex. 13.20 | Q 7 | Page 32

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

Ex. 13.20 | Q 8 | Page 32

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

Ex. 13.20 | Q 9 | Page 32

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

Ex. 13.20 | 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.

Ex. 13.20 | 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.

 
Ex. 13.20 | 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).

Ex. 13.20 | Q 13 | Page 32

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

Ex. 13.20 | Q 14 | Page 32

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

Ex. 13.20 | Q 15 | Page 32

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

Ex. 13.20 | Q 16.1 | Page 32

Evaluate the following:

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

Ex. 13.20 | Q 16.2 | Page 32

Evaluate the following:

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

Ex. 13.20 | Q 16.3 | Page 32

Evaluate the following:

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

Ex. 13.20 | Q 16.4 | Page 32

Evaluate the following:

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

Ex. 13.20 | Q 16.5 | Page 32

Evaluate the following:

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

Ex. 13.20 | Q 17 | Page 32

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

Ex. 13.20 | Q 18 | Page 33

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

Ex. 13.20 | Q 19 | Page 33

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

Ex. 13.20 | Q 20 | Page 33

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

Ex. 13.20 | 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.

Ex. 13.20 | Q 22 | Page 33

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

Ex. 13.20 | Q 23 | Page 33

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

Ex. 13.20 | Q 24 | Page 33

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

Ex. 13.20 | 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 \[\left| z_1 + z_2 + z_3 \right|\] .

Ex. 13.20 | Q 26 | Page 33

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

Chapter 13: Complex Numbers Exercise 13.30 solutions [Page 39]

Ex. 13.30 | Q 1.1 | Page 39

Find the square root of the following complex number:

−5 + 12i

Ex. 13.30 | Q 1.2 | Page 39

Find the square root of the following complex number:

−7 − 24i

Ex. 13.30 | Q 1.3 | Page 39

Find the square root of the following complex number:

1 − i

Ex. 13.30 | Q 1.4 | Page 39

Find the square root of the following complex number:

 −8 − 6i

Ex. 13.30 | Q 1.5 | Page 39

Find the square root of the following complex number:

8 −15i

Ex. 13.30 | Q 1.6 | Page 39

Find the square root of the following complex number:

\[- 11 - 60\sqrt{- 1}\]

Ex. 13.30 | Q 1.7 | Page 39

Find the square root of the following complex number:

 \[1 + 4\sqrt{- 3}\]

Ex. 13.30 | Q 1.8 | Page 39

Find the square root of the following complex number:

 4i

Ex. 13.30 | Q 1.9 | Page 39

Find the square root of the following complex number:

i

Chapter 13: Complex Numbers Exercise 13.40 solutions [Pages 57 - 58]

Ex. 13.40 | Q 1.1 | Page 57

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

1 + i

Ex. 13.40 | Q 1.2 | Page 57

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

\[\sqrt{3} + i\]

Ex. 13.40 | Q 1.3 | Page 57

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

1 − i

Ex. 13.40 | Q 1.4 | Page 57

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

\[\frac{1 - i}{1 + i}\]

Ex. 13.40 | Q 1.5 | Page 57

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

\[\frac{1}{1 + i}\]

Ex. 13.40 | Q 1.6 | Page 57

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

\[\frac{1 + 2i}{1 - 3i}\]

Ex. 13.40 | 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° 

Ex. 13.40 | Q 1.8 | Page 57

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

 \[\frac{- 16}{1 + i\sqrt{3}}\]

Ex. 13.40 | Q 2 | Page 57

Write (i25)3 in polar form.

Ex. 13.40 | Q 3.1 | Page 57

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

Ex. 13.40 | Q 3.2 | Page 57

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

 tan α − i

Ex. 13.40 | Q 3.3 | Page 57

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

1 − sin α + i cos α

Ex. 13.40 | Q 3.4 | Page 57

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

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

Ex. 13.40 | 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 \[z_1 = - \bar{{z_2}}\].

Ex. 13.40 | Q 5 | Page 57

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

Ex. 13.40 | Q 6 | Page 58

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

Chapter 13: Complex Numbers solutions [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 \[\left( 1 + \sqrt{3} \right)\].

Chapter 13: Complex Numbers solutions [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

Ex. 13.10Ex. 13.20Ex. 13.30Ex. 13.40Others

RD Sharma Mathematics Class 11

Mathematics Class 11

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

RD Sharma solutions for Class 11 Maths 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 Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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

Using RD Sharma Class 11 solutions Complex Numbers exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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