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Chapters
Chapter 2: Relations
Chapter 3: Functions
Chapter 4: Measurement of Angles
Chapter 5: Trigonometric Functions
Chapter 6: Graphs of Trigonometric Functions
Chapter 7: Values of Trigonometric function at sum or difference of angles
Chapter 8: Transformation formulae
Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle
Chapter 10: Sine and cosine formulae and their applications
Chapter 11: Trigonometric equations
Chapter 12: Mathematical Induction
Chapter 13: Complex Numbers
Chapter 14: Quadratic Equations
Chapter 15: Linear Inequations
Chapter 16: Permutations
Chapter 17: Combinations
Chapter 18: Binomial Theorem
Chapter 19: Arithmetic Progression
Chapter 20: Geometric Progression
Chapter 21: Some special series
Chapter 22: Brief review of cartesian system of rectangular co-ordinates
Chapter 23: The straight lines
Chapter 24: The circle
Chapter 25: Parabola
Chapter 26: Ellipse
Chapter 27: Hyperbola
Chapter 28: Introduction to three dimensional coordinate geometry
Chapter 29: Limits
Chapter 30: Derivatives
Chapter 31: Mathematical reasoning
Chapter 32: Statistics
Chapter 33: Probability

Chapter 13: Complex Numbers
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers Exercise 13.1 [Pages 3 - 13]
Evaluate the following:
i457
Evaluate the following:
(ii) i528
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\]
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
i30 + i80 + i120
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
i5 + i10 + i15
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Find the value of the following expression:
(1 + i)6 + (1 − i)3
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers Exercise 13.2 [Pages 31 - 33]
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{1 - 4i}{5 + i} \right)\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the real value of x and y, if
\[\frac{(1 + i)x - 2i}{3 + i} + \frac{(2 - 3i)y + i}{3 - i}\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the conjugate of the following complex number:
4 − 5 i
Find the conjugate of the following complex number:
\[\frac{1}{3 + 5i}\]
Find the conjugate of the following complex number:
\[\frac{1}{1 + i}\]
Find the conjugate of the following complex number:
\[\frac{(3 - i )^2}{2 + i}\]
Find the conjugate of the following complex number:
\[\frac{(1 + i)(2 + i)}{3 + i}\]
Find the conjugate of the following complex number:
\[\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}\]
Find the multiplicative inverse of the following complex number:
1 − i
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Find the multiplicative inverse of the following complex number:
4 − 3i
Find the multiplicative inverse of the following complex number:
\[\sqrt{5} + 3i\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
Find the modulus of \[\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}\].
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
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.
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Solve the equation \[\left| z \right| = z + 1 + 2i\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
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 \[\left| z_1 + z_2 + z_3 \right|\] .
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers Exercise 13.3 [Page 39]
Find the square root of the following complex number:
−5 + 12i
Find the square root of the following complex number:
−7 − 24i
Find the square root of the following complex number:
1 − i
Find the square root of the following complex number:
−8 − 6i
Find the square root of the following complex number:
8 −15i
Find the square root of the following complex number:
\[- 11 - 60\sqrt{- 1}\]
Find the square root of the following complex number:
\[1 + 4\sqrt{- 3}\]
Find the square root of the following complex number:
4i
Find the square root of the following complex number:
−i
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers Exercise 13.4 [Pages 57 - 58]
Find the modulus and argument of the following complex number and hence express in the polar form:
1 + i
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\sqrt{3} + i\]
Find the modulus and argument of the following complex number and hence express in the polar form:
1 − i
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 - i}{1 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1}{1 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 + 2i}{1 - 3i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
sin 120° - i cos 120°
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{- 16}{1 + i\sqrt{3}}\]
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
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 \[z_1 = - \bar{{z_2}}\].
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, prove that \[\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0\].
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers [Pages 62 - 63]
Write the values of the square root of i.
Write the values of the square root of −i.
If x + iy =\[\sqrt{\frac{a + ib}{c + id}}\] then write the value of (x2 + y2)2.
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write 1 − i in polar form.
Write −1 + i \[\sqrt{3}\] in polar form .
Write the argument of −i.
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
Write the conjugate of \[\frac{2 - i}{\left( 1 - 2i \right)^2}\] .
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
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 \[\left( 1 + \sqrt{3} \right)\].
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 13 Complex Numbers [Pages 63 - 66]
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
2
0
1
i
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`
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
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
\[\sqrt{x^2 - y^2}\]
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
\[\left| z \right| = 1, \text { arg }(z) = \frac{\pi}{4}\]
\[\left| z \right| = 1, \text { arg }(z) = \frac{\pi}{6}\]
\[\left| z \right| = \frac{\sqrt{3}}{2},\text { arg }(z) = \frac{5\pi}{24}\]
\[\left| z \right| = \frac{\sqrt{3}}{2}, \text { arg }(z) = \tan^{- 1} \frac{1}{\sqrt{2}}\]
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
\[\cos\frac{\pi}{2} - i \sin\frac{\pi}{2}\]
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
1
−1
i
0
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
π
\[\frac{\pi}{3}\]
\[\frac{2\pi}{3}\]
\[\frac{\pi}{4}\]
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
\[\cot\frac{\theta}{2}\]
cot θ
\[i \cot\frac{\theta}{2}\]
\[i \tan\frac{\theta}{2}\]
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
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
The principal value of the amplitude of (1 + i) is
\[\frac{\pi}{4}\]
\[\frac{\pi}{12}\]
\[\frac{3\pi}{4}\]
π
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
16
8
4
2
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
If a = 1 + i, then a2 equals
1 − i
2i
(1 + i) (1 − i)
i − 1.
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
0
1
−1
none of these
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
\[\sqrt{6}\]
\[- \sqrt{6}\]
\[i\sqrt{6}\]
none of these.
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
60°
120°
210°
240°
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
1
−1
0
none of these
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
0
\[\frac{\pi}{2}\]
π
none of these.
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
\[\frac{1}{13}\]
\[\frac{1}{5}\]
\[\frac{1}{12}\]
none of these
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
1
\[1/\sqrt{26}\]
\[5/\sqrt{26}\]
none of these
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
\[2 \sin\frac{\theta}{2}\]
\[2 \cos\frac{\theta}{2}\]
\[2\left| \sin\frac{\theta}{2} \right|\]
\[2\left| \cos\frac{\theta}{2} \right|\]
If \[x + iy = (1 + i)(1 + 2i)(1 + 3i)\],then x2 + y2 =
0
1
100
none of these
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
0
\[\frac{1}{2}\]
\[\cot\frac{\theta}{2}\]
\[\frac{1}{2}\cot\frac{\theta}{2}\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
9/85
−9/85
53/85
none of these
If \[\frac{1 - ix}{1 + ix} = a + ib\] then \[a^2 + b^2\]
1
-1
0
none of these
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
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) = \[\frac{3\pi}{4}\]
The amplitude of \[\frac{1}{i}\] is equal to
0
\[\frac{\pi}{2}\]
\[- \frac{\pi}{2}\]
π
The argument of \[\frac{1 - i}{1 + i}\] is
\[- \frac{\pi}{2}\]
\[\frac{\pi}{2}\]
\[\frac{3\pi}{2}\]
\[\frac{5\pi}{2}\]
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
\[\frac{\pi}{3}\]
\[- \frac{\pi}{3}\]
\[\frac{\pi}{6}\]
\[- \frac{\pi}{6}\]
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
\[\frac{1}{2}(1 + i)\]
\[\frac{1}{2}(1 - i)\]
1
\[\frac{1}{2}\]
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
i
-1
\[-\]i
4
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
The value of \[(1 + i )^4 + (1 - i )^4\] is
8
4
-8
-4
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
\[a > b > 0\]
\[a < b < 0\]
\[b < a < 0\]
\[b > a > 0\]
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
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
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
If z is a complex number, then
\[\left| z \right|^2 > \left| z \right|^2\]
\[\left| z \right|^2 = \left| z \right|^2\]
\[\left| z \right|^2 < \left| z \right|^2\]
\[\left| z \right|^2 \geq \left| z \right|^2\]
Which of the following is correct for any two complex numbers z1 and z2?
\[\left| z_1 z_2 \right| = \left| z_1 \right|\left| z_2 \right|\]
\[\arg\left( z_1 z_2 \right) = \arg\left( z_1 \right) \arg\left( z_2 \right)\]
\[\left| z_1 + z_2 \right| = \left| z_1 \right| + \left| z_2 \right|\]
\[\left| z_1 + z_2 \right| \geq \left| z_1 \right| + \left| z_2 \right|\]
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

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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