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

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

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:

\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]

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:

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

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 \[z^2 + \left| z \right|^2 = 0\].

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

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.

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 \[\left( 1 + \sqrt{3} \right)\].

The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.

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

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

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

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

The polar form of (i²⁵)³ is

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

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

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

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

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

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

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

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

If a = 1 + i, then a² equals

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

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

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

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

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

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

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

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

If \[x + iy = (1 + i)(1 + 2i)(1 + 3i)\],then x² + y² =

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

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

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

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

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

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

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

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

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

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

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 

The value of \[(1 + i )^4 + (1 - i )^4\] is

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

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

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

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

If z is a complex number, then

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

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