Science (English Medium) Class 11 - CBSE Question Bank Solutions

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 \[z_1 = - \bar{{z_2}}\].

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

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 \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].

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