PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

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

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

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

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.