PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions

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 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 \[z^2 + \left| z \right|^2 = 0\].