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

Find the multiplicative inverse of the following complex number:

1 − i

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that \[\left( A \cap B \right)' = A' \cup B'\] 

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))`

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 \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].