PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions

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 `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`

Find the real value of x and y, if

\[(1 + i)(x + iy) = 2 - 5i\]

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

Find \[\left( A \cap C \right)'\]

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.