PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

Find the value of the following expression:

1+ i² + i⁴ + i⁶ + i⁸ + ... + i²⁰

Prove that: 

\[A \subseteq B, B \subseteq C \text{ and } C \subseteq A \Rightarrow A = C .\] 

Find the value of the following expression:

(1 + i)⁶ + (1 − i)³

Express the following complex number in the standard form a + i b:

\[\frac{3 + 2i}{- 2 + i}\]

Express the following complex number in the standard form a + i b:

\[\frac{1}{(2 + i )^2}\]

Express the following complex number in the standard form a + i b:

\[(1 + i)(1 + 2i)\]

Express the following complex number in the standard form a + i b:

\[\frac{1 - i}{1 + i}\]

Express the following complex number in the standard form a + ib:

\[\frac{(2 + i )^3}{2 + 3i}\]

Express the following complex number in the standard form a + i b:

\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .

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