PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

Write down all possible subsets of each of the following set:

{1, {1}}, 

Find the value of the following expression:

i⁴⁹ + i⁶⁸ + i⁸⁹ + i¹¹⁰

Write down all possible proper subsets each of the following set:

{1, 2},

Find the value of the following expression:

i³⁰ + i⁸⁰ + i¹²⁰

Write down all possible proper subsets each of the following set:  

{1, 2, 3}

Find the value of the following expression:

i + i² + i³ + i⁴

Write down all possible proper subsets each of the following set: 

{1}. 

Find the value of the following expression:

i⁵ + i¹⁰ + i¹⁵

What is the total number of proper subsets of a set consisting of n elements? 

Find the value of the following expression:

\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]

If A is any set, prove that: \[A \subseteq \phi \Leftrightarrow A = \phi .\] 

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:

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