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

Evaluate the following:

 \[i^{30} + i^{40} + i^{60}\]

Let \[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\]Which of the following are true? \[\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A\] 

Evaluate the following:

\[i^{49} + i^{68} + i^{89} + i^{110}\]

Let \[\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A\] Which of the following are true? \[\left\{ \left\{ 2 \right\}, \left\{ 1 \right\} \right\} \not\subset A\]

Let \[A = \left\{ \phi, \left\{ \phi \right\}, 1, \left\{ 1, \phi \right\}, 2 \right\}\] Which of the following are true?\[\left\{ \left\{ \phi \right\} \right\} \subset A\]

Show that 1 + i¹⁰ + i²⁰ + i³⁰ is a real number.

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

 {a} 

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

 {0, 1}, 

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

{a, b, c}, 

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: \[A \subseteq \phi \Leftrightarrow A = \phi .\]