PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions

Prove the following identities
\[\frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} + \frac{\sin^3 x - \cos^3 x}{\sin x - \cos x} = 2\]

Prove the following identities
\[\left( \sec x \sec y + \tan x \tan y \right)^2 - \left( \sec x \tan y + \tan x \sec y \right)^2 = 1\]

Prove the following identities
\[\frac{\cos x}{1 - \sin x} = \frac{1 + \cos x + \sin x}{1 + \cos x - \sin x}\]

Prove the following identities

Prove the following identities
\[1 - \frac{\sin^2 x}{1 + \cot x} - \frac{\cos^2 x}{1 + \tan x} = \sin x \cos x\]

Prove the following identities

Prove the following identities
\[\left( 1 + \tan \alpha \tan \beta \right)^2 + \left( \tan \alpha - \tan \beta \right)^2 = \sec^2 \alpha \sec^2 \beta\]

Prove the following identities:

Prove the following identities 

Prove the following identities

If \[X = \left\{ 8^n - 7n - 1: n \in N \right\} \text{ and } Y = \left\{ 49\left( n - 1 \right): n \in N \right\}\] \[X \subseteq Y .\]

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that: 

\[\left( A \cup B \right)' = A' \cap B'\] 

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:

\[\left( A \cap B \right)' = A'B' .\]

For any two sets A and B, prove that 

 B ⊂ A ∪ B         

For any two sets A and B, prove that 

A ∩ B ⊂ A             

For any two sets A and B, prove that A ⊂ B ⇒ A ∩ B = A 

For any two sets A and B, show that the following statements are equivalent:

(i) \[A \subset B\] 

(ii) \[A \subset B\]=ϕ 

(iii) \[A \cup B = B\]

(iv) \[A \cap B = A .\] 

For three sets A, B and C, show that \[A \cap B = A \cap C\]

For three sets A, B and C, show that \[A \subset B \Rightarrow C - B \subset C - A\]