Commerce (English Medium) इयत्ता ११ - CBSE Question Bank Solutions for Mathematics

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 \[A \subset B \Rightarrow C - B \subset C - A\] 

For any two sets, prove that: 

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

For any two sets, prove that: 

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

Find sets A, B and C such that \[A \cap B, A \cap C \text{ and } B \cap C\]are non-empty sets and\[A \cap B \cap C = \phi\]

For any two sets A and B, prove that: \[A \cap B = \phi \Rightarrow A \subseteq B'\] 

If A and B are sets, then prove that  \[A - B, A \cap B \text{ and } B - A\] are pair wise disjoint. 

Using properties of sets, show that for any two sets A and B,\[\left( A \cup B \right) \cap \left( A \cap B' \right) = A\]