Science (English Medium) Class 11 - CBSE Question Bank Solutions

Prove the:
\[ \sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}} = - \frac{2}{\cos x},\text{ where }\frac{\pi}{2} < x < \pi\]

If \[T_n = \sin^n x + \cos^n x\], prove that \[\frac{T_3 - T_5}{T_1} = \frac{T_5 - T_7}{T_3}\]

If \[T_n = \sin^n x + \cos^n x\], prove that  \[2 T_6 - 3 T_4 + 1 = 0\]

If \[T_n = \sin^n x + \cos^n x\], prove that \[6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0\]

Prove that:  tan 225° cot 405° + tan 765° cot 675° = 0

Prove that:

If a set contains n elements, then write the number of elements in its power set. 

Write the number of elements in the power set of null set. 

Let A = {x : x ∈ N, x is a multiple of 3} and B = {x : x ∈ N and x is a multiple of 5}. Write \[A \cap B\] 

Let A and B be two sets having 3 and 6 elements respectively. Write the minimum number of elements that \[A \cup B\] 

If A = {x ∈ C : x² = 1} and B = {x ∈ C : x⁴ = 1}, then write A − B and B − A. 

Prove that: cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = \[\frac{1}{2}\]

If A and B are two sets such that \[A \subset B\], then write B' − A' in terms of A and B.

Let A and B be two sets having 4 and 7 elements respectively. Then write the maximum number of elements that \[A \cup B\] can have. 

If \[A = \left\{ \left( x, y \right) : y = \frac{1}{x}, 0 \neq x \in R \right\}\]and\[B = \left\{ \left( x, y \right) : y = - x, x \in R \right\}\] then write\[A \cap B\]

If \[A = \left\{ \left( x, y \right) : y = e^x , x \in R \right\} and B = \left\{ \left( x, y \right) : y = e^{- x} , x \in R \right\}\]write\[A \cap B\] 

If A and B are two sets such that \[n \left( A \right) = 20, n \left( B \right) = 25\]\text{ and } \[n \left( A \cup B \right) = 40\], then write \[n \left( A \cap B \right)\] 

If A and B are two sets such that \[n \left( A \right) = 115, n \left( B \right) = 326, n \left( A - B \right) = 47,\] then write \[n \left( A \cup B \right)\] 

Prove that: tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0

The number of subsets of a set containing n elements is