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

If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in

If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =

If \[\tan\alpha = \frac{x}{x + 1}\] and 

State whether the following pairs of sets are disjoint.

{1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6}

State whether the following pairs of sets are disjoint.

{a, e, i, o, u} and {c, d, e, f}

State whether the following pairs of sets are disjoint.

{x : x is an even integer} and {x : x is an odd integer}

State whether the following statement is true or false. Justify your answer.

{2, 3, 4, 5} and {3, 6} are disjoint sets.

State whether the following statement is true or false. Justify your answer.

{a, e, i, o, u } and {a, b, c, d} are disjoint sets.

State whether the following statement is true or false. Justify your answer.

{2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.

State whether the following statement is true or false. Justify your answer.

{2, 6, 10} and {3, 7, 11} are disjoint sets.

\[\cap\] If A and B are two disjoint sets, then \[n \left( A \cup B \right)\]is equal to 

Prove that:  \[\sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}} = \tan x\]

Prove that:  \[\frac{\sin 2x}{1 - \cos 2x} = cot x\]

Prove that:  \[\frac{\sin 2x}{1 + \cos 2x} = \tan x\]

Prove that: \[\sqrt{2 + \sqrt{2 + 2 \cos 4x}} = 2 \text{ cos } x\]

Prove that:  \[\frac{1 - \cos 2x + \sin 2x}{1 + \cos 2x + \sin 2x} = \tan x\]

Prove that:  \[\frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} = \tan x\]

Prove that:  \[\frac{\cos 2 x}{1 + \sin 2 x} = \tan \left( \frac{\pi}{4} - x \right)\]

Prove that:  \[\frac{\cos x}{1 - \sin x} = \tan \left( \frac{\pi}{4} + \frac{x}{2} \right)\]

Prove that: \[\cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \frac{5\pi}{8} + \cos^2 \frac{7\pi}{8} = 2\]