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

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?

{1, 2, 3, 4, 5, 6, 7, 8}

(i) If \[\left( \frac{a}{3} + 1, b - \frac{2}{3} \right) = \left( \frac{5}{3}, \frac{1}{3} \right)\] find the values of a and b. 

(ii) If (x + 1, 1) = (3, y − 2), find the values of x and y.

Prove the following identites

sec⁴x - sec²x = tan⁴x + tan²x

Prove the following identities
\[\sin^6 x + \cos^6 x = 1 - 3 \sin^2 x \cos^2 x\]

Prove the following identities
\[\left( cosec x - \sin x \right) \left( \sec x - \cos x \right) \left( \tan x + \cot x \right) = 1\]

Prove the following identities 
\[cosec x \left( \sec x - 1 \right) - \cot x \left( 1 - \cos x \right) = \tan x - \sin x\]

Prove the following identities
\[\frac{1 - \sin x \cos x}{\cos x \left( \sec x - cosec x \right)} \cdot \frac{\sin^2 x - \cos^2 x}{\sin^3 x + \cos^3 x} = \sin x\]

Prove the following identitie

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