Sets, Relations & Functions
Functions
nta_pyq_2025_jan
Grade 11
Question:
If f (x) = x x 2 ,x \in R , then \sum 81 k=1 f ( k 82 ) is equal to 2 +\sqrt2
Step-by-Step Solution
Key Concept: Apply the core result for domains, ranges and functional equations and simplify using the given constraints.
f (x) = 2 x (4) 2 + \sqrt2 x 1-x 2 2 f (x) + f (1 - x) = + x 1-x 2 + \sqrt2 2 + \sqrt2 x x 2 2 2 + \sqrt2 = + = = 1 x x x 2 + \sqrt2 2 + \sqrt22 2 + \sqrt2 81 k 1 2 81 Now, \sum f ( ) = f ( ) + f ( ) + \ldots\ldots + f ( ) 82 82 82 82 k=1 1 1 2 1 = f ( ) + f ( ) + \ldots \ldots + f (1 - ) + f (1 - ) 82 82 82 82 1 1 2 2 41 [f ( ) + f (1 - )] + [f ( ) + f (1 - )] + \ldots .40 cases + f ( ) 82 82 82 82 82 1/2 2 (1 + 1 + \ldots . +1)40 times + 1/2 1/2 2 + 2 1 81 40 + = 2 2
Correct Answer: 4