Maths-
General
Easy

Question

Let f : [-10, 10] →R, where f(x) = sin x + open square brackets fraction numerator x to the power of 2 end exponent over denominator a end fraction close square brackets be an odd function. Then set of values of parameter a is/are

  1. (-10, 10) – {0}    
  2. (0, 10)    
  3. [100, ¥)    
  4. (100, ¥)    

The correct answer is: (100, ¥)


    f colon open square brackets negative 10 comma 10 close square brackets rightwards arrow R
f open parentheses x close parentheses equals sin x plus open parentheses x squared over a close parentheses
I t space i s space g i v e n space t h a t space f open parentheses x close parentheses space i s space a n space o d d space f u n c t i o n. space therefore f open parentheses negative x close parentheses equals negative f open parentheses x close parentheses space space space space space....... open parentheses 1 close parentheses
f open parentheses negative x close parentheses equals sin open parentheses negative x close parentheses plus open parentheses x squared over a close parentheses
f open parentheses negative x close parentheses equals negative sin x plus open parentheses x squared over a close parentheses space space space space space space space space...... open parentheses 2 close parentheses
F r o m space e q u a t i o n space open parentheses 1 close parentheses space a n d space open parentheses 2 close parentheses comma
minus open square brackets sin x plus open parentheses x squared over a close parentheses close square brackets equals negative sin x plus open parentheses x squared over a close parentheses
minus sin x minus open parentheses x squared over a close parentheses equals negative sin x plus open parentheses x squared over a close parentheses
minus open parentheses x squared over a close parentheses equals open parentheses x squared over a close parentheses
open parentheses fraction numerator 2 x squared over denominator a end fraction close parentheses equals 0
x squared over a equals 0
therefore 0 less or equal than x squared less than a

W e space k n o w space t h a t comma
x element of open square brackets negative 10 comma 10 close square brackets
therefore x squared element of open square brackets 0 comma 100 close square brackets
0 less or equal than x squared less than 100
0 less or equal than x squared less than a
therefore a greater than 100
therefore a element of open parentheses 100 comma infinity close parentheses

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