Ques. 1JEE Main 2025, 8 April (Evening Shift)
Given below are two statements: In the light of the above statements, choose the correct answer from the options given below:
- Statement I is false but Statement II is true
- Both Statement I and Statement II are false
- Both Statement I and Statement II are true
- Statement I is true but Statement II is false
Answer
(3) Both Statement I and Statement II are true
Solution
Statement I: using the series expansions near x = 0,
-1x = x - x33 + x55 -
e 1+x1-x = 12 [ e(1+x) - e(1-x) ] = x + x33 + x55 +
-1x + e 1+x1-x - 2x = 2x55 + (terms in x7, x9, )
x 0 -1x + e 1+x1-x - 2xx5 = 25
so Statement I is true.
Statement II: the limit is of the 1 form, so
x 1x 21-x = e x 1(x - 1) 21-x = e-2 = 1e2
and Statement II is also true.
Answer: (3)