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Ques. 1JEE Main 2024, 1 February (Morning Shift)
Let \x\ denote the fractional part of x and f(x)= -1 (1-\x\2 ) -1(1-\x\)\x\-\x\3, x 0. If L and R respectively denotes the left hand limit and the right hand limit of f(x) at x=0, then 32 2 (L2+R2 ) is equal to .
Answer
18
Solution
R.H.L. (x=h 0+, \x\=h):
R= h 0 -1(1-h2)h -1(1-h)1-h2
Put -1(1-h2)= h2=1- =2 2 2 h= 2 2
h= 2 ( /2) 2 R= 2 2= 2
L.H.L. (x=-h, \x\=1-h): 1-\x\2=2h-h2, 1-\x\=h
L= h 0 -1(2h-h2) -1h(1-h)(2h-h2)= 2 12= 4
∴ 32 2 ( 216+ 22 )=2+16=18
Answer: 18