Verified Solution • 2025–26 Session moderate

QJEEM21Mac64e: JEE Main 2021 Mathematics - Class 12 JEEM21Mac64e Solution: Let a, b R, b 0, Define a function f ( x ) = { a sin ⁡ π 2 ( x…

Let a, b ∈\in R, b ∈\in 0, Define a function

f(x)={asin⁡π2(x−1),forx≤0tan⁡2x−sin⁡2xbx3,forx>0.

If f is continuous at x = 0, then 10 −- ab is equal to ________________.

Step-by-Step Verified Solution

f(x)={asin⁡π2(x−1),forx≤0tan⁡2x−sin⁡2xbx3,forx>0

For continuity at '0'

lim⁡x→0+f(x)=f(0)\mathop {\lim }\limits_{x \to {0^ + }} f(x) = f(0)

⇒lim⁡x→0+tan⁡2x−sin⁡2xbx3=−a\Rightarrow \mathop {\lim }\limits_{x \to {0^ + }} {{\tan 2x - \sin 2x} \over {b{x^3}}} = - a

⇒lim⁡x→0+8x33+8x33!bx3=−a\Rightarrow \mathop {\lim }\limits_{x \to {0^ + }} {{{{8{x^3}} \over 3} + {{8{x^3}} \over {3!}}} \over {b{x^3}}} = - a

⇒8(13+13!)=−ab\Rightarrow 8\left( {{1 \over 3} + {1 \over {3!}}} \right) = - ab

⇒4=−ab\Rightarrow 4 = - ab

⇒10−ab=14\Rightarrow 10 - ab = 14

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