JEE Main 2026 22 January Evening Shift
Exam JEE Main
Year 2026
Date 22 January
Shift Evening Shift
Subject Mathematics
Chapter Limits JEE Main 2026 · 22 January · Evening Shift · Mathematics · Limits
If _ x arrow 0 e^ (a-1) x +2 ~b x+(c-2) e^ -x x x- _ e (1+x) =2 ,...
If
lim x → 0 e ( a − 1 ) x + 2 cos b x + ( c − 2 ) e − x x cos x − log e ( 1 + x ) = 2 \lim _{x \rightarrow 0} \frac{\mathrm{e}^{(\mathrm{a}-1) x}+2 \cos \mathrm{~b} x+(\mathrm{c}-2) \mathrm{e}^{-x}}{x \cos x-\log _{\mathrm{e}}(1+x)}=2 lim x → 0 x c o s x − l o g e ( 1 + x ) e ( a − 1 ) x + 2 c o s b x + ( c − 2 ) e − x = 2 , then
a 2 + b 2 + c 2 \mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2} a 2 + b 2 + c 2 is equal to :
Step-by-step solution
lim x → 0 ( 1 + ( a − 1 ) x + ( a − 1 ) 2 x 2 2 ! ) + 2 ( 1 − b 2 x 2 2 ! ) + ( c − 2 ) ( 1 − x + x 2 2 ! ) x ( 1 − x 2 2 ! ) − ( x − x 2 2 … ) = 2 \lim _{x \rightarrow 0} \frac{\left(1+(a-1) x+\frac{(a-1)^2 x^2}{2!}\right)+2\left(1-\frac{b^2 x^2}{2!}\right)+(c-2)\left(1-x+\frac{x^2}{2!}\right)}{x\left(1-\frac{x^2}{2!}\right)-\left(x-\frac{x^2}{2} \ldots\right)}=2 lim x → 0 x ( 1 − 2 ! x 2 ) − ( x − 2 x 2 … ) ( 1 + ( a − 1 ) x + 2 ! ( a − 1 ) 2 x 2 ) + 2 ( 1 − 2 ! b 2 x 2 ) + ( c − 2 ) ( 1 − x + 2 ! x 2 ) = 2 lim x → 0 ( 1 + 2 + c − 2 ) + x ( a − 1 − c + 2 ) + x 2 ( ( a − 1 ) 2 2 − b 2 + ( c − 2 2 ) ) x 2 2 − x 3 2 ! + … = 2 \lim _{x \rightarrow 0} \frac{(1+2+c-2)+x(a-1-c+2)+x^2\left(\frac{(a-1)^2}{2}-b^2+\left(\frac{c-2}{2}\right)\right)}{\frac{x^2}{2}-\frac{x^3}{2!}+\ldots}=2 lim x → 0 2 x 2 − 2 ! x 3 + … ( 1 + 2 + c − 2 ) + x ( a − 1 − c + 2 ) + x 2 ( 2 ( a − 1 ) 2 − b 2 + ( 2 c − 2 ) ) = 2 For which
∵ c + 1 = 0 ⇒ c = − 1 \because c+1=0 \Rightarrow c=-1 ∵ c + 1 = 0 ⇒ c = − 1 ∵ a − c = − 1 ⇒ a = − 2 \because a-c=-1 \Rightarrow a=-2 ∵ a − c = − 1 ⇒ a = − 2 ∵ ( a − 1 ) 2 2 − b 2 + ( c − 2 2 ) = 1 \because \frac{(a-1)^2}{2}-b^2+\left(\frac{c-2}{2}\right)=1 ∵ 2 ( a − 1 ) 2 − b 2 + ( 2 c − 2 ) = 1 9 2 − b 2 − 3 2 = 1 ⇒ b 2 = 2 \frac{9}{2}-b^2-\frac{3}{2}=1 \Rightarrow b^2=2 2 9 − b 2 − 2 3 = 1 ⇒ b 2 = 2 a 2 + b 2 + c 2 = 4 + 2 + 1 = 7 a^2+b^2+c^2=4+2+1=7 a 2 + b 2 + c 2 = 4 + 2 + 1 = 7
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