Let a function ⨍ : R→R be defined as ⨍ x=sinx-exif x≤0a+-xif 0<x<12x-bif x≥1 where x is the greatest integer less than or equal to x if ⨍ is continuous on R then a+b is equal to :

fx=sin x-ex ; x≤0a+-x ; 0<x<12x-b ; x≥1 is continuous function⇒limx→0-fx=limx→0+fx ⇒limx→0- sin -e2=limx→0+a+-x⇒-1=a-1⇒a=0 limx→1-a+-x=limx→1+2x-b⇒a-1=2-b ∴a+b=3

Let a function ⨍ : R→R be defined as ⨍ x=sinx-exif x≤0a+-xif 0<x<12x-bif x≥1 where x is the greatest integer less than or equal to x if ⨍ is continuous on R then a+b is equal to :

fx=sin x-ex ; x≤0a+-x ; 0<x<12x-b ; x≥1 is continuous function⇒limx→0-fx=limx→0+fx ⇒limx→0- sin -e2=limx→0+a+-x⇒-1=a-1⇒a=0 limx→1-a+-x=limx→1+2x-b⇒a-1=2-b ∴a+b=3

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Let a function $⨍ : R→R$ be defined as
$⨍ (x) = \{\begin{matrix} \sin {x-e}^{x} & if x \leq 0 \\ a + [- x] & if 0 < x < 1 \\ 2 x - b & if x \geq 1 \end{matrix}$ where $[x]$ is the
greatest integer less than or equal to $x$ if $⨍$ is continuous on $R$ then $a + b$ is equal to :

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answer is B.

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Detailed Solution

$f (x) = \{\begin{array}{l} \sin x - e^{x}; x \leq 0 \\ a + [- x]; 0 < x < 1 \\ 2 x - b; x \geq 1 \end{array} is continuous function$ $\Rightarrow \lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{+}} f (x) \Rightarrow \lim_{x \to 0^{-}} \sin - e^{2} = \lim_{x \to 0^{+}} a + [- x] \Rightarrow - 1 = a - 1 \Rightarrow a = 0 \lim_{x \to 1^{-}} a + [- x] = \lim_{x \to 1^{+}} 2 x - b \Rightarrow a - 1 = 2 - b ∴ a + b = 3$