JEE MAIN - Mathematics (2021 - 20th July Morning Shift - No. 13)
Let a function f : R $$\to$$ R be defined as $$f(x) = \left\{ {\matrix{
{\sin x - {e^x}} & {if} & {x \le 0} \cr
{a + [ - x]} & {if} & {0 < x < 1} \cr
{2x - b} & {if} & {x \ge 1} \cr
} } \right.$$
where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:
where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:
4
3
2
5
Explanation
Continuous x = 0
f(0+) = f(0$$-$$) $$\Rightarrow$$ a $$-$$ 1 = 0 $$-$$ e0
$$\Rightarrow$$ a = 0
Continuous at x = 1
f(1+) = f(1$$-$$)
$$\Rightarrow$$ 2(1) $$-$$ b = a + ($$-$$1)
$$\Rightarrow$$ b = 2 $$-$$ a + 1 $$\Rightarrow$$ b = 3
$$\therefore$$ a + b = 3
f(0+) = f(0$$-$$) $$\Rightarrow$$ a $$-$$ 1 = 0 $$-$$ e0
$$\Rightarrow$$ a = 0
Continuous at x = 1
f(1+) = f(1$$-$$)
$$\Rightarrow$$ 2(1) $$-$$ b = a + ($$-$$1)
$$\Rightarrow$$ b = 2 $$-$$ a + 1 $$\Rightarrow$$ b = 3
$$\therefore$$ a + b = 3
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