JEE Main 2022 · 28 Jun · Shift 1 · Mathematics Q8

← JEE Main 2022 · 28 Jun · Shift 1

MathematicsSingle correct+4 / -1Question 8 of 90

Let $f:\mathbb R\to\mathbb R$ be defined as
$f(x)=\begin{cases}\left[e^x\right], & x<0\\ ae^x+[x-1], & 0\le x<1\\ b+[\sin(\pi x)], & 1\le x<2\\ \left[e^{-x}\right]-c, & x\ge2\end{cases}$
where $a,b,c\in\mathbb R$ and $[t]$ denotes greatest integer less than or equal to $t$ . Then, which of the following statements is true ?

There exists

a,b,c\in\mathbb R

such that

f

is continuous on

\mathbb R

f

is discontinuous at exactly one point, then

a+b+c=1

f

is discontinuous at exactly one point, then

a+b+c\ne1

f

is discontinuous at atleast two points, for any values of

a

b

and

c

Answer and worked solution

Official answer: C

At $x=1$ . $f(1)=b$ but $f=b-1$ just to the right (there $\sin\pi x<0$ ), so $f$ is always discontinuous at 1.

Elsewhere. Continuity at 0 needs $a-1=0$ ; at 2 needs $b-1=-c$ .

Exactly one discontinuity means $a=1$ and $b+c=1$ , so $a+b+c=2\ne1$ .

(C)

Topics:Continuity Graphs of functions

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JEE Main 2022 · 28 Jun · Shift 1, Mathematics question 8