1
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions such that

$$ g \circ f(x)= \begin{cases}x^2, & \text { if } x \geq 0 \\ e^x-1, & \text { if } x<0\end{cases} $$

Then which of the following statements is correct?

A
$g$ is onto.
B
$f$ is onto.
C
$g$ is one-one.
D
$f$ is one-one.
2
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1

$F(x)=\int_0^{e^x}\left(t^3+2 t^2-t-2\right) d t$, then for how many real numbers $x$ does $F^{\prime}(x)=0$ ?

A
0
B
3
C
2
D
1
3
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1

Let $C$ be a circle which passes through $(-2,1)$ and $(0,3)$ and whose centre lies on the line $y-x=1$. Then which of the following points lies on $C$ ?

A
$(\sqrt{2}+1,1)$
B
$(1, \sqrt{2}+1)$
C
$(1, \sqrt{3}+1)$
D
$(1, \sqrt{3}-1)$
4
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1
For what value of $t$ does $f(x)=x^3+t x^2+(2-t) x-3$ have only real roots?
A
$t=0$.
B
$t=4$.
C
$t=2$.
D
$t=-1$
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