1
IAT (IISER) 2022
MCQ (Single Correct Answer)
+4
-1

The value of the integral

$$ \int_1^{100} \frac{[x]}{x} d x $$

where $[x]$ is the greatest integer less than or equal to $x$ for any real number $x$, is

A
$\log \left(\frac{100^{98}}{98!}\right)$
B
$\log \left(\frac{100^{99}}{98!}\right)$
C
$\log \left(\frac{100^{98}}{99!}\right)$
D
$\log \left(\frac{100^{99}}{99!}\right)$
2
IAT (IISER) 2020
MCQ (Single Correct Answer)
+4
-1

If $p(t)=\frac{t(t-1) \cdots(t-2019)}{2019!}$, then the value of

$$ \int_0^1\left(\frac{1}{t+1}+\frac{1}{t+2}+\cdots+\frac{1}{t+2020}\right) p(-t-1) d t $$

is:

A
$2019^2$
B
2019
C
$2020^2$
D
2020
3
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
IAT (IISER) Subjects
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