1
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $I=\int_{e^{-\pi / 2}}^{e^{\pi / 2}}\left(\sin ^2(\log (x))+\sin \left(\log \left(x^2\right)\right)\right) d x$. What is the value of $I$ ?
A
0
B
$\frac{\pi e^{\frac{\pi}{2}}}{2}$
C
$e^{\pi / 2}-e^{-\pi / 2}$
D
$e^\pi-1$
2
IAT (IISER) 2023
MCQ (Single Correct Answer)
+4
-1
Let $f: \mathbf{R} \rightarrow(0, \infty)$ be a continuous decreasing function. Suppose $f(0), \dot{f}(1), \ldots, f(10)$ are in a geometric progression with common ratio $\frac{1}{5}$. In which of the following intervals does the value of $\int_0^{10} f(x) d x$ lie?
A
$(0,2 f(0))$
B
$(4 f(0), 6 f(0))$
C
$(8 f(0), 10 f(0))$
D
$(12 f(0), 14 f(0))$
3
IAT (IISER) 2023
MCQ (Single Correct Answer)
+4
-1
Let $f:(-1,2) \rightarrow \mathbf{R}$ be a differentiable function such that $f^{\prime}(x)=\frac{2}{x^2-5}$ and $f(0)=0$. Then in which of the following intervals does $f(1)$ lie?
A
$(-\infty, 0)$
B
$(0,2)$
C
$(2,4)$
D
$(4, \infty)$
4
IAT (IISER) 2022
MCQ (Single Correct Answer)
+4
-1
For a natural number $n$, let $C_n$ be the curve in the $X Y$-plane given by $y=x^n$, where $0 \leq$ $x \leq 1$. Let $A_n$ denote the area of the region bounded between $C_n$ and $C_n+1$. Then the largest value of $A_n$ is
A
$1 / 2$
B
$1 / 3$
C
$1 / 6$
D
$1 / 12$
IAT (IISER) Subjects
EXAM MAP