1
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a strictly decreasing function with $|f(t)|<\pi / 2$ for all $t \in \mathbf{R}$. Let $g:[0, \pi] \rightarrow$ R be a function defined by $g(t)=\sin (f(t))$. Which one of the following statements is Correct?
A
$g$ is increasing on $[0, \pi]$.
B
$g$ is decreasing on $[0, \pi]$.
C
$g$ is increasing on $(0, \pi / 2)$ and decreasing on $(\pi / 2, \pi)$.
D
$g$ is decreasing on $(0, \pi / 2)$ and increasing on $(\pi / 2, \pi)$.
2
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $f, g: R \rightarrow R$ be functions. If $g$ is continuous, then which one of the following cases implies that $f$ is continuous?
A
$g(x)=(f(x))^2$
B
$g(x)=|f(x)|$
C
$g(x)=(f(x))^3$
D
$g(x)=\sin (f(x))$
3
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
What is the largest area of a rectangle, whose sides are parallel to the coordinate axes, that can be inscribed under the graph of the curve $y=1-x^2$ and above the $x$-axis?
A
$\frac{2}{3 \sqrt{3}}$
B
$\frac{4}{3 \sqrt{3}}$
C
$\frac{1}{3}$
D
$\frac{4}{3}$
4
IAT (IISER) 2024
MCQ (Single Correct Answer)
+4
-1
Let $M$ be the set of all $3 \times 3$ matrices with real entries. Consider the relation $R$ on $M$ given by $R=\{(A, B) \in M \times M: \operatorname{det}(A-B)$ is an integer $\}$. Which one of the following statements is Correct?
A
$R$ is reflexive and symmetric, but not transitive.
B
$R$ is reflexive, but neither symmetric nor transitive.
C
$R$ is an equivalence relation.
D
$R$ is symmetric and transitive, but not reflexive.
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