IAT (IISER) 2024 | Application of Derivatives Question 1 | Mathematics

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?

$g$ is increasing on $[0, \pi]$.

$g$ is decreasing on $[0, \pi]$.

$g$ is increasing on $(0, \pi / 2)$ and decreasing on $(\pi / 2, \pi)$.

$g$ is decreasing on $(0, \pi / 2)$ and increasing on $(\pi / 2, \pi)$.

IAT (IISER) 2024

MCQ (Single Correct Answer)

-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?

$\frac{2}{3 \sqrt{3}}$

$\frac{4}{3 \sqrt{3}}$

$\frac{1}{3}$

$\frac{4}{3}$

IAT (IISER) 2023

MCQ (Single Correct Answer)

-1

Let $\alpha$ be a real number. What is the total number of distinct point(s) of intersection between the parabola $y=x^2+4 x \sin \alpha+6$ and the pair of lines $y^2=1$ ?

Zero

One

Two

Four

IAT (IISER) 2023

MCQ (Single Correct Answer)

-1

Let $f(x)=\sin (3 x), x \in\left[0, \frac{\pi}{2}\right]$. Which of the following statements is true

$f$ is increasing on $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$.

$f$ is decreasing on $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

$f$ is increasing on $\left(\frac{\pi}{4}, \frac{\pi}{3}\right)$ and decreasing on $\left(\frac{\pi}{3}, \frac{\pi}{2}\right)$.

$f$ is decreasing on $\left(\frac{\pi}{4}, \frac{\pi}{3}\right)$ and increasing on $\left(\frac{\pi}{3}, \frac{\pi}{2}\right)$.

IAT (IISER) Subjects