Application Of Derivatives - Calculus - Mathematics Previous Year Questions

1

2022 · Mathematics · Calculus · Application Of Derivatives

IAT IISER 2022

Let $a$ be a nonzero real number and $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function such that $f^{\prime}(x)>0$ for all $x \in R$. Consider $g(x)=f\left(2 a^2 x-a x^2\right)$. Then $g$ has

A

Local maxima at $x=a$ if $a>0$

B

Local maxima at $x=a$ if $a<0$

C

Local minima at $x=a$ if $a>0$

D

A point of inflection at $x=a$

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2

2022 · Mathematics · Calculus · Application Of Derivatives

IAT IISER 2022

The function given by $f(x)=2 x^3-15 x^2+36 x-5$ is

A

Increasing on the interval $(0,2)$

B

Decreasing on the interval $(-3,0)$

C

Increasing on the interval $(2,3)$

D

Decreasing on the interval $(3, \infty)$

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3

2022 · Mathematics · Calculus · Application Of Derivatives

IAT IISER 2022

Let $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_1 x+a_0$ be a polynomial. Suppose that $f(0)=0$,

$$\left.\left.\frac{d f}{d x}\right]_{x=0}=1, \frac{d^2 f}{d x^2}\right]_{x=0}=4$$

and

$$\frac{d^3 f}{d x^3}=\frac{d^5 f}{d x^5}$$

Then $f(5)=$

A

25

B

35

C

55

D

105

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4

2023 · Mathematics · Calculus · Application Of Derivatives

IAT IISER 2023

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

A

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

B

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

C

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

D

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

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5

2023 · Mathematics · Calculus · Application Of Derivatives

IAT IISER 2023

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 $y^2=1$ ?

A

Zero

B

One

C

Two

D

Four

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6

2024 · Mathematics · Calculus · Application Of Derivatives

IAT IISER 2024

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

$\frac{4}{3}$

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7

2024 · Mathematics · Calculus · Application Of Derivatives

IAT IISER 2024

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 $(\pi / 2, \pi)$.

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