Calculus

Application of Derivatives

Hard

core

In a sphere of radius $r$ , a right circular cone of height $h$ having a maximum curved surface area is inscribed. The expression for the curved surface area of the cone is

$\sqrt{2}\pi\sqrt{(2h^2r-h^3)}$

$\sqrt{2r}\pi\sqrt{(2hr-h^2)}$

$\sqrt{2r}\pi\sqrt{(2h^2r-h^3)}$

$\sqrt{2}\pi r\sqrt{(2h^2r-h^3)}$

✅ Correct Option: 3

If $R$ is the base radius, then $R^2=r^2-(h-r)^2=2rh-h^2$ , and the slant height is $l=\sqrt{R^2+h^2}=\sqrt{2rh}$ . So CSA $=\pi Rl=\pi\sqrt{(2rh-h^2)(2rh)}=\sqrt{2r}\,\pi\sqrt{2h^2r-h^3}$ .

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Consider the function $f(x) = x^3 - 3x$ . Then

Match List-I with List-II

List-I	List-II
(A) Point of local Maxima	(I) 1
(B) Point of local Minima	(II) -1
(C) Local maximum value	(III) 2
(D) Local minimum value	(IV) -2

Choose the correct answer from the options given below:

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If $f(x) = 2x^3 - 15x^2 + 36x + 1$ , $x \in [1, 5]$ , then the absolute minimum value of $f(x)$ is:

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Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$y = x^2$	I.	decreasing on $(0, \infty)$
B.	$y = -x^2$	II.	increasing on $(0, \infty)$
C.	$y = x$	III.	decreasing on $(-\infty, \infty)$
D.	$y = -x$	IV.	increasing on $(-\infty, \infty)$

Choose the correct answer from the options given below:

2025: 16 May Shift 1

The maximum value of f(x) = $\left(\frac{1}{x}\right)^x$ is

In a sphere of radius rrr, a right circular cone of height hhh having a maximum curved surface area is inscribed. The expression for the curved surface area of the cone is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

In a sphere of radius $r$ , a right circular cone of height $h$ having a maximum curved surface area is inscribed. The expression for the curved surface area of the cone is