Application of Derivatives

JEE Main 2025 (Online) 23rd January Morning ShiftNumericalMedium📈 Topic in 32% papers+4

Ques. 1JEE Main 2025, 23 January (Morning Shift)

If the set of all values of a, for which the equation 5x3-15x-a=0 has three distinct real roots, is the interval ( , ), then -2 is equal to .

Answer

Solution

Roots of 5x3-15x=a are the points where y=a cuts f(x)=5x3-15x.

f'(x)=15 (x2-1 )=0 x= 1: f(-1)=10 (maximum), f(1)=-10 (minimum)

[figure]

Three distinct roots when the line cuts the curve three times: -10<a<10

-2 =10+20=30

Answer: 30

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Ques. 2JEE Main 2025, 23 January (Evening Shift)

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm3/min and the thickness of the ice-cream layer decreases at the rate of 14 cm/min. The surface area (in cm2) of the chocolate ball (without the ice-cream layer) is :

Ques. 3JEE Main 2025, 22 January (Evening Shift)

Let f(x)= 0x2 t2-8t+15et dt, x . Then the numbers of local maximum and local minimum points of f, respectively, are :

3 and 2
2 and 2
2 and 3
1 and 3

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