Limits and Derivatives

JEE Main 2025 (Online) 4th April Evening ShiftMCQMedium📈 Topic in 12% papers+4−1

Ques. 1JEE Main 2025, 4 April (Evening Shift)

Let f be a differentiable function on R such that f(2)=1, f'(2)=4. Let x 0(f(2+x))3/x=e . Then the number of times the curve y=4x3-4x2-4( -7)x- meets x-axis is :

Answer

(3) 2

Solution

f(2) = 1, the limit is of the form 1 , so

x 0(f(2 + x))3/x = e x 0 3(f(2 + x) - 1)x = e3f'(2) = e12

Hence = 12, and the curve becomes

y = 4x3 - 4x2 - 20x - 12 = 4(x3 - x2 - 5x - 3)

At x = -1: -1 - 1 + 5 - 3 = 0, so (x + 1) is a factor and

y = 4(x + 1)(x2 - 2x - 3) = 4(x + 1)2(x - 3)

∴ The curve touches the x-axis at x = -1 and cuts it at x = 3, i.e. it meets the x-axis at 2 points.

Answer: (3)

Keep practising — Limits and Derivatives

Ques. 2JEE Main 2025, 3 April (Evening Shift)

If x 0 ( xx ) 1x2 = p, then 96 e p is equal to

Ques. 3JEE Main 2025, 2 April (Morning Shift)

For , , R, if x 0 x2 x + ( - 1)ex2 2x - x = 3, then + - is equal to :

-1
4
6
7

Ques. 4JEE Main 2025, 2 April (Evening Shift)

If x 0 (2x) + a (4x) - bx4 is finite, then (a + b) is equal to:

0
34
-1
12

Ques. 5JEE Main 2025, 29 January (Morning Shift)

Let [t] be the greatest integer less than or equal to t. Then the least value of p N for which x 0+ (x ( [ 1x ] + [ 2x ] + + [ px ] ) - x2 ( [ 1x2 ] + [ 22x2 ] + + [ 92x2 ] ) ) 1 is equal to .

Ques. 6JEE Main 2025, 28 January (Evening Shift)

Let f(x) = n r=0n ( (x/2r+1 ) + 3 (x/2r+1 )1 - 2 (x/2r+1 ) ). Then x 0 ex - ef(x)(x - f(x)) is equal to .

Ques. 7JEE Main 2025, 24 January (Morning Shift)

x 0 cosecx ( 2 2x+3 x- 2x+ x+4 ) is