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Limits and Derivatives

JEE Main 2025 (Online) 29th January Morning ShiftNumericalMedium📈 Topic in 5% papers+4
Ques. 1JEE 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 .

Answer

24

Solution

[ kx ] = kx - \ kx \, where 0 \ \ < 1

x [ kx ] = k - x \ kx \ k as x 0+

Similarly, x2 [ k2x2 ] = k2 - x2 \ k2x2 \ k2

limit = (1 + 2 + + p) - (12 + 22 + + 92)

= p(p + 1)2 - 9 10 196 = p(p + 1)2 - 285

Condition: p(p + 1)2 - 285 1 p(p + 1) 572

p = 23: 23 24 = 552 < 572; p = 24: 24 25 = 600 572

least value of p = 24

Answer: 24

Keep practising — Limits and Derivatives

Ques. 2JEE 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. 3JEE Main 2025, 24 January (Morning Shift)

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

  1. 1 15
  2. 12 5
  3. 0
  4. - 12 5
Ques. 4JEE Main 2025, 23 January (Evening Shift)

x (2x2-3x+5 )(3x-1) x2 (3x2+5x+4 ) (3x+2)x is equal to :

  1. 2e3
  2. 23 e
  3. 2e 3
  4. 2 3e
Ques. 5JEE Main 2025, 22 January (Evening Shift)

If x ( ( e1-e ) ( 1e- x1+x ) )x= , then the value of e 1+ e equals :

  1. e-2
  2. e2
  3. e
  4. e-1
Ques. 6JEE Main 2024, 9 April (Evening Shift)

x 0 e - (1 + 2x) 12xx is equal to

  1. -2e
  2. e - e2
  3. 0
  4. e
Ques. 7JEE Main 2024, 8 April (Morning Shift)

The value of x 0 2 ( 1 - x 2x [3] 3x [10] 10xx2 ) is .

Ques. 8JEE Main 2024, 8 April (Evening Shift)

If = x 0+ ( e x - e x x - x ) and = x 0(1 + x) 12 x are the roots of the quadratic equation ax2 + bx - e = 0, then 12 e(a + b) is equal to .

Ques. 9JEE Main 2024, 5 April (Evening Shift)

Let a > 0 be a root of the equation 2x2 + x - 2 = 0. If x 1a 16 (1 - (2 + x - 2x2 ) )(1 - ax)2 = + 17, where , Z, then + is equal to .

Ques. 10JEE Main 2024, 4 April (Morning Shift)

If x 1 (5x+1)1/3-(x+5)1/3(2x+3)1/2-(x+4)1/2= m 5n(2n)2/3, where (m, n)=1, then 8m+12n is equal to .

Ques. 11JEE Main 2024, 4 April (Evening Shift)

Let f(x) = 0x (t + (1 - et ) )dt, x R. Then, x 0 f(x)x3 is equal to

  1. 16
  2. - 16
  3. 23
  4. - 23

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