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Limits and Derivatives

JEE Main 2026 (Online) 6th April Evening ShiftMCQMedium📈 Topic in 14% papers+4−1
Ques. 1JEE Main 2026, 6 April (Evening Shift)

Let x 2 ( (x-2) ) (rx2+(p-2)x-2p )(x-2)2=5 for some r, p R. If the set of all possible values of q, such that the roots of the equation rx2-px+q=0 lie in (0, 2), be the interval ( , ], then 4( + ) equals :

  1. 11
  2. 13
  3. 17
  4. 21

Answer

(3) 17

Solution

[figure]

x 2 (x - 2)x - 2 = 1, the given limit becomes x 2 rx2 + (p - 2)x - 2px - 2 = 5.

For a finite limit the numerator must vanish at x = 2:

4r + 2(p - 2) - 2p = 0 4r - 4 = 0 r = 1

Then x2 + (p - 2)x - 2p = (x - 2)(x + p), so

x 2(x + p) = 2 + p = 5 p = 3

The equation is x2 - 3x + q = 0; let h(x) = x2 - 3x + q. For both roots to lie in (0, 2) we need:

(i) D 0: 9 - 4q 0 q 94

(ii) h(0) > 0: q > 0

(iii) h(2) > 0: 4 - 6 + q > 0 q > 2

(iv) the vertex x = 32 lies in (0, 2), which is true.

Hence q (2, 94 ], so = 2 and = 94.

∴ 4( + ) = 8 + 9 = 17

Answer: (3)

Keep practising — Limits and Derivatives

Ques. 2JEE Main 2026, 5 April (Morning Shift)

The product of all possible values of , for which x 0 ( 1- ( x) (( +1)x) (( +2)x) 2(( +1)x) )=2, is :

  1. -2
  2. 1
  3. -1
  4. 54
Ques. 3JEE Main 2026, 5 April (Evening Shift)

Let f(x)= y 0 (1- (xy)) (xy)y3. Then the number of solutions of the equation f(x)= x, x R is :

  1. 0
  2. 2
  3. 3
  4. 1
Ques. 4JEE Main 2026, 2 April (Morning Shift)

If x 2 (x3-5x2+ax+b ) ( x-1-1 ) e(x-1)=m, then a+b+m is equal to :

  1. 5
  2. 6
  3. 8
  4. 10
Ques. 5JEE Main 2026, 28 January (Morning Shift)

The value of x 0 e ( (ex) (e2x) (e10x) )e2 - e2 x is equal to

  1. (e10 - 1)2e2(e2 - 1)
  2. (e20 - 1)2e2(e2 - 1)
  3. (e10 - 1)2(e2 - 1)
  4. (e20 - 1)2(e2 - 1)
Ques. 6JEE Main 2026, 24 January (Morning Shift)

If the function f(x) = ex (e x - x - 1 ) + e( x + x) - x x - x is continuous at x = 0, then the value of f(0) is equal to

  1. 23
  2. 12
  3. 2
  4. 32
Ques. 7JEE Main 2026, 22 January (Evening Shift)

If x 0 e(a - 1)x + 2 bx + (c - 2)e-xx x - e(1 + x) = 2, then a2 + b2 + c2 is equal to :

  1. 3
  2. 5
  3. 9
  4. 7
Ques. 8JEE Main 2026, 21 January (Morning Shift)

Let f : R (0, ) be a twice differentiable function such that f(3) = 18, f'(3) = 0 and f''(3) = 4. Then x 1 e ( f(2 + x)f(3) ) 18(x - 1)2 is equal to :

  1. 9
  2. 18
  3. 1
  4. 2
Ques. 9JEE Main 2026, 21 January (Evening Shift)

Let f(x) = x3 + x2 f'(1) + 2x f''(2) + f'''(3), x R. Then the value of f'(5) is :

  1. 6575
  2. 1175
  3. 25
  4. 625
Ques. 10JEE Main 2025, 8 April (Evening Shift)

Given below are two statements: In the light of the above statements, choose the correct answer from the options given below:

  1. Statement I is false but Statement II is true
  2. Both Statement I and Statement II are false
  3. Both Statement I and Statement II are true
  4. Statement I is true but Statement II is false
Ques. 11JEE Main 2025, 7 April (Morning Shift)

x 0+ (5(x) 13 ) e (1 + 3x2 ) ( -13 x )2 (e5(x) 43 - 1 ) is equal to

  1. 53
  2. 1
  3. 13
  4. 115

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