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Relations and Functions · Question 9

JEE Main 2026 (Online) 22nd January Evening ShiftMCQMedium📈 Topic in 3% papers+4−1
1

Let f(x) = [x]2 - [x + 3] - 3, x R, where [.] is the greatest integer function. Then

  1. f(x) = 0 for finitely many values of x
  2. f(x) < 0 only for x [-1, 3)
  3. 02 f(x) dx = -6
  4. f(x) > 0 only for x [4, )

Answer

(2) f(x) < 0 only for x [-1, 3)

Solution

[x + 3] = [x] + 3

f(x) = [x]2 - [x] - 6 = ([x] - 3)([x] + 2)

(1) f(x) = 0 [x] = 3 or [x] = -2 x [3, 4) [-2, -1) ⇒ infinitely many values ⇒ false

(2) f(x) < 0 -2 < [x] < 3 [x] \-1, 0, 1, 2\ x [-1, 3) ⇒ true

(3) 02 f(x) dx = 01 (-6) dx + 12 (-6) dx = -12 ⇒ false

(4) f(x) > 0 [x] > 3 or [x] < -2 x (- , -2) [4, ) ⇒ not only [4, ) ⇒ false

Answer: (2)

JEE Main 2026 (Online) 22nd January Evening Shift

Keep practising — Relations and Functions

2

Let the domain of the function f(x) = 3 5 (7 - 2 (x2 - 10x + 85 ) ) + -1 ( | 3x - 717 - x | ) be ( , ]. Then + is equal to :

  1. 12
  2. 8
  3. 10
  4. 9
JEE Main 2026 (Online) 22nd January Evening Shift
3

The number of elements in the relation R = \(x, y) : 4x2 + y2 < 52, x, y Z \ is

  1. 86
  2. 67
  3. 89
  4. 77
JEE Main 2026 (Online) 22nd January Evening Shift
4

Let the domain of the function f(x) = -1 ( 4x + 53x - 7 ) be [ , ] and the domain of g(x) = 2 (2 - 6 27(2x + 5) ) be ( , ). Then |7( + ) + 4( + )| is equal to .

JEE Main 2025 (Online) 8th April Evening Shift
5

If the range of the function f(x) = 5-xx2-3x+2, x 1, 2, is (- , ] [ , ), then 2 + 2 is equal to :

  1. 188
  2. 192
  3. 190
  4. 194
JEE Main 2025 (Online) 7th April Evening Shift
6

Let the domains of the functions f(x) = 4 3 7 (8 - 2 (x2 + 4x + 5 ) ) and g(x) = -1 ( 7x + 10x - 2 ) be ( , ) and [ , ], respectively. Then 2 + 2 + 2 + 2 is equal to :

  1. 15
  2. 13
  3. 16
  4. 14
JEE Main 2025 (Online) 4th April Evening Shift
7

If the domain of the function f(x) = e ( 2x-35+4x ) + -1 ( 4+3x2-x ) is [ , ), then 2 + 4 is equal to

  1. 4
  2. 3
  3. 7
  4. 5
JEE Main 2025 (Online) 3rd April Morning Shift
8

If the domain of the function f(x) = 7 (1 - 4 (x2 - 9x + 18 ) ) is ( , ) ( , ), then + + + is equal to

  1. 17
  2. 15
  3. 16
  4. 18
JEE Main 2025 (Online) 3rd April Evening Shift
9

Let A = \1, 2, 3, , 100\ and R be a relation on A such that R = \(a, b) : a = 2b + 1\. Let (a1, a2), (a2, a3), (a3, a4), , (ak, ak+1) be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k, for which such a sequence exists, is equal to :

  1. 6
  2. 8
  3. 7
  4. 5
JEE Main 2025 (Online) 2nd April Evening Shift
10

If the domain of the function f(x) = 1 10 + 3x - x2 + 1 x + |x| is (a, b), then (1 + a)2 + b2 is equal to :

  1. 29
  2. 30
  3. 25
  4. 26
JEE Main 2025 (Online) 2nd April Evening Shift
11

Let f : R R be a function defined by f(x) = (2 + 3a)x2 + ( a+2a-1 )x + b, a 1. If f(x + y) = f(x) + f(y) + 1 - 27xy, then the value of 28 i=15|f(i)| is

  1. 735
  2. 675
  3. 715
  4. 545
JEE Main 2025 (Online) 28th January Morning Shift

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