Greatest Integer Functions MCQ Quiz - Objective Question with Answer for Greatest Integer Functions - Download Free PDF

Last updated on Jul 3, 2025

Latest Greatest Integer Functions MCQ Objective Questions

Greatest Integer Functions Question 1:

Comprehension:

Consider the following for the two (02) items that follow: Let the function , where [] is the greatest integer function and .

What is limx0f(x)g(x) equal to?

  1. sin1
  2. sin1
  3. 0
  4. Limit does not exist

Answer (Detailed Solution Below)

Option 4 : Limit does not exist

Greatest Integer Functions Question 1 Detailed Solution

Calculation:

Given,

The function is f(x)=sin(x), where x is the greatest integer function, and g(x) = |x| , the absolute value function.

We are tasked with finding:

limx0f(x)g(x)

For g(x)=|x|, we know that:

limx0g(x)=0

For f(x)=sin(x), we know that:

For x0+, x=0, so f(x) = sin(0) = 0 .

For x0, x=1, so f(x)=sin(1), which is a nonzero constant.

Evaluating the limit:

For x0+, f(x)g(x)=0x=0

For x0, f(x)g(x)=sin(1)x, which becomes undefined as x0because the denominator approaches 0, but the numerator remains a nonzero constant.

∴ Since the left-hand and right-hand limits do not match, the limit does not exist.

The correct answer is Option (4):

Greatest Integer Functions Question 2:

Comprehension:

Consider the following for the two (02) items that follow: Let the function , where [] is the greatest integer function and .

What limx0f(x)g(x) is  equal to?

  1. -1
  2. 0
  3. 1
  4. Limit does not exist

Answer (Detailed Solution Below)

Option 2 : 0

Greatest Integer Functions Question 2 Detailed Solution

Calculation:

Given,

The function isf(x)=sin(x), wherexis the greatest integer function, and g(x) = |x| , the absolute value function.

We are tasked with finding:

limx0f(x)g(x)

For g(x)=|x|, we know that:

limx0g(x)=0

For f(x)=sin(x), we know that:

For x0+), x=0, so f(x) = sin(0) = 0 .

For x0, x=1, so f(x)=sin(1), which is a nonzero constant.

Evaluating the limit:

Forx0+,f(x)g(x)=0×x=0

Forx0, f(x)g(x)=sin(1)×(x), which approaches 0 as x0..

∴ The value of limx0f(x)g(x)is 0.

The correct answer is Option (2).

Greatest Integer Functions Question 3:

Let [.] denote the greatest integer function. If 03[1ex1]dx=αloge2, then α3 is equal to ____.

Answer (Detailed Solution Below) 8

Greatest Integer Functions Question 3 Detailed Solution

Concept:

Greatest Integer Function and Definite Integral:

  • The greatest integer function, denoted by [x], gives the largest integer less than or equal to x.
  • To integrate a greatest integer function, divide the integral into intervals where the function is constant.
  • The function inside the integral is f(x) = [1 / ex−1] = [e1−x].
  • We need to evaluate ∫₀³ [e1−x] dx = α − logₑ2.

 

Calculation:

f(x) = [e1−x] is a decreasing function

f(0) = [e1] = [2.718] = 2

f(1−ln2) = e1−(1−ln2) = eln2 = 2

⇒ boundary point

f(x) = 2 for x ∈ [0, 1−ln2)

f(1) = [e0] = [1] = 1

f(x) = 1 for x ∈ [1−ln2, 1)

f(x) < 1 for x ≥ 1 ⇒ [f(x)] = 0

Now break the integral accordingly:

∫₀³ [e1−x] dx = ∫₀1−ln2 2 dx + ∫1−ln21 1 dx + ∫₁³ 0 dx

⇒ 2(1 − ln2) + (1 − (1 − ln2)) + 0

⇒ 2 − 2ln2 + ln2 = 2 − ln2

Given: ∫₀³ [e1−x] dx = α − ln2

Comparing both sides:

α − ln2 = 2 − ln2 ⇒ α = 2

Now, α3 = 23 = 8

∴ The value of α3 is 8.

Greatest Integer Functions Question 4:

Let z = [y] and y = [x] - x, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z? 

  1. -1
  2. 0
  3. 1
  4. 2

Answer (Detailed Solution Below)

Option 1 : -1

Greatest Integer Functions Question 4 Detailed Solution

Explanation:

Since, {x}+[x] = x

⇒x – [x] = {x}

⇒ 0≤ x – [x] < 1 (0 ≤ {x} < 1)

⇒ –1 ≤ [x] –x ≤ 0

But x is positive and non-integer; then

⇒ –1 < [x]–x < 0

⇒ –1 < y < 0

⇒ [y] = –1 

∴ Option (a) is correct.

Greatest Integer Functions Question 5:

At which of the points is the function f(x) = [x], where [x] is the greatest integer function continuous?

  1. 3.6
  2. -4
  3. 3
  4. 0

Answer (Detailed Solution Below)

Option 1 : 3.6

Greatest Integer Functions Question 5 Detailed Solution

Explanation:

To find the point of discontinuity of the function f(x) = [x], draw the graph of the function f(x) = [x].

Greatest Integer Function: (Floor function)

The function f (x) = [x] is called the greatest integer function and means the greatest integer less than or equal to x i.e [x] ≤ x.

The domain of [x] is R and range is I, where R is the set of real numbers and I is the set of integers.

F1 A.K 9.4.20 Pallavi D1

 

From the graph, we can say that the function is discontinuous at every integer.

The function f ( x )= [x] is continous for all x , except integral values of x . 

∴ It is continuous at x = 3.6, which is not an integer. 

The correct option is 1.

Top Greatest Integer Functions MCQ Objective Questions

Let z = [y] and y = [x] − x, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z ?

  1. −1
  2. 0
  3. 1
  4. 2

Answer (Detailed Solution Below)

Option 1 : −1

Greatest Integer Functions Question 6 Detailed Solution

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Concept:

  • The greatest integer function of a number rounds off the number to the integer less than the number
  • Every integer x can be witten as x = [x] + {x}, where [x] is the integer part of x and {x} is the fractional part of x
  • 0 ≤ {x} < 1
  • If x is an integer, then {x} = 0

Property of greatest integer function:

  • [-x] = -[x] , if x ∈ Z
  •  [-x] = -[x] - 1, if x ∉ Z

Calculation:

Given, z = [y] and y = [x] − x, where [.] is the greatest integer function

 If x is not an integer but positive, 

⇒ x > 0 and x ∉ Z

⇒ y = [x] − x =  [x] − ([x] + {x})

⇒ y = -{x} 

Now given z = [y] 

Putting the value of y,

⇒ z =[-{x}]

⇒ z = -[{x}] - 1   ({x} ∉ Z)

⇒ z = -0 - 1  (∵ 0 ≤ {x} < 1))

⇒ z = -1 

∴ The correct option is (1).

If [x]2 – 5 [x] + 6 = 0, where [ . ] denote the greatest integer function, then 

  1. x ∈ [3, 4]
  2. x ∈ (2, 3] 
  3. x ∈ [2, 3]
  4. x ∈ [2, 4)

Answer (Detailed Solution Below)

Option 4 : x ∈ [2, 4)

Greatest Integer Functions Question 7 Detailed Solution

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Concept:

Greatest Integer Function:

The greatest integer function is denoted by [x], for any real function. The function rounds off the real number down to the integer less than the number.

properties of the greatest integer function

  • [x] = x, where x is an integer 
  • [x + n] = [x] + n, where n ∈ Z
  • [ - x] = - [x], if x ∈ Z
  • [ - x] = - [x] - 1, if x ∉ Z
  • If [ f(x)] ≥ Y, then f(x) ≥ Y, where Z is the set on integers

Calculation:

Given, [x]2 – 5 [x] + 6 = 0

⇒ [x]2 - 3 [x] - 2[x] + 6 = 0

⇒ [×]([x] - 3) - 2([x] - 3) = 0

⇒ ([x] - 3)([x] - 2) = 0

⇒ [x] - 3 = 0 or [x] - 2 = 0

⇒ [x] = 3 or [x] = 2

⇒ [x] = 2, 3

For [x] = 2, x ∈ [2, 3)

For [x] = 3, x = [3 ,4)

∴ The required value of x ∈ [2, 3) ∪ [3,4)

⇒ x ∈ [2, 4)

∴ x ∈ [2, 4).

If f(x) = 2|x| and g(x) = [x] where [.] denotes greatest integer function then find the value of f o g (- 17/2) ?

  1. 1512
  2. 1256
  3. 256
  4. 512

Answer (Detailed Solution Below)

Option 4 : 512

Greatest Integer Functions Question 8 Detailed Solution

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Concept:

Greatest Integer Function: (Floor function)

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

Domain of [x] is R and range is I.

If f :A → B and g : C → D. Then (fog) (x) will exist if and only if co-domain of g = domain of f i.e D = A and (gof) (x) will exist if and only if co-domain of f = domain of g i.e B = C.

Calculation:

Given: f(x) = 2|x| and g(x) = [x] where [.] denotes greatest integer function 

Here, we have to find out the value of f o g (- 3/2)

⇒ f o g (- 17/2) = f( g(- 17/2))

∵ g(x) = [x], so g(- 17/2) = [- 17/2] = - 9

⇒ f o g(- 17/2) = f(- 9)

∵ f(x) = 2|x| so, f(- 9) = 2|- 9| = 29 = 512

Hence, f o g (- 17/2) = 512

Calculate the value of p if function f (x) = [1.2] p + [-2.23] p + p = 3[1.2], where [.] show the greatest integer function.

  1. 3
  2. -3
  3. 8
  4. -4

Answer (Detailed Solution Below)

Option 2 : -3

Greatest Integer Functions Question 9 Detailed Solution

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CONCEPT:

The real function f: R → R defined by f (x) = [x], x ∈R picks the value of the greatest integer less than or equal to x, is called the greatest integer function.

Thus f (x) = [x] = – 1 for – 1 ≤ x < 0

And f (x) = [x] = 0 for 0 ≤ x < 1

[x] = 1 for 1 ≤ x < 2

[x] = 2 for 2 ≤ x < 3 and so on     

CALCULATION:

Given function is f (x) = [1.2] p + [-2.23] p + p = 3[1.2]

⇒ [1.2] = 1, [-2.23] = -3

∴ p – 3p + p = 3 × 1

⇒ p = -3

The function f(x) = [x] where [x] the greatest integer function is continuous at

  1. -2
  2. 1
  3. 4
  4. 1.5

Answer (Detailed Solution Below)

Option 4 : 1.5

Greatest Integer Functions Question 10 Detailed Solution

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Explanation:

To find the point of discontinuity of the function f(x) = [x], draw the graph of the function f(x) = [x].

Greatest Integer Function: (Floor function)

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

The domain of [x] is R and range is I, where R is the set of real numbers and I is the set of integers.

F1 A.K 9.4.20 Pallavi D1

 

From the graph, we can say that the function is discontinuous at every integer.

The function f ( x )= [x] is continous for all x , except integral valaues of x . 

∴ It is continuous at x = 1.5, which is not an integer. 

The correct option is 4.

Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x = -3.5?

  1. -4
  2. -3.5
  3. -3
  4. 0

Answer (Detailed Solution Below)

Option 4 : 0

Greatest Integer Functions Question 11 Detailed Solution

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Concept:

Greatest Integer Function: (Floor function)

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

The domain of [x] is R and the range is I.

Note:

Any function is differentiable only if it is continuous.

The floor function f(x) = ⌊x⌋ is differentiable in every open interval between integers, (n, n + 1) for any integer n.

Calculation:

Given that,

y = [x + 1], -4 < x < -3

We have to determine the derivative at y = [x + 1] at x = -3.5

We know that the floor function is differentiable at all points except integer points.

Hence, y = [x + 1] is differentiable at x = -3.5

⇒ y = [-3.5 + 1] = [-2.5] = -3

⇒ dy/dx = 0

∴ The derivative of y with respect to x at x = -3.5 is 0.

The greatest integer function defined by f(x) = [x], 0 < x < 2 is

  1. Neither continuous nor differentiable at x = 1
  2. Continuous but not differentiable at x = 1
  3. Not continuous but differentiable at x = 1
  4. Both continuous and differentiable at x = 1

Answer (Detailed Solution Below)

Option 1 : Neither continuous nor differentiable at x = 1

Greatest Integer Functions Question 12 Detailed Solution

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Calculation:-

[x] function can be drawn as

F1 A.K 9.4.20 Pallavi D1

Let us check the continuity

f(x) at x + 1 = 0

∴ The function is not continuous, which is also evident from the graph as well.

As, the function is not continuous,

∴ it will not be differentiable as well.

Find the domain of function f(x)=2.3x[x], where [.] represents the greatest integer function.

  1. R - Z
  2. R
  3. (-∞, 0) (0, ∞)
  4. None of these

Answer (Detailed Solution Below)

Option 1 : R - Z

Greatest Integer Functions Question 13 Detailed Solution

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CONCEPT:

The real function f: R → R defined by f (x) = [x], x ∈R picks the value of the greatest integer less than or equal to x, is called the greatest integer function.

Thus f (x) = [x] = – 1 for – 1 ≤ x < 0

And f (x) = [x] = 0 for 0 ≤ x < 1

[x] = 1 for 1 ≤ x < 2

[x] = 2 for 2 ≤ x < 3 and so on

CALCULATIONS:

Given function is f(x)=2.3x[x]

As we know that 0x[x]<1,xR

For x ∈ Z, x – [x] = 0

So, domain is R – Z 

If f(x) = [x] is a greatest integer function and g(x) = x2 then find g o f(- 1.3) ?

  1. 4
  2. 2
  3. 1
  4. None of these

Answer (Detailed Solution Below)

Option 1 : 4

Greatest Integer Functions Question 14 Detailed Solution

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Concept:

Greatest Integer Function: (Floor function)

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

Domain of [x] is R and range is I.

If f :A → B and g : C → D. Then (fog) (x) will exist if and only if co-domain of g = domain of f i.e D = A and (gof) (x) will exist if and only if co-domain of f = domain of g i.e B = C.

Calculation:

Given: f(x) = [x] is a greatest integer function and g(x) = x2 

Here, we have to find the value of g o f(- 1.3)

⇒ g o f(- 1.3) = g( f (- 1.3))

∵ f(x) = [x] so, f(- 1. 3) = [- 1.3] = - 2

⇒ g o f(- 1.3) = g(- 2)

∵ g(x) = x2 so, g(- 2) = 4.

Hence, g o f(- 1.3) = 4

If f(x) = (x)[x] where [.] denotes greatest integer function the find the value of f(5/2) ?

  1. 25/4
  2. 125/8
  3. 25/8
  4. 125/4

Answer (Detailed Solution Below)

Option 1 : 25/4

Greatest Integer Functions Question 15 Detailed Solution

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Concept:

 

Greatest Integer Function: (Floor function)

The function f (x) = [x] is called the greatest integer function and means greatest integer less than or equal to x i.e [x] ≤ x.

Domain of [x] is R and range is I.

Calculation:

Given: f(x) = (x)[x]

Where [.] denotes greatest integer function

Here, we have to find the value of f(5/2)

⇒ f(5/2) = (5/2)[5/2]

As we know that. [5/2] = 2

⇒ f(5/2) = (5/2)2 = 25/4

∴ The value of f(5/2) is 25/4.

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