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:
What is
Answer (Detailed Solution Below)
Greatest Integer Functions Question 1 Detailed Solution
Calculation:
Given,
The function is
We are tasked with finding:
For
For
For
For
Evaluating the limit:
For
For
∴ 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:
What
Answer (Detailed Solution Below)
Greatest Integer Functions Question 2 Detailed Solution
Calculation:
Given,
The function is
We are tasked with finding:
For
For
For
For
Evaluating the limit:
For
For
∴ The value of
The correct answer is Option (2).
Greatest Integer Functions Question 3:
Let [.] denote the greatest integer function. If
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?
Answer (Detailed Solution Below)
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?
Answer (Detailed Solution Below)
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.
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 ?
Answer (Detailed Solution Below)
Greatest Integer Functions Question 6 Detailed Solution
Download Solution PDFConcept:
- 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
Answer (Detailed Solution Below)
Greatest Integer Functions Question 7 Detailed Solution
Download Solution PDFConcept:
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) ?
Answer (Detailed Solution Below)
Greatest Integer Functions Question 8 Detailed Solution
Download Solution PDFConcept:
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.
Answer (Detailed Solution Below)
Greatest Integer Functions Question 9 Detailed Solution
Download Solution PDFCONCEPT:
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 = -3The function f(x) = [x] where [x] the greatest integer function is continuous at
Answer (Detailed Solution Below)
Greatest Integer Functions Question 10 Detailed Solution
Download Solution PDFExplanation:
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.
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?
Answer (Detailed Solution Below)
Greatest Integer Functions Question 11 Detailed Solution
Download Solution PDFConcept:
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
Answer (Detailed Solution Below)
Greatest Integer Functions Question 12 Detailed Solution
Download Solution PDFCalculation:-
[x] function can be drawn as
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
Answer (Detailed Solution Below)
Greatest Integer Functions Question 13 Detailed Solution
Download Solution PDFCONCEPT:
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
As we know that
For x ∈ Z, x – [x] = 0
So, domain is R – ZIf f(x) = [x] is a greatest integer function and g(x) = x2 then find g o f(- 1.3) ?
Answer (Detailed Solution Below)
Greatest Integer Functions Question 14 Detailed Solution
Download Solution PDFConcept:
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) ?
Answer (Detailed Solution Below)
Greatest Integer Functions Question 15 Detailed Solution
Download Solution PDFConcept:
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.