Greatest Integer Functions MCQ Quiz in తెలుగు - Objective Question with Answer for Greatest Integer Functions - ముఫ్త్ [PDF] డౌన్లోడ్ కరెన్
Last updated on Apr 4, 2025
Latest Greatest Integer Functions MCQ Objective Questions
Top Greatest Integer Functions MCQ Objective Questions
Greatest Integer Functions Question 1:
Let \(\mathrm{x}=(8 \sqrt{3}+13)^{13} \) and \( \mathrm{y}=(7 \sqrt{2}+9)^{9} \). If [t] denotes the greatest integer ≤ t, then
Answer (Detailed Solution Below)
Greatest Integer Functions Question 1 Detailed Solution
Calculation:
\(\mathrm{x}=(8 \sqrt{3}+13)={ }^{13} \mathrm{C}_{0} \cdot(8 \sqrt{3})^{13}+{ }^{13} \mathrm{C}_{1}(8 \sqrt{3})^{12}(13)^{1}+\ldots \)
\(\mathrm{x}^{\prime}=(8 \sqrt{3}-13)^{13}={ }^{13} \mathrm{C}_{0}(8 \sqrt{3})^{13}-{ }^{13} \mathrm{C}_{1}(8 \sqrt{3})^{12}(13)^{1}+\ldots \)
\(\mathrm{x}-\mathrm{x}^{\prime}=2\left[{ }^{13} \mathrm{C}_{1} \cdot(8 \sqrt{3})^{12}(13)^{1}+{ }^{13} \mathrm{C}_{3}(8 \sqrt{3})^{10} \cdot(13)^{3} \ldots\right]\)
∴ , x − x' is even integer, hence [x] is even
Now, \( \mathrm{y}=(7 \sqrt{2}+9)^{9}={ }^{9} \mathrm{C}_{0}(7 \sqrt{2})^{9}+\) \({ }^{9} \mathrm{C}_{1}(7 \sqrt{2})^{8}(9)^{1} +{ }^{9} \mathrm{C}_{2}(7 \sqrt{2})^{7}(9)^{2} \ldots \ldots\)
\(\mathrm{y}^{\prime}=(7 \sqrt{2}-9)^{9}={ }^{9} \mathrm{C}_{0}(7 \sqrt{2})^{9}-\) \({ }^{9} \mathrm{C}_{1}(7 \sqrt{2})^{8}(9)^{1} +{ }^{9} \mathrm{C}_{2}(7 \sqrt{2})^{7}(9)^{2} \ldots \ldots\)
\(\rm y-y^{\prime}=2\left[{ }^{9} C_{1}(7 \sqrt{2})^{8}(9)^{1}+{ }^{9} C_{3}(7 \sqrt{2})^{6}(9)^{3}+\ldots\right]\)
y − y' = Even integer, hence [y] is even
∴ [x] + [y] is even
Hence, the correct answer is Option 1.
Greatest Integer Functions Question 2:
Let [.] denote the greatest integer function. If \(\int_{0}^{3} \left[ \frac{1}{e^{x-1}} \right] dx = \alpha - \log_e 2, \text{ then } \alpha^3 \text{ is equal to } \_\_\_\_.\)
Answer (Detailed Solution Below) 8
Greatest Integer Functions Question 2 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 3:
Consider the function f(x) = [x + 1] - (sin\(\frac{\pi }{2}\)[x]) for x ϵ R. where [x] denotes the greatest integer less than or equal to x. Let l1 = limx→0-f(x) and l2 = limx→0+ f(x).It follows that
Answer (Detailed Solution Below)
Greatest Integer Functions Question 3 Detailed Solution
Concept :
⇒ f(x) = [x] denotes a step function whose graph is as follows :
⇒ Thus by the graph we can depict any value, for example [2.93] = 2, [-0.5] = -1, ...
Calculation :
Given the function f(x) = [x + 1] - (sin\(\frac{π }{2}\)[x]).
It is given that l1 = limx→0-f(x).
⇒ l1 = limx→0-f(x) = limx→0- {[x + 1] - (sin\(\frac{π }{2}\)[x])} = {[0-+1] - (sin\(\frac{π }{2}\)[0-])} = {[1-] - (sin\(\frac{π }{2}\)[0-])}.
⇒ l1 = {[1-] - (sin\(\frac{-π }{2}\))} = {0- (-1)} = 1.
It is given that l2 = limx→0+f(x).
⇒ l2 = limx→0+f(x) = limx→0+ {[x + 1] - (sin\(\frac{π }{2}\)[x])} = {[0++1] - (sin\(\frac{π }{2}\)[0+])} = {[1+] - (sin\(\frac{π }{2}\)[0+])}.
⇒ l2 = {[1+] - (sin(0))} = {1- 0} = 1.
Thus l1 = l2 = 1.
Mistake Points
Student often gets mistaken in two points when solving these type of problems :
- Observation and representation of step bracket wherever necessary.
- sin\(\frac{π }{2}\)(0-) = sin\(\frac{π }{2}\)(0+) = 0, but their Difference when a step function is used on them.
⇒ sin\(\frac{π }{2}\)[0-] = sin\(\frac{-π }{2}\) = -1 and sin\(\frac{π }{2}\)[0+] = sin0 = 0.
Greatest Integer Functions Question 4:
If f(x) = (x)[x] where [.] denotes greatest integer function and g(x) = x2 then find the value of g o f(5/2) ?
Answer (Detailed Solution Below)
Greatest Integer Functions Question 4 Detailed Solution
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)[x] where [.] denotes greatest integer function and g(x) = x2
Here, we have to find the value of g o f(5/2)
⇒ g o f(5/2) = g( f(5/2))
∵ f(x) = (x)[x] where [.] denotes greatest integer function
⇒ f(5/2) = (5/2)[5/2]
As we know that [5/2] = 2
⇒ f(5/2) = (5/2)2 = 25/4
⇒ g o f(5/2) = g(25/4)
∵ g(x) = x2 so, g(25/4) = 625/16
Hence, g o f(5/2) = 625/16