English
Hindi

Question

Download Solution PDF

What is  equal to, where [⋅] is the greatest integer function?

This question was previously asked in
NDA (Held On: 17 April 2016) Maths Previous Year paper
Attempt in App
View all NDA Papers >
  1. 0
  2. 1
  3. 2
  4. 4

Answer (Detailed Solution Below)

Option 3 : 2
Free Tests
View all Free tests >
Free
NDA 01/2025: English Subject Test
30 Qs. 120 Marks 30 Mins
Start Now

Detailed Solution

Download Solution PDF

Concept:

1. If f(x) is periodic function with period T then,


2. Fractional part of x: This is the difference between x and its greatest integer part, [x]

  • The fractional part of x is given by: {x} = x – [x]
  • {x} = x for 0 ≤ x < 1
  • Period of the fractional part of x is one.


Graph of the fractional part of x:

Calculation:

We have to find the value of 

                        (∵ x – [x] = {x})

As we know that period of {x} is one.

Download Solution PDF
Latest NDA Updates

Last updated on Jun 18, 2025

->UPSC has extended the UPSC NDA 2 Registration Date till 20th June 2025.

-> A total of 406 vacancies have been announced for NDA 2 Exam 2025.

->The NDA exam date 2025 has been announced. The written examination will be held on 14th September 2025.

-> The selection process for the NDA exam includes a Written Exam and SSB Interview.

-> Candidates who get successful selection under UPSC NDA will get a salary range between Rs. 15,600 to Rs. 39,100. 

-> Candidates must go through the NDA previous year question paper. Attempting the NDA mock test is also essential. 

Exams Tests Super Practice

More Definite Integrals Questions

Q1.The value of the integral, \(\int_1^3\left[x^2-2 x-2\right] d x\) where [x] denotes the greatest integer less than or equal to x , is :
Q2.For any real number x, let [x] denote the largest integer less than equal to x. Let f be a real valued function defined on the interval [–10, 10] by \(f(x)=\left\{\begin{array}{c} x-[x], \text { if }[x] \text { is odd } \\ 1+[x]-x, \text { if }[x] \text { is even } \end{array}\right.\) Then the value of \(\frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos \pi x d x\) is
Q3.The value of \(\rm \int_{-\pi}^\pi \frac{2y(1+\sin y)}{1+\cos^2y}dy\) is:
Q4.The integral \(\rm \int_0^{\frac{\pi}{4}}\frac{136\sin x}{3\sin x+5\cos x}\) is equal to:
Q5.If the value of the integral \(\int_{-1}^1 \frac{\cos α x}{1+3^x} d x\) is \(\frac{2}{\pi}\). Then, a value of α is
Q6.\(\int_0^{\pi/4}\sin^3 \theta d\theta=?\)
Q7.Evaluate the definite integration \(\rm\displaystyle\int_0^4 \)(x + 1) dx and choose the right answer:
Q8.If f is an odd function then \(\int_{-a}^{a}\frac{f(sin x)}{f(cos x)+ f(sin^2x)}dx\)
Q9.If x satisfies the equation \(\rm 4^{8^x}=256\), then x is equal to ______.
Q10.The value of \(\rm \displaystyle\int_0^{16} \frac{x^{\frac{1}{4}}}{{1+ \sqrt{x}}}dx\) is:

More Integral Calculus Questions

Q1.If ∫(cos x - sin x)/\(\rm \sqrt{8-\sin 2x)}\) dx = a sin-1(sin x + cos x)/b + c where c is a constant of integration, then the ordered pair (a, b) is equal to:
Q2.The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola y2 = 9x, is:
Q3.The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = x is
Q4.The area of the region: R{(x, y): 5x2 ≤ y ≤ 2x2 + 9} is:
Q5.The value of the integral, \(\int_1^3\left[x^2-2 x-2\right] d x\) where [x] denotes the greatest integer less than or equal to x , is :
Q6.Let the locus of the centre (α, β), β > 0, of the circle which touches the circle x2 + (y – 1)2 = 1 externally and also touches the x - axis be L. Then the area bounded by L and the line y = 4 is :
Q7.For any real number x, let [x] denote the largest integer less than equal to x. Let f be a real valued function defined on the interval [–10, 10] by \(f(x)=\left\{\begin{array}{c} x-[x], \text { if }[x] \text { is odd } \\ 1+[x]-x, \text { if }[x] \text { is even } \end{array}\right.\) Then the value of \(\frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos \pi x d x\) is
Q8.The area of the region given by A = {(x, y ); x2 ≤ y ≤ min {x + 2, 4 - 3x}} is
Q9.Let \(f(x) = \left\lbrace \begin{matrix} -2, & -2 \le x \le 0 \\\ x - 2, & 0 < x \le 2 \end{matrix} \right.\) and h(x) = f(|x|) + |f(x)|. Then \(\int_{-2}^2 h(x) dx\) is equal to:
Q10.One of the points of intersection of the curves y = 1 + 3x - 2x2 and \(y = \frac{1}{x}\) is \(\left(\frac{1}{2}, 2 \right).\) Let the area of the region enclosed by these curves be \(\frac{1}{24}(ℓ \sqrt{5}+m) - n\log_e (1+\sqrt{5})\), where ℓ, m, n, ∈ N. Then ℓ + m + n is equal to
Suggested Test Series
View All >
Current Affairs (CA) 2025 Mega Pack for SSC/Railways/State Exam Mock Test
431 Total Tests with 4 Free Tests
Start Free Test
Current Affairs (CA) 2024 Mega Pack for SSC/Railways/State Exam Mock Test
427 Total Tests with   5 Free Tests
Start Free Test

NDA Important Links

ResultAdmit CardApplication FormEligibilitySalarySyllabus

More Mathematics Questions

Q1.log10 10000 = _______?
Q2.Consider a system equation \(2x+y-z=0\), \(4x - py + 4z = 4\) and \(x-y+z=q\) where \(p, q \in I\) and \(p, q \in [1, 10]\), then identify the correct statement(s).   List-I List-II (I) Number of ordered pairs \((p, q)\) for which system of equation has unique solution is (P) 1 (II) Number of ordered pairs \((p, q)\) for which system of equation has no solution is (Q) 9 (III) Number of ordered pairs \((p, q)\) for which system of equation has infinite solution is (R) 91 (IV) Number of ordered pairs \((p, q)\) for which system of equation has atleast one solution is (S) 90
Q3.Match List I with List II:   List - I List - II (I) The number of solutions of the equation\((1 - \cos 2x)(\cos 2x + \cot^2 x) = 0\) where \(0 \le x \le 2\pi\) is (P) 4 (II) \(\tan 81^\circ - \tan 63^\circ - \tan 27^\circ + \tan 9^\circ\) equals (Q) 3 (III) The number of solutions of (x, y), which satisfy |y| = sin x and y = cos-1(cos x), where -2π ≤ x ≤ 2π is (R) 2 (IV) For the equation \(\cos^{-1}x + \cos^{-1}2x + \pi = 0\), the number of real solutions is (S) 1     (T) 0  
Q4.If Σn = 36, find n
Q5.If log 2 = 0.2614, log 3 = 0.3521, log 6 = ?
Q6.The growth model for a population is Nt = N0ert where Nt is the population at time t, N0 is the initial population and r is the per capita growth rate. How long does it take for the population numbers to double?
Q7.Suppose that the growth rate of the number of individuals (N) in a population in an environment with a carrying capacity K is modelled as \(\frac{d N}{d t}=r N \frac{K-N}{K}\) where r is the per capita growth rate and t is time; r and K are positive constants. At which of the following values of N will the population growth rate be the maximum?
Q8.The shadow of a vertical pole, formed due to a nearby lamp which is at a height of 12 m, measures 4 m. If the lamp is moved horizontally 2 m further away from the pole, the length of the shadow measures 8 m. What is the height (in m) of the pole?
Q9.A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Out of these 16 number of cards, two cards are drawn at random. Find the probability that at least one of them is an ace :
Q10.If the equations 2x2 + kx - 5 = 0 and x2 - 3x - 4 = 0 have one root common, then find the value of k.
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
support@testbook.com
Toll Free:1800 833 0800
Office Hours: 10 AM to 7 PM (all 7 days)
Company
About usCareers We are hiringTeach Online on TestbookPartnersMediaSitemap
Products
Test SeriesTestbook PassOnline CoursesOnline VideosPracticeBlogRefer & EarnBooks
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Hot Links： teen patti master download teen patti gold new version 2024 all teen patti master teen patti gold downloadable content