Question

Download Solution PDF

Consider the set A = {x ∈ ℚ : 0 < (√2 - 1)x < √2 + 1} as a subset of ℝ. Which of the following statements is true?

This question was previously asked in
CSIR-UGC (NET) Mathematical Science: Held on (2024 June)
Attempt in App
View all CSIR NET Papers >
  1. sup A = 2 + 2√3
  2. sup A = 3 + 2√2
  3. inf A = 2 + 2√3
  4. inf A = 3 + 2√2

Answer (Detailed Solution Below)

Option 2 : sup A = 3 + 2√2
Free Tests
View all Free tests >
Free
Seating Arrangement
10 Qs. 20 Marks 15 Mins
Start Now

Detailed Solution

Download Solution PDF

Concept:

The supremum  of the set is the least upper bound for  . The infimum  is the greatest lower bound of 

Explanation:

 

We are given the set } as a subset of , and we need to determine

the correct supremum (sup) and infimum (inf) of this set.

We are asked to solve for    in the inequality .

⇒  2.

Multiply the numerator and denominator by  

So, the inequality becomes 

The set  consists of rational numbers  that satisfy this inequality. Therefore,

The supremum  of this set is  , which is the least upper bound for  .

The infimum  is 0 , since  approaches but never reaches 0 from the positive side.

The correct statement is 

Hence option 2) is correct.

Download Solution PDF
Latest CSIR NET Updates

Last updated on Jun 23, 2025

-> The last date for CSIR NET Application Form 2025 submission has been extended to 26th June 2025.

-> The CSIR UGC NET is conducted in five subjects -Chemical Sciences, Earth Sciences, Life Sciences, Mathematical Sciences, and Physical Sciences. 

-> Postgraduates in the relevant streams can apply for this exam.

-> Candidates must download and practice questions from the CSIR NET Previous year papers. Attempting the CSIR NET mock tests are also very helpful in preparation.

‹‹ Previous Ques
Next Ques ››
Exams Tests Super Practice

More Supremum & Infimum Questions

Q1.Which of the following assertions is correct?

More Analysis Questions

Q1.Let u = u(x, t) be the solution of the following initial value problem  \(\rm \left\{\begin{matrix}u_t+2024u_x=0,&x ∈ R, t>0\\\ u(x, 0)=u_0(x), &x ∈ R\end{matrix}\right.\) where u0 : ℝ → ℝ s an arbitrary C1 function. Consider the following statements:  S1 : If At = {x ∈ ℝ : u(x, t) < 1} and |At| denotes the Lebesgue measure of A, for every t ≥ 0, then |At| = |A0|, ∀t > 0 S2 : If u0 is Lebesgue integrable, then for every t > 0, the function x → u(x, t) is Lebesgue integrable. 
Q2.Let (an)n≥1 be a bounded sequence in ℝ. Which of the following statements is FALSE? 
Q3.Consider the set A = {x ∈ ℚ : 0 < (√2 - 1)x < √2 + 1} as a subset of ℝ. Which of the following statements is true?
Q4.For each n ≥ 1 define fn : ℝ → ℝ by \(\rm f_n(x)=\frac{x^2}{√{x^2+\frac{1}{n}}}, \) x ∈ ℝ where √ denotes the non-negative square root. Wherever \(\rm \lim_{n \rightarrow \infty}f_n(x)\) exists, denote it by f(x). Which of the following statements is true? 
Q5.Let \(\rm S =\left\{x \in R:x >1 \ and \ \frac{1-x^4}{1-x^3}>22\right\}\) Which of the following is true about S? 
Q6.Let S be a dense subset of R and f : ℝ → ℝ a given function. Define g : S → ℝ by g(x) = f(x). Which of the following statements is necessarily true? 
Q7.Let C be the collection of all sets S such that the power set of S is countably infinite. Which of the following statements is true? 
Q8.Let 𝑐00 = {(𝑥1, 𝑥2, 𝑥3, … ) ∶ 𝑥𝑖 ∈ ℝ, 𝑖 ∈ ℕ, 𝑥𝑖 ≠ 0 only for finitely many indices 𝑖}. For (𝑥1, 𝑥2, 𝑥3, … ) ∈ 𝑐00, let ‖(𝑥1, 𝑥2, 𝑥3, … )‖∞ = sup{|𝑥𝑖 | ∶ 𝑖 ∈ ℕ}. Define 𝐹, 𝐺 ∶ (𝑐00, ‖⋅‖∞) → (𝑐00, ‖⋅‖∞) by 𝐹((𝑥1, 𝑥2, … , 𝑥𝑛, … )) = \(\rm \left((1+x)_{x_1}, (2+\frac{1}{2})x_2,..., (n+\frac{1}{n})x_n,...\right)\) 𝐺((𝑥1, 𝑥2, … , 𝑥𝑛, … )) = \(\rm \left(\frac{x_1}{1+1},\frac{x_2}{2+\frac{1}{2}},..., \frac{x_n}{n+\frac{1}{n}},..\right)\) for all (𝑥1, 𝑥2, … , 𝑥𝑛, … ) ∈ 𝑐00. Then
Q9.Let ℓ2 = {(𝑥1, 𝑥2, 𝑥3, … ) ∶ 𝑥𝑛 ∈ ℝ for all 𝑛 ∈ ℕ and \(\rm \Sigma_{n=1}^\infty x_n^2<\infty \}\) For a sequence (𝑥1, 𝑥2, 𝑥3, … ) ∈ ℓ2 , define ‖(𝑥1, 𝑥2, 𝑥3, … )‖2 = \(\rm (\Sigma_{n=1}^\infty x_n^2)^{\frac{1}{2}}\) Let 𝑆 ∶ (ℓ2 , ‖⋅‖2) → (ℓ2 , ‖⋅‖2) and 𝑇 ∶ (ℓ2 , ‖⋅‖2) → (ℓ2 , ‖⋅‖2) be defined by 𝑆(𝑥1, 𝑥2, 𝑥3, … ) = (𝑦1, 𝑦2, 𝑦3, … ), where 𝑦𝑛 = \(\rm \left\{\begin{matrix}0,&n=1\\\ x_{n-1},& n\ge2\end{matrix}\right.\) 𝑇(𝑥1, 𝑥2, 𝑥3, … ) = (𝑦1, 𝑦2, 𝑦3, … ), where 𝑦𝑛 = \(\rm \left\{\begin{matrix}0,&n\ is\ odd\\\ x_{n},& n\ is\ even\end{matrix}\right.\) Then

More Mathematical Science Questions

Q1.The only function among the following that satisfies Cauchy's Riemann (C-R) equations is : 
Q2.What is the cardinality of the set of real solutions of ex + x = 1?
Q3.Let A be a 10 x 10 real matrix. Assume that the rank of A is 7. Which of the following statements is necessarily true? 
Q4.Let a point P be chosen at random on the line segment AB of length α. Let Z1 and Z2 denote the lengths of line segments AP and BP respectively. Then the value of E(|Z1 - Z2|) Is 
Q5.Consider the contour γ given by \(\rm \gamma (\theta)=\left\{\begin{matrix}e^{2i \theta}&for\ \theta \in [0, \pi /2]\\\ 1+2e^{2i\theta}&for\ \theta \in [\pi/2, 3\pi/2]\\\ e^{2i\theta}&for\ \theta \in [3\pi/2, 2\pi]\end{matrix}\right.\) Then what is the value of \(\rm \int_{\gamma}\frac{dz}{z(z-2)}\)?
Q6.Let X1....X10 be a random sample from a distribution with the probability density function \(\rm f(x|θ)=\left\{\begin{matrix}θ x^{θ-1}, & if \:0<x<1\\\ 0, & otherwise,\end{matrix}\right.\) where θ > 0 Is an unknown parameter. The prior distribution of θ is given by   \(\rm \pi (θ)=\left\{\begin{matrix}θ e^{-θ}, & if\ θ>0\\\ 0, & otherwise,\end{matrix}\right.\) The Bayes estimator of θ under squared error loss is  
Q7.For a quadratic form f(x, y, z) ∈ ℝ[x, y z], we say that (a, b, c) ∈ ℝ3 is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)? 
Q8.Let A : ℝm → ℝn be a non-zero linear transformation. Which of the following statements is true? 
Q9.Let X1, X2 be a random sample from a population having probability density function f ∈ { f0, f1} where  \(\rm f_0(x)=\left\{\begin{matrix}\frac{1}{2}& if\ 0\le x\le 2\\\ 0&otherwise\end{matrix}\right.\ and \ \rm f_1(x)=\left\{\begin{matrix}\frac{1}{4}& if\ 0\le x\le 4\\\ 0&otherwise\end{matrix}\right.\) For testing the null hypothesis H0 : f = f0 against the alternate hypothesis H1 : f = f1, the power of a most powerful test of size α = 0.05 is equal to  
Q10.Let V be the real vector space of 2 x 2 matrices with entries in ℝ. Let T : V → V denote the linear transformation defined by T(B) = AB for all B ∈ V, where \(\rm A=\begin{pmatrix}2&0\\\ 0&1\end{pmatrix}\). What is the characteristic polynomial of T? 
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 baaz teen patti master official teen patti gold download apk teen patti master new version real cash teen patti