English
Hindi

Question

Download Solution PDF

Given f, g are continuous functions on [0, 1] such that f(0) = f(1) = 0; g(0) = g(1) = 1 and f(1/2) > g(1/2). Which of the following statements is true?

This question was previously asked in
CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
Download PDF Attempt in App
View all CSIR NET Papers >
  1. There is no t ∈ [0, 1] such that f(t) = g(t)
  2. There is exactly one t ∈ [0, 1] such that f(t) = g(t)
  3. There are at least two t ∈ [0, 1] such that  f(t) = g(t)
  4. There are always infinitely many t ∈ [0, 1] such that  f(t) = g(t)

Answer (Detailed Solution Below)

Option 3 : There are at least two t ∈ [0, 1] such that  f(t) = g(t)
Free Tests
View all Free tests >
Free
Seating Arrangement
10 Qs. 20 Marks 15 Mins
Start Now

Detailed Solution

Download Solution PDF

Explanation:

Let 

h(x) = f(x) - g(x) 

h(0) = f(0) - g(0) = -1 < 0 . . . . . . . . . 1

h(1/2) = f(1/2) - g(1/2) > 0   . . . . . . . . 2

h(1) = f(1) - g(1) = -1 < 0   . . . . . . .  3

by equation 1 and 2 , there is one t  [0 , 1]

by equation 2 and 3 , there is one t  [ 0 ,1 ]

Hence option (3) is correct 

 

Download Solution PDF
Latest CSIR NET Updates

Last updated on Jul 8, 2025

-> The CSIR NET June 2025 Exam Schedule has been released on its official website.The exam will be held on 28th July 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 Elementary Set Theory Questions

Q1.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? 
Q2.Let x, y ∈ [0, 1] be such that x ≠ y. Which of the following statements is true for every ϵ > 0?
Q3.How many real roots does the polynomial x3 + 3x − 2023 have?
Q4.Which of the following statements is true?
Q5.f ∶ \(\mathbb{N} \rightarrow \mathbb{N}\) be a bounded function. Which of the following statements is NOT true?
Q6.Let {En} be a sequence of subsets of \(\mathbb{R}\). Define \(\limsup _n E_n=\bigcap_{k=1}^{\infty} \bigcup_{n=k}^{\infty} E_n\) \(\liminf _n E_n=\bigcup_{k=1}^{\infty} \bigcap_{n=k}^{\infty} E_n\) Which of the following statements is true?

More Analysis Questions

Q1.What is the cardinality of the set of real solutions of ex + x = 1?
Q2.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. 
Q3.Let (an)n≥1 be a bounded sequence in ℝ. Which of the following statements is FALSE? 
Q4.Consider the set A = {x ∈ ℚ : 0 < (√2 - 1)x < √2 + 1} as a subset of ℝ. Which of the following statements is true?
Q5.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? 
Q6.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? 
Q7.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? 
Q8.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? 
Q9.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
Q10.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 100 bonus teen patti real cash game mpl teen patti