NCERT Solutions for Class 10 Maths<\/a> Chapter 1 Real Numbers Ex 1.3 Questions and Answers are prepared by our highly skilled subject experts.<\/p>\nNCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.3<\/h2>\n <\/p>\n
Question 1. \nProve that \\(\\sqrt{5}\\) is irrational. \nSolution: \nLet us assume, to the contrary, that \\(\\sqrt{5}\\) is irrational that is we can find integers a and b (b \u2260 0) such that \\(\\sqrt{5}\\) = \\(\\frac { a }{ b }\\). suppose a and b have a common factor other than 1 then we can divide by the common factor and assume that a and b are coprime. \nSo b\\(\\sqrt{5}\\) = a \nSquaring on both sides, and rearranging we get 5b\u00b2 = a\u00b3. \nThus for a\u00b2 is divisible by 5, it follows that a is also divisible by 5. \nSo, we can write a 5c for some integer c. \nSubstituting for a, we get \n5b\u00b2 = 25c\u00b2 \nb\u00b2 = 5c\u00b2 \nThis means that b2 is divisible by 5 and so b is also divisible by 5. \nTherefore, a and b have at least 5 as common factor. \nBut this contradicts the fact that a and b are coprime. \nThis contradiction has arisen because of our incorrect assumption that \\(\\sqrt{5}\\) is irrational. \nSo we conclude that \\(\\sqrt{5}\\) is irrational.<\/p>\n
Question 2. \nProve that 3 + 2A\\(\\sqrt{5}\\) is irrational. \nSolution: \nLet us assume, to contrary, that 3 + 2\\(\\sqrt{5}\\) is rational. \nThat is, we can find coprime a and b (b \u2260 0) \nsuch that 3 + 2\\(\\sqrt{5}\\) = \\(\\frac { a }{ b }\\) \nTherefore 3 – \\(\\frac { a }{ b }\\) = – 2\\(\\sqrt{5}\\) \nRearranging this equation we get 2\\(\\sqrt{5}\\) = \\(\\frac { a }{ b }\\) – 3 = \\(\\frac { a – 3b }{ b }\\) \nSince a and b are integers, \\(\\frac { a }{ b }\\) – 3 we get is \nrational and so 2\\(\\sqrt{5}\\) is rational and so \\(\\sqrt{5}\\) is rational. \nBut this contradicts the fact \\(\\sqrt{5}\\) is irrational. \nThis contadiction has arise because of our incorrect assumption 3 + 2\\(\\sqrt{5}\\) is rational. \nThus, we conclude that 3 + 2\\(\\sqrt{5}\\) is irrational.<\/p>\n
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Question 3. \nProve that the following are irrationals: \n(i) \\(\\frac{1}{\\sqrt{2}}\\) \n(ii) 7\\(\\sqrt{5}\\) \n(iii) 6 + \\(\\sqrt{2}\\) \nSolution: \nLet us assume to the contrary, that \\(\\frac{1}{\\sqrt{2}}\\) is rational that is, we can find coprime a and b(b \u2260 0) such that = \\(\\frac{1}{\\sqrt{2}}\\) = \\(\\frac { a }{ b }\\) \nSince a and b are integers so \\(\\frac { a }{ b }\\) is rational and so \\(\\sqrt{2}\\) is rational. \nBut this contradicts the fact that \\(\\sqrt{2}\\) is irrational. \nThus, we conclude that \\(\\frac{1}{\\sqrt{2}}\\) is irrational<\/p>\n
(ii) 7\\(\\sqrt{5}\\) \nLet us assume, to the contrary that 7\\(\\sqrt{5}\\) is rational that is, we can find coprime a and b (\u2260 0) \nsuch that 7\\(\\sqrt{5}\\) = \\(\\frac { a }{ b }\\) Rearranging, we get \\(\\sqrt{5}\\) = \\(\\frac { a }{ b }\\) \nSince 7, a and b are integers, \\(\\frac { a }{ 7b }\\) is rational and so \\(\\sqrt{5}\\) is rational \nBut this contradicts the fact that \\(\\sqrt{5}\\) is irrational. So, we conclude that 7\\(\\sqrt{5}\\) is irrational.<\/p>\n
(iii) 6 + \\(\\sqrt{2}\\) \nLet us assume, to the contrary, that 6 + \\(\\sqrt{2}\\) is irrational. \nThat is, we can find co prime a and b (* 0) such that 6 + \\(\\sqrt{2}\\) = 7 b \nRearranging, we get \\(\\sqrt{2}\\) = \\(\\frac { a-6b }{ b }\\) \nSince a, b and 6 are integers, so \\(\\frac { a-6b }{ b }\\) is rational and so is rational. \nBut this contradicts the fact that \\(\\sqrt{2}\\) is \nirrational. So we conclude that 6 + \\(\\sqrt{2}\\) is irrational.<\/p>\n
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These NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.3 Questions and Answers are prepared by our highly skilled subject experts. NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.3 Question 1. Prove that is irrational. Solution: Let us assume, to the contrary, that is irrational that is we …<\/p>\n
NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.3<\/span> Read More »<\/a><\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"default","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","spay_email":""},"categories":[2],"tags":[],"yoast_head":"\nNCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.3 - MCQ Questions<\/title>\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\t \n\t \n\t \n