{"id":1215,"date":"2020-09-15T06:38:12","date_gmt":"2020-09-15T01:08:12","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=1215"},"modified":"2021-07-02T15:01:51","modified_gmt":"2021-07-02T09:31:51","slug":"kseeb-sslc-class-10-maths-solutions-chapter-8-ex-8-3","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/kseeb-sslc-class-10-maths-solutions-chapter-8-ex-8-3\/","title":{"rendered":"KSEEB SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Ex 8.3"},"content":{"rendered":"

KSEEB SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Ex 8.3<\/strong> are part of KSEEB SSLC Class 10 Maths Solutions<\/a>. Here we have given Karnataka SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Exercise 8.3.<\/p>\n

Karnataka SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Exercise 8.3<\/h2>\n

Question 1.
\nProve that \\(\\sqrt{5}\\) is irrational.
\nSolution:
\nLet us assume, to the contrary, that \\(\\sqrt{5}\\) is rational.
\n\u2234 \\(\\sqrt{5}=\\frac{a}{b}\\)
\n\u2234 b \u00d7 \\(\\sqrt{5}\\) = a
\nBy Squaring on both sides,
\n5b2<\/sup> = a2<\/sup> …………. (i)
\n\u2234 5 divides a2<\/sup>.
\n5 divides a.
\n\u2234 We can write a = 5c.
\nSubstituting the value of ‘a’ in eqn. (i),
\n5b2<\/sup> = (5c)2<\/sup> = 25c2<\/sup>
\nb2<\/sup> = 5c2<\/sup>
\nIt means 5 divides b2<\/sup>.
\n\u2234 5 divides b.
\n\u2234 ‘a’ and ‘b’ have at least 5 as a common factor.
\nBut this contradicts the fact that a’ and ‘b’ are prime numbers.
\n\u2234 \\(\\sqrt{5}\\) is an irrational number.<\/p>\n

Question 2.
\nProve that \\(3+2 \\sqrt{5}\\) is irrational.
\nSolution:
\nLet us assume that \\(3+2 \\sqrt{5}\\) is an irrational number.
\nHere, p, q, \u2208 z, q \u2260 0
\n\"KSEEB
\n\\(\\sqrt{5}\\) is rational number.
\n\u2235 \\(\\frac{p-3 q}{2 q}\\) is rational number.
\nBut \\(\\sqrt{5}\\) is not a rational number.
\nThis contradicts the fact that,
\n\u2234 \\(3+2 \\sqrt{5}\\) is an irrational number.<\/p>\n

Question 3.
\nProve that the following are irrationals :
\ni) \\(\\frac{1}{\\sqrt{2}}\\)
\nii) \\(7 \\sqrt{5}\\)
\niii) \\(6+\\sqrt{2}\\)
\nSolution:
\ni) Let \\(\\frac{1}{\\sqrt{2}}\\) is a rational number.
\n\\(\\frac{1}{\\sqrt{2}}=\\frac{\\mathrm{p}}{\\mathrm{q}}\\)
\n\\(\\sqrt{2}=\\frac{q}{p}\\)
\nBy Squaring on both sides,
\n2 \u00d7 p2<\/sup> = q2<\/sup>
\n2, divides q2<\/sup>.
\n\u2234 2, divides q
\n\u2235 q is an even number.
\nSimilarly ‘p’ is an even number.
\n\u2234 p and q are even numbers.
\n\u2234 Common factor of p and q is 2.
\nThis contradicts the fact that p and q also irrational.
\n\u2234 \\(\\sqrt{2}\\) is an irrational number.
\n\u2234 \\(\\frac{1}{\\sqrt{2}}\\) is an irrational number.<\/p>\n

ii) Let \\(7 \\sqrt{5}\\)is a rational number.
\n\u2234 \\(7 \\sqrt{5}=\\frac{p}{q}\\)
\n\\(\\sqrt{5}=\\frac{p}{7 q}\\)
\nHere,\\(\\frac{p}{7 q}\\) is one rational number.
\nIt means \\(\\sqrt{5}\\) which is equal also a rational number.
\nThis contradicts to the fact that \\(\\sqrt{5}\\) is an irrational number.
\nThis contradicts to the fact that \\(7 \\sqrt{5}\\) is rational number.
\n\u2234 \\(7 \\sqrt{5}\\) is a rational number.<\/p>\n

iii) Let \\(6+\\sqrt{2}\\) is a rational number.
\n\"KSEEB
\n\\(\\frac{a-6 b}{b}\\) is a rational number, b
\n\u2234 \\(\\sqrt{2}\\) is also rational number.
\nThis contradicts to the fact that \\(\\sqrt{2}\\) is an irrational number.
\nThis contradicts to the fact that \\(6+\\sqrt{2}\\) is a rational number.
\n\u2234 \\(6+\\sqrt{2}\\) is an irrational number.<\/p>\n

We hope the given KSEEB SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Ex 8.3 will help you. If you have any query regarding Karnataka SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Exercise 8.3, drop a comment below and we will get back to you at the earliest.<\/p>\n","protected":false},"excerpt":{"rendered":"

KSEEB SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Ex 8.3 are part of KSEEB SSLC Class 10 Maths Solutions. Here we have given Karnataka SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Exercise 8.3. Karnataka SSLC Class 10 Maths Solutions Chapter 8 Real Numbers Exercise 8.3 Question 1. Prove that is irrational. …<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/1215"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=1215"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/1215\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=1215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=1215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=1215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}