{"id":11519,"date":"2024-03-09T11:33:50","date_gmt":"2024-03-09T06:03:50","guid":{"rendered":"https:\/\/tsboardsolutions.in\/?p=11519"},"modified":"2024-03-12T17:49:09","modified_gmt":"2024-03-12T12:19:09","slug":"ts-inter-2nd-year-maths-2a-de-moivres-theorem-important-questions","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.in\/ts-inter-2nd-year-maths-2a-de-moivres-theorem-important-questions\/","title":{"rendered":"TS Inter 2nd Year Maths 2A De Moivre\u2019s Theorem Important Questions"},"content":{"rendered":"

Students must practice these\u00a0TS Inter 2nd Year Maths 2A Important Questions<\/a> Chapter 2 De Moivre\u2019s Theorem to help strengthen their preparations for exams.<\/p>\n

TS Inter 2nd Year Maths 2A De Moivre\u2019s Theorem Important Questions<\/h2>\n

Question 1.
\nSimplify \\(\\frac{(\\cos \\alpha+i \\sin \\alpha)^4}{(\\sin \\beta+i \\cos \\beta)^8}\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 2.
\nIf m,n are integers and x = cos \u03b1 + i sin \u03b1, y = cos \u03b2 + i sin \u03b2 then prove that
\nxm<\/sup> yn <\/sup>+ \\(\\frac{1}{x^m y^n}\\) = cos (m\u03b1 +n\u03b2) and
\nxm<\/sup> yn <\/sup>– \\(\\frac{1}{x^m y^n}\\) = 2i sin (m\u03b1 +n\u03b2)
\nSolution:
\n\"TS
\nQuestion 3.
\nIf n is a positive Integer, show that \\((1+i)^n+(1-i)^n=2^{\\frac{n+2}{2}} \\cos \\left(\\frac{n \\pi}{4}\\right)\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 4.
\nIf n is an Integer then show that
\n(1 + cos \u03b8 + i sin \u03b8)n<\/sup> + (1 + cos \u03b8 – i sin \u03b8)n<\/sup> \\(=2^{n+1} \\cos ^n\\left(\\frac{\\theta}{2}\\right) \\cos \\left(\\frac{n \\theta}{2}\\right)\\)
\nSolution:
\nL.H.S
\n(1 + cos \u03b8 + i sin \u03b8)n<\/sup> + (1 + cos \u03b8 – i sin \u03b8)n<\/sup>
\n\"TS<\/p>\n

Question 5.
\nIf cos \u03b1+cos \u03b2 + cos \u03b3 = 0 = sin \u03b1 + sin \u03b2 + sin \u03b3, Prove that cos2<\/sup> \u03b1 +cos2<\/sup> \u03b2 +cos \u03b3 = \\(\\frac{3}{2}\\) sin2<\/sup> \u03b1 + sin2 <\/sup>\u03b2 + sin2<\/sup> \u03b3.
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 6.
\nFind all the values of \\((\\sqrt{3}+i)^{1 \/ 4}\\)
\nSolution:
\nThe modulus amplitude form of
\n\"TS
\nQuestion 7.
\nFind all the roots of the equation
\nx11 <\/sup>– x7 <\/sup>+ x4 <\/sup>-1 = 0
\nSolution:
\nx11 <\/sup>– x7 <\/sup>+ x4 <\/sup>-1\u00a0 = x7<\/sup>(x4<\/sup>-1) +1 (x4<\/sup>– 1) = (x4<\/sup>-1)(x7<\/sup>. 1)
\nTherefore the roots of the given equations are precisely the roots of unity and 7th roots of – 1.
\nThey are cis = \\(\\frac{2 \\mathrm{k} \\pi}{4} \\) = cis \\(\\frac{\\mathrm{k} \\pi}{4}\\) k\u2208{0,1,2,3} and
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 8.
\nIf 1, \u03c9, \u03c92<\/sup> are the cube roots of unity, prove that
\n\"TS
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 9.
\nIf \u03b1, \u03b2 are the roots of the equation x2<\/sup> + x + 1 = 0 then prove that \u03b14<\/sup> + \u03b24<\/sup> + \u03b1-1<\/sup> = \u03b2-1<\/sup>
\nSolution:
\nSince \u03b1, \u03b2 are the complex cube roots of unity,
\nwe may take \u03b1 = \u03c9, \u03b2 = \u03c92
\n\"TS<\/sup><\/p>\n","protected":false},"excerpt":{"rendered":"

Students must practice these\u00a0TS Inter 2nd Year Maths 2A Important Questions Chapter 2 De Moivre\u2019s Theorem to help strengthen their preparations for exams. TS Inter 2nd Year Maths 2A De Moivre\u2019s Theorem Important Questions Question 1. Simplify Solution: Question 2. If m,n are integers and x = cos \u03b1 + i sin \u03b1, y = … Read more<\/a><\/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_featured_media_url":"","_links":{"self":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/11519"}],"collection":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/comments?post=11519"}],"version-history":[{"count":4,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/11519\/revisions"}],"predecessor-version":[{"id":11609,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/11519\/revisions\/11609"}],"wp:attachment":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/media?parent=11519"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/categories?post=11519"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/tags?post=11519"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}