{"id":4267,"date":"2023-12-21T05:56:39","date_gmt":"2023-12-21T00:26:39","guid":{"rendered":"https:\/\/wordpress-505192-1602719.cloudwaysapps.com\/?p=4267"},"modified":"2023-12-21T16:44:37","modified_gmt":"2023-12-21T11:14:37","slug":"samacheer-kalvi-12th-maths-solutions-chapter-3-ex-3-2","status":"publish","type":"post","link":"https:\/\/samacheerguru.com\/samacheer-kalvi-12th-maths-solutions-chapter-3-ex-3-2\/","title":{"rendered":"Samacheer Kalvi 12th Maths Solutions Chapter 3 Theory of Equations Ex 3.2"},"content":{"rendered":"

You can Download Samacheer Kalvi 12th Maths Book Solutions<\/a> Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.<\/p>\n

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 3 Theory of Equations Ex 3.2<\/h2>\n

Question 1.
\nIf k is real, discuss the nature of the roots of the polynomial equation 2x2<\/sup> + kx + k = 0, in terms of k.
\nSolution:
\nThe given quadratic equation is 2x2<\/sup> + kx + k = 0
\na = 2, b = k, c = k
\n\u2206 = b2<\/sup> – 4ac = k2<\/sup> – 4(2) k = k2<\/sup> – 8k
\n(i) If the roots are equal
\nk2<\/sup> – 8k = 0
\n\u21d2 k(k – 8) = 0
\n\u21d2 k = 0, k = 8
\n(ii) If the roots are real
\nk2<\/sup> – 8k > 0
\nk(k – 8) > 0
\nk \u2208 (-\u221e, 0) \u222a (8, \u221e)
\n(iii) If this roots are imaginary
\nk2<\/sup> – 8k < 0
\n\u21d2 k \u2208 (0, 8)<\/p>\n

Question 2.
\nFind a polynomial equation of minimum degree with rational coefficients, having 2 + \u221a3 i as a root.
\nSolution:
\nGiven roots is (2 + \u221a3 i)
\nThe other root is (2 – \u221a3 i), since the imaginary roots with real co-efficient occur as conjugate pairs.
\nx2<\/sup> – x(S.O.R) + P.O.R = 0
\n\u21d2 x2<\/sup> – x(4) + (4 + 3) = 0
\n\u21d2 x2<\/sup> – 4x + 7 = 0<\/p>\n

Question 3.
\nFind a polynomial equation of minimum degree with rational coefficients, having 2i + 3 as a root.
\nSolution:
\nGiven roots is (3 + 2i), the other root is (3 – 2i);
\nSince imaginary roots occur in with real co-efficient occurring conjugate pairs.
\nx2<\/sup> – x(S.O.R) + P.O.R = 0
\n\u21d2 x2<\/sup> – x(6) + (9 + 4) = 0
\n\u21d2 x2<\/sup> – 6x + 13 = 0<\/p>\n

\"Samacheer<\/p>\n

Question 4.
\nFind a polynomial equation of minimum degree with rational coefficients, having \u221a5 – \u221a3 as a root.
\nSolution:
\nThe given one roots of the polynomial equation are (\u221a5 – \u221a3)
\nThe other roots are (\u221a5 + \u221a3), (-\u221a5 + \u221a3) and (-\u221a5 – \u221a3).
\nThe quadratic factor with roots (\u221a5 – \u221a3) and (\u221a5 + \u221a3) is
\n= x2<\/sup> – x(S.O.R) + P.O.R
\n= x2<\/sup> – x(2\u221a5) + (\u221a5 – \u221a3) (\u221a5 + \u221a3)
\n= x2<\/sup> – 2\u221a5 x + 2
\nThe other quadratic factors with roots (-\u221a5 + \u221a3) (-\u221a5 – \u221a3) is
\n= x2<\/sup> – x (S.O.R) + P.O.R
\n= x2<\/sup> – x (-2\u221a5 ) + (5 – 3)
\n= x2<\/sup> + 2\u221a5x + 2
\nTo rationalize the co-efficients with minimum degree
\n(x2<\/sup> – 2\u221a5 x + 2) (x2<\/sup> + 2\u221a5 x + 2) = 0
\n\u21d2 (x2<\/sup> + 2)2<\/sup> – (2\u221a5 x)2<\/sup> = 0
\n\u21d2 x4<\/sup> + 4 + 4x2<\/sup> – 20x2<\/sup> = 0
\n\u21d2 x4<\/sup> – 16x2<\/sup> + 4 = 0<\/p>\n

Question 5.
\nProve that a straight line and parabola cannot intersect at more than two points.
\nSolution:
\nLet the standard equation of parabola y2<\/sup> = 4ax …..(1)
\nEquation of line be y = mx + c …(2)
\nSolving (1) & (2)
\n(mx + c)2<\/sup> = 4ax
\n\u21d2 mx2<\/sup> + 2mcx + c2<\/sup> – 4ax = 0
\n\u21d2 mx2<\/sup> + 2x(mc – 2a) + c2<\/sup> = 0
\nThis equation can not have more than two solutions and hence a line and parabola cannot intersect at more than two points.<\/p>\n

Samacheer Kalvi 12th Maths Solutions Chapter 3 Theory of Equations Ex 3.2 Additional Problems<\/h3>\n

Question 1.
\nFind a polynomial equation of minimum degree with rational co-efficients having 1 – i as a root.
\nSolution:
\nGiven root is 1 – i
\nThe other root is 1 + i
\nSum of the roots: 1 – i + 1 + i = 2
\nproduct of the roots: (1 – i) (1 + i) = (1)2<\/sup> + (1)2<\/sup> \u21d2 1 + 1 = 2
\n\u2234 The required polynomial equation of minimum degree with rational coefficients is
\nx2<\/sup> – x (S.R.) + (P.R.) = 0
\nx2<\/sup> – 2x + 2 = 0<\/p>\n

Question 2.
\nFind a polynomial equation of minimum degree with rational co-efficients having \\(\\sqrt{3}+\\sqrt{7}\\) as a root.
\nSolution:
\n\"Samacheer
\nThe required polynomial equation of minimum degree
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 3.
\nIf the roots of the equation x3<\/sup> + px2<\/sup> + qx + r = 0 are in A.P then show that 2p3<\/sup> – 9pq + 27 r = 0.
\nSolution:
\nLet the roots of the given equation is a – d, a, a + d
\n\"Samacheer<\/p>\n

Question 4.
\nSolve 27x3<\/sup> + 42x2<\/sup> – 28x -8 = 0 given that its roots are in geometric progressive.
\nSolution:
\n\"Samacheer<\/p>\n

Question 5.
\nSolve the equation 15x3<\/sup> – 23x2<\/sup> + 9x – 1 = 0. Where roots are in H.P.
\nSolution:
\n\"Samacheer
\n\"Samacheer<\/p>\n","protected":false},"excerpt":{"rendered":"

You can Download Samacheer Kalvi 12th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations. Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 3 Theory of Equations Ex 3.2 Question 1. If k is real, discuss the nature of the roots of the […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","footnotes":""},"categories":[1],"tags":[],"blocksy_meta":"","jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/posts\/4267"}],"collection":[{"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/comments?post=4267"}],"version-history":[{"count":1,"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/posts\/4267\/revisions"}],"predecessor-version":[{"id":55304,"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/posts\/4267\/revisions\/55304"}],"wp:attachment":[{"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/media?parent=4267"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/categories?post=4267"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/samacheerguru.com\/wp-json\/wp\/v2\/tags?post=4267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}