DEPARTMENT OF MATHEMATICS â&#x20AC;&#x153;T3 Examination, May-2018â&#x20AC;? Semester:2nd Subject:Algebra Branch: Maths Course Type:Core Time: 3 Hours Max.Marks: 80

Date of Exam:21/05/2018 Subject Code:MAH115-T Session: I Course Nature:Hard Program: B.Sc Signature: HOD/Associate HOD:

Note: All questions are compulsory from part A (2*10 = 20 Marks). Attempt any two questions from Part B (15 Marks each). Attempt any two Questions from Part -C (15 Marks each). PART -A Q.1 (a) Explain descarte Rule of sign for roots of equation f(x) = 0. (b) Find the nature of roots of the equation đ?&#x2018;&#x201C; (đ?&#x2018;Ľ) = đ?&#x2018;Ľ 4 + 15đ?&#x2018;Ľ 2 + 7đ?&#x2018;Ľ â&#x2C6;&#x2019; 11 = 0 (c) State Fundamental Theorem of Algebra. (d) Form a quadratic equation whose one root is 1+i. (e) If Îą, Î˛, Îł are roots of the equation 4đ?&#x2018;Ľ 3 + 16 đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 9đ?&#x2018;Ľ â&#x2C6;&#x2019; 36 = 0. Then find ÎąÎ˛ + Î˛Îł + ÎłÎą 1 1 1

(f) If Îą, Î˛, Îł are roots of the equationđ?&#x2018;Ľ 3 + đ?&#x2018;&#x17E;đ?&#x2018;Ľ + đ?&#x2018;&#x; = 0. Then find cubic equation whose roots aređ?&#x203A;ź , đ?&#x203A;˝ , đ?&#x203A;ž. (g) Factorizeđ?&#x2018;Ľ 4 + 64. (h) Find least possible number of imaginary roots of the equationđ?&#x2018;Ľ 9 â&#x2C6;&#x2019; đ?&#x2018;Ľ 5 + đ?&#x2018;Ľ 4 + đ?&#x2018;Ľ 2 + 1 = 0. (i) Explain Descarteâ&#x20AC;&#x2122;s Method of solving the biquadratic equation. (j) Diminish the roots of the equation đ?&#x2018;Ľ 5 + 4đ?&#x2018;Ľ 3 â&#x2C6;&#x2019; đ?&#x2018;Ľ 2 + 11 = 0 by 3. PART - B Q.2 (a) Solve the equationđ?&#x2018;Ľ 5 + đ?&#x2018;Ľ 4 â&#x2C6;&#x2019; 7đ?&#x2018;Ľ 3 + 11đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 4đ?&#x2018;Ľ â&#x2C6;&#x2019; 2 = 0, it being given that one of its roots is â&#x2C6;&#x2019;2 + â&#x2C6;&#x161;3

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(b) Solve the equation đ?&#x2018;Ľ 3 â&#x2C6;&#x2019; 12đ?&#x2018;Ľ 2 + 39đ?&#x2018;Ľ â&#x2C6;&#x2019; 28 = 0, the roots being in A.P

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Q.3 (a) Find the condition that sum of two roots of the equation đ?&#x2018;Ľ 4 + đ?&#x2018;?đ?&#x2018;Ľ 3 + đ?&#x2018;&#x17E;đ?&#x2018;Ľ 2 + đ?&#x2018;&#x;đ?&#x2018;Ľ + đ?&#x2018; = 0 is equal to zero. (8) (b) Find the condition that two roots of the equation đ?&#x2018;Ľ 3 â&#x2C6;&#x2019; đ?&#x2018;?đ?&#x2018;Ľ 2 + đ?&#x2018;?đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x2018;&#x2018; = 0be equal. *******

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Q.4 (a) Solve the equation đ?&#x2018;Ľ 4 â&#x2C6;&#x2019; 9đ?&#x2018;Ľ 2 + 4đ?&#x2018;Ľ + 12 = 0, given that it has a multiple root. (b)Solve the equation 4đ?&#x2018;Ľ 3 â&#x2C6;&#x2019; 4đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 15đ?&#x2018;Ľ + 18 = 0,two of its roots being equal.

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PART -C Q.5 (a) Solve the equation 6đ?&#x2018;Ľ 3 â&#x2C6;&#x2019; 11đ?&#x2018;Ľ 2 + 6đ?&#x2018;Ľ â&#x2C6;&#x2019; 1 = 0 if roots are in H.P. (b) Find the equation whose roots are the squares of the roots of đ?&#x2018;Ľ 3 + đ?&#x2018;&#x17E;đ?&#x2018;Ľ + đ?&#x2018;&#x; = 0.

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Q.6 (a) Find the equation whose roots are squared differences of the roots of the equation đ?&#x2018;Ľ 3 + 6đ?&#x2018;Ľ 2 + 2 = 0. (7) (b) If Îą, Î˛, Îł are roots of the equation đ?&#x2018;Ľ 3 â&#x2C6;&#x2019; đ?&#x2018;&#x17D;đ?&#x2018;Ľ 2 + đ?&#x2018;?đ?&#x2018;Ľ â&#x2C6;&#x2019; đ?&#x2018;? = 0. Form an equation whose roots are Î˛+Îą, Îł+Îą, Îą+Î˛

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Q.7(a) Solve by Cardanâ&#x20AC;&#x2122;s Methodđ?&#x2018;Ľ 3 â&#x2C6;&#x2019; 12đ?&#x2018;Ľ â&#x2C6;&#x2019; 65 = 0 (b) Solve by Descarteâ&#x20AC;&#x2122;s method đ?&#x2018;Ľ 4 â&#x2C6;&#x2019; 3đ?&#x2018;Ľ 2 â&#x2C6;&#x2019; 42đ?&#x2018;Ľ â&#x2C6;&#x2019; 40 = 0

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