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DEPARTMENT OF MATHEMATICS “T3 Examination, May-2018� 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?‘“ (đ?‘Ľ) = đ?‘Ľ 4 + 15đ?‘Ľ 2 + 7đ?‘Ľ − 11 = 0 (c) State Fundamental Theorem of Algebra. (d) Form a quadratic equation whose one root is 1+i. (e) If Îą, β, Îł are roots of the equation 4đ?‘Ľ 3 + 16 đ?‘Ľ 2 − 9đ?‘Ľ − 36 = 0. Then find ιβ + βγ + γι 1 1 1

(f) If Îą, β, Îł are roots of the equationđ?‘Ľ 3 + đ?‘žđ?‘Ľ + đ?‘&#x; = 0. Then find cubic equation whose roots aređ?›ź , đ?›˝ , đ?›ž. (g) Factorizeđ?‘Ľ 4 + 64. (h) Find least possible number of imaginary roots of the equationđ?‘Ľ 9 − đ?‘Ľ 5 + đ?‘Ľ 4 + đ?‘Ľ 2 + 1 = 0. (i) Explain Descarte’s Method of solving the biquadratic equation. (j) Diminish the roots of the equation đ?‘Ľ 5 + 4đ?‘Ľ 3 − đ?‘Ľ 2 + 11 = 0 by 3. PART - B Q.2 (a) Solve the equationđ?‘Ľ 5 + đ?‘Ľ 4 − 7đ?‘Ľ 3 + 11đ?‘Ľ 2 − 4đ?‘Ľ − 2 = 0, it being given that one of its roots is −2 + √3

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(b) Solve the equation đ?‘Ľ 3 − 12đ?‘Ľ 2 + 39đ?‘Ľ − 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?‘Ľ 4 + đ?‘?đ?‘Ľ 3 + đ?‘žđ?‘Ľ 2 + đ?‘&#x;đ?‘Ľ + đ?‘ = 0 is equal to zero. (8) (b) Find the condition that two roots of the equation đ?‘Ľ 3 − đ?‘?đ?‘Ľ 2 + đ?‘?đ?‘Ľ − đ?‘‘ = 0be equal. *******

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Q.4 (a) Solve the equation đ?‘Ľ 4 − 9đ?‘Ľ 2 + 4đ?‘Ľ + 12 = 0, given that it has a multiple root. (b)Solve the equation 4đ?‘Ľ 3 − 4đ?‘Ľ 2 − 15đ?‘Ľ + 18 = 0,two of its roots being equal.

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

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Q.6 (a) Find the equation whose roots are squared differences of the roots of the equation đ?‘Ľ 3 + 6đ?‘Ľ 2 + 2 = 0. (7) (b) If Îą, β, Îł are roots of the equation đ?‘Ľ 3 − đ?‘Žđ?‘Ľ 2 + đ?‘?đ?‘Ľ − đ?‘? = 0. Form an equation whose roots are β+Îą, Îł+Îą, Îą+β

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Q.7(a) Solve by Cardan’s Methodđ?‘Ľ 3 − 12đ?‘Ľ − 65 = 0 (b) Solve by Descarte’s method đ?‘Ľ 4 − 3đ?‘Ľ 2 − 42đ?‘Ľ − 40 = 0

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