Kurukshetra University 2008 B.A Mathematics Algebra and Trignometry - Question Paper
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BAE/A08 ALGEBRA AND TRIGONOMETRY Paper-BM-101 (B.A.: 30 : jB.Sc.: 45 (Maximum Marks rune : Three Hours) Note : Aucmpt any five questions, selecting at least one question from each scction. SKCTION-I Define equivalence relation in a set. In the set ,of integrals, let a relation R be defined as iff a - b is even. Prove that R is an equivalence relation. 3(414) For what values of a and b the equations .t + y + 5z - 6 a 0; x + 2> 4- 3aj - b = 0; x + 3y + az - 1=0 have (i) no solution, (ii) unique solution, (iii) infinite number of solutions ? 3(414) I. (a) (b) Define Symmetric and Skew-symmetric matrices. Show that the value of the determinant of a skew-symmetric matrix of odd order is always zero. 3(414) (a) I I \ I -1 -I .3 1 *. P and Q such tha hence find rank of A. then find non-singular matrices <b) If A PAQ is in the normal form and 3(4W) (P.T.O. 3I/8.000/KD/398 |
X (a) The rank of ihe product o' iwo matrices cannot cxcccd the rank of either matrix Prove it 3(4t4) (h) find the characteristic roots of the matrix 10 2' 31 A = O 2 I 2 0 I | Find also the corresponding characteristic vcctors of A. 3(4V4) SECTION-II 4. (a) Solve the equation l-S-M.r'+St- 1=3. given that the roots are in II.P 3(4W) 1 bj Solve the equation a*1 - 6.t + 9 - 0 by (union's method. 3(4,4) 5. (a) II the product of two loots of the cqtMiiun X* + pX* + tj.X2 * be equal in magnitude hut opposite in sign of the product of the other two. show that p's + r* s 4tfs. 3(4!<4) (h) Find Ihe equation of squared differences of the roots of the equation r * 3.i * 2 - A 3f4,-4) SK(TION-lll 6. (a) Show that the set G = (fl+//2 ,/>Q| * ="> infinite abelian group w.r.t. usual addition. 3(4'/) (h) Define Kernel of homomorphism. Let 0 be a homomorphism from group G into group G\ Show that the homomofphism 0 is an isomorphism of G into G' if ami only if Ker 6 = <r}. 3(414) 3I/8.000/KD/398 2 |
7. (a) Prove that the order of each subgroup of a finite group
divides the order of the group. 3(4W)
(b) Define Permutation. Cyclic permutation. Even and Odd-permuation. t.ci S 11. 2. 3. 4, 5) and/s(2. 3), ij 4) then show that fog gof- 3(414)
8. (a) Define Ring and Subring. Show that the set of matrices
where o. be: is a subring of the ring of all 2 x2 matrices over integers under usual addition and multiplication of matrices.
(b) Prove that an Arbitrary intersection of subrings is a subring. 3(4Vi)
SECTION-IV
9. (a) State and prove Demoivre's Theorem. 3(4V'j) (b) Sum the series
I + sin a. cos jl+ cos 2P+ cos 30+.....to
3(4 W)
10. (a) Prove hat
<*) * cos 8-/ sin0
3(4*4)
whcic 0 <4 j '
(b) Prove that
logum| + -/j = itan''(sinh.t).
3(4*)
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