(6 pages) OCTOBER 2013 U / I D 4 6 4 3 1 /U C C ATime: Three hours Maximum: 100 marks

PART A (10 x 3 = 30 marks). Answer any TEN questions.

All questions carry equal marks.1. Find the inverse of the function : -* R,

GjBiTLDrrgu srrswra.

2. State the pigeonhole principle. iSlggluSlar Gjatotro)

5. Define reflection of .R2 - sir iS^ rrsjr tS)0 CT65T j)g)|&|.

7. Find the angle between the pair of lines. 12x 2 - 10:cy + 2/ + 1 lx - 5;y + 2 = 0.12x2 - lOxy + 2 y 2 + l lx - 5y + 2 = 0 ctotjd 9 0 Ggrruj.Grr(0^ l6if)i_Cuj -OTrr C(T6wrib rrwr.

8. Find the eccentricity of the ellipse 3x2 + 4y2 - 12x + 24y + 36 = 0.3x2 + 4y2 - 12x + 24y + 36 = 0 ermp jgdrojiLl&Ssjt eccentricity arrafer*.

9. Find -|-(x*).

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10. Find D2(sinx.cosjc).D2(sinx.cosx) mu arrears.

11. Write the half sine series of the function f{x) in(o,*).(0,;r) crairjD @a)L_Q6uofluSlaj. f(x) erarp rS^ enijoffn Qjsrn_5>tj CT(ipg].

12. Solve p 2 +4p + 5 = 0. j|nr p2 + 4p + 5 = 0.

PART B ( 5x6 = 30 marks)Answer any FIVE questions.

All questions carry equal marks./13. Prove that (A u f i) = A 'n B' for any two sets.fgGjSgiiib @ 0 awrmi(g50 ( ) = n B ' gran

jftjpiQjs.14. Show that composition of two onto functions is onto.

ggijafer Gica) aFrrrrqaflT Q^rr^uqu) $ 0 (Sldco s=rr(Ti_|ctft j)g)j&ja>.

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15. Find the inverse of the matrix A = 1 - 1 2 0 1 30 - 2 4A = 1 - 1 2 0 1 30 - 2 4 CTQiTfD e^sofluSlsrr Gr&iTLDftp arrears.

16. Find the standard matrix of reflection about x-axis on R 2.

R2 - arr iS r^rotr x QurrjD#$ i% $Iu sSIul9ot pysirotrs.

17. Obtain the polar equation of circle.&JiLi_$$ljT QurrcomT ffiDefrumlenL. srrawra.

18. Find 4~(*x ") ax$duj$ rrRjr.

19. Solve ( D2 + 4D + 13)y = sin2 x .ip 2 + 4 0 + 13)^ = Sin2 - 65T ^ ITL| rt6WT.

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PART C (4 x 10 = 40 marks) Answer any FOUR questions.

All questions carry equal marks.20. Find the dual canonical form of x.y.z'+[{x + z)(x +y')].

x.y.z' + [(x + z)(x + y')] - QuiTienir j)ujld6t ^ aDLDuaoua arrawra.4 5 - 6 721. Express A =matrices.

4 5-6 7^arflsoflaiT Qu0saeorrs CTQpgia.

as product of elementary

A = ot(T(D cSiexiflemij Q^tn_ f)sr>co

22. Find the evolute of an ellipse. j$t)jl_l-$)6ot evolute >uj rrcwr.

K\23. Find the reduction formula for / = I cos" xdx.o

K\I n - Jcos" x dx - dst6urriuuurril()L_ arrewra.o

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24. Find the shortest distance betweenx - 4 _ y - 5 _ z - 3 _ 3 _ 3 - 4__ - an 7 " 8 - 5 'Also find the equation of the line of shortest distance.

x - 4 y - 5 z - 3 . . x + 2 y + 3 3 - 4_ = _ o w n = = OTatrjD GiT(Sln)j uilff CSamlu^ du ffioaTLjrrQib

25. Solve (2 x 2D 2 + 3 xD + 7)y = log*.jgiir (2*2D2 + 3 xD + l )y = log*.

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