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Ky } rd) Pi. LBA HVE VHANM. DOT I EXAMINATION Firat Your HW MEDLAL MATHEMATICS (HWutive fram the Hdtitied bateh of 2008-2009) Nimes d Howes Max. Marks: 70 PART .A (10x 2 = 20) All questions are tompulsory. Waoh question carries 3 marks. Rio Oo a c l, (A) Mind the determinate of the matrix | 2 ; : ec (bh) Prove that (tan 04 e6t0)’ = sec? 0+ cosec’6. (0) Vind the value of k, if the straight lines (ve l0y+h=0 and kx =5y+8=0 are parallel. x ,.. 6 =8in#x=1 1) Compute in —————, (dl) iy im ¥ (0) /(W) @ log(eeow +tanx), Find f(x). dy (0 If ve qaeos"l, y=asin’t find. ain’ # (w) lvaluate Fre recat [P.7.0.] " With epee os ve > ae na iriangle ABO, If a oA ebeoal, then —™ ghow that the triangle ie elther isoscalen ov pight anaglod, i) (i) Show that tan TH tan HO an 75" (li) Evaluate cow 240° + win 420° « cones 60° 4 tan 30°, (a) Find the equation of the straight line passing through the point (<8,10) and parpandicular to dv4+ 2941060, (b) Find the equation of the circle which passes through (- 71) and has centre (=4)-4), Find also the radiue of that circle, coun Jeain’» (a) Evaluate (i) dx (ii) | whe dx, (b) Evaluate (i) Lt tsanl (i) ba If y= sot and xe at", State the Buler's theorem on homogeneous functions of two variables, Verify this theorem for = x? +3x"y- axy' + y". (b) If ye Tan "x, (+ x*bty 12x00, (a) that thin Prove Difforentiate Laibnitz’s theorem, [PD a, 1.6A} [PD - 1.64] I/VI Pharm. D DEGREE EXAMINATION. First Year REMEDIAL MATHEMATICS Mtfective from the admitted batch of 2008-2009) Time : Three hours et Maximum : 70 marks PART A — (10 x 2 = 20 marks), Answer ALL questions 1 Nae Each question carries 2 marks. la+b oa Bil kieaah isi) 1. (a) Showthat| a@ a+e ec |=4abc. b c bte} (b) In a AABC, if A+C = 8% ‘ghow: that tan Atan BtanC = tan B-tanA~tanC . If the points (kl 1),(2,15) and (-3,-5) are _ collinear, then find the value of k. 5 x3 +8 Byadpate le x92 X— 20 dy dx Tf y= Ioete™* ai sin Ax), then find — 0) (g) (h) (i) G) (a) (b) \ @) spo nae 4 ; : : oe We «N Differentiate 3¥x rm with respect to » 9 Sho oct O “an 08 22: dy pve ° If y =x" sinx then find me : | @) “a 30°: | tio? | e eae Evaluate j cae (hth | pint ch che po" l+sinx ) (a) prove -0= | 2 Find the order and degree of the differential xo 3* equ! dy)’ (d*y) Find i" : equation (2) ($3) + 6x=0. | (p) grove padi? Find the Laplace transform of t°e™. yuate (3) wai PART B — (5 x 10 = 50 marks) (a) ¥ 1 Answer any FIVE questions. ) gua? ( Each question carries 10 marks. =, or : ee If a,b,c are all different from zero and a thi a a’ l+a® (a) s ctio™ 6. iw b b? 14+b°|/=0 , then prove that abc=1. for “ e¢ o° “De® : oF 2 Solve the system of equations x+2y+3z=6, iad Qx+4y+z=7 and 3x+2y+9z=14 using equ matrix inversion method. [PD - = =, He TE A= 6 Be =k(a-b)(b-c)(e-a) then le. et ' that : find k mf : 4) Find the circum radius of the triangle with sides 13,14 and 15. 5) In AABC, if b+e:e+a:a+b=11:12:18 a then find cos A: cos B: CosC. .. PRvalsate lim sin x(sec.x —1) x90 a3 P= Da h 1+ x? ay dy (b) Tf y=Tan (ee) then find a + “Ifx=asec@ and y=btand, then find = =sin'"x Prove that (1-x?)y, —xy,=0 (1 . "yeas —(2n +1)2¥ p41 -n*y, =0. rate 4 6. (a) Show that [asin’ ada =—. o (b) Evaluate (i) [xlogxdx i! 1 (i) leven (a) Solve the differential equation (e* + Dydy +(y + dx =0. (b) Solve the differential equation 9.. ay cos” ¥.—+y=tanx, dire (a) Find the Laplace transform of the function F(t) =(S int-Cost)?. (b) Evaluate L{F()} if F(t) -{@- to1 0 O eet find Ae 5), | p (a) Using Euler's theorem show that at ps Yup : : fy xu, + Uy = Stanx for the function i} . ()} oe xt+y of C4) “a Vx+Jy)' : x (b) Compute im hae I. Ns 2 ny ; wi 6. (a) Evaluate fxsin Iydx ; 8 x (b) Evaluate ee : inx+bcosx 7. (a) Evaluate See. : sinx+cosx (b) Evaluate [log (1+tan0)H9. feo 3 Fo tatiecde “> tye c + En a ty > = * ee es | ey, : 6. (a) Evaluate | = ae dx. cos’ \xe (b) Evaluate J xlog xd. nla cos’? x 7. a) Evaluate | - —dx . ss J sin’? x+cos”’ x ' —— . (b) Evaluate ioe Road dx. ‘ 4 1+x he F 4 d ; ee 8. (a) _ Solve (cos x)ot ysinx =tanx. i he . § (b) Find L(te* +92 +5cost+7i° + 5sin3t+ 2). =. | 5 4 Ss 
