(ii) Q.2 Reduce the following matrix to normal form and find its rank.

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1 B. E. Semester- (All Brach ecept Aero. Questio Bak Each questio has marks. Q. Fid the rak o 9 Q. Reduce the ollowig matri to ormal orm ad id its rak. Q. Fid iverse o a matri b Gauss Jorda Reductio method Q. Eamie or cosistec the ollowig equatios ad solve them i the are cosistet: 9 9 Q. Fid the eigevalues & eigevectors o the matri 8 Q. Evaluate d cos d si Q. Prove that cos d si d Q.8 Evaluate d cos si Q.9 I d ta show that ad the deduce the value o. Q. I,( cot d show that.hece evaluate d cot. Q. Fid the legth o the arc o the parabola a rom the verte to oe etremit o the latus rectum. Q. Fid the perimeter o the cardioid a cos r

2 B. E. Semester- (All Brach ecept Aero. Questio Bak Each questio has marks. Q. Fid the whole legth o lemiscate r a cos. Q. Fid the area betwee the curve a ad its asmptote. Q. Fid the area o the loop o the curve a. Q. e si Evaluate lim lim Q. ta Evaluate lim lim a b Q.8 Evaluate lim log ta ta lim cot cot Q.9 Evaluate lim log lim cot Q. Q. I Fid the, id. th derivative o ta si Q. I acos(log bsi(log, prove that ( (. Q. Q. I I cot prove that ( (. ( (,prove that ( (. Q. I e mcos show that ( ( ( m. Q. Epad ( sec i power o up to b maclauri`s series. Q. si Prove that =... Q.8 Test the covergece o the Sequeces l,

3 B. E. Semester- (All Brach ecept Aero. Questio Bak Each questio has marks. Q.9 Fid the sum o the series i it coverges Q. Test the covergece o the series,... Q. Discuss the covergece o, Q. Discuss the covergece o p-series. Q. Discuss the covergece o (a l (b ta. Q. Discuss the covergece o (a (b!e (c l Q. Discuss the covergece o (a (b Q. Veri that w w Where w e l l, w =si.cos + cos.si Q. Epress u ad v as uctio o u ad v both b usig the Chai Rule e l, l( ucos v, usi v

4 B. E. Semester- (All Brach ecept Aero. Questio Bak Each questio has marks. Q.8 Epress u ad v as uctio o u ad both b usig the Chai Rule v ta, u cos v, u si v Q.9 Epress u ad as uctio o u ad v both b usig the Chai Rule v w, u v, u v, uv; ( u, v /, Q. Fid the taget plae ormal lie at the poit P o the give suraces. P (,, P (,, Q. Fid the taget plae ormal lie at the poit P o the give surace. cos e P (,, Q. Fid a equatio or the plae that is taget to the give surace at the give poit. l(, (,, ( e, (,, Q. The volumev r h o a right circular clider is to be calculated rom measured values o r ad h. Suppose that r is measured with a error o o more tha % ad h with a error o o more tha.%. Estimate the resultig possible percetage error i the calculatio ov. Q. Fid the lieariatio L(, o the uctio at each poit. (, at (, (, e cos at (, / Q. Fid the local etreme values o the uctio (, Q. Fid local maima & local miima ad saddle poits o the uctio (, Q. Fid local maima & local miima ad saddle poits o the uctio

5 B. E. Semester- (All Brach ecept Aero. Questio Bak Each questio has marks. (, Q.8 Fid local maima & local miima ad saddle poits o the uctio (, Q.9 (a Epress the ollowig comple umbers ito Polar orm a + i b+ +i (b I ad are real,solve the ollowig equatio: i i i Q. (cos i si (cos i si Simpli (cos i si (cos i si cos i si Prove that ( si i cos cos8 i si 8 Q. si i cos Prove that ( cos( isi( si i cos Q. Prove that th roots o uit orm a G.P. ad sum o the roots is ero ad product o the roots is (- - Q. Solve the equatios + + +=, = usig De Moivre s theorem. Q. d (a Fid the solutio to the dieretial equatio ( ( give that = e d whe = d (b Show that =acos+si is asolutio o cos si d Q. Q. (a Solve : d d Solve ( e d e ( d, (b Solve : d ( d Q. d d (a Solve : (b Solve d log, d Q.8 (a Evaluate / e - d (b Evaluate ( d

6 Each questio has marks. GUJARAT UNIVERSITY B. E. Semester- (All Brach ecept Aero. Questio Bak Q.9 Fid Γ(-½

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