ENGINEERING MATHEMATICS I QUESTION BANK. Module Using the Leibnitz theorem find the nth derivative of the following : log. e x log d.

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1 ENGINEERING MATHEMATICS I QUESTION BANK Modle Usig the Leibit theoem id the th deivtive o the ollowig : b si c e d e Show tht d d! Usig the Leibit theoem pove the ollowig : I si b the pove tht b I si show tht c I b the pove tht d I sih the pove tht e I si the pove tht I the pove tht g I the pove tht h I mt d mt si the pove tht m i I m m show tht m j I m pove tht m d hece show tht m

2 Pol Cves Fid the gle betwee the dis vecto d the tget o the ollowig cves m m m si m l e Fid the gle betwee the dis vecto d the tget d lso id the slope o the tget s idicted o the ollowig cves t b c si t Show tht the ollowig pis o cves itesect ech othe othogoll, b b si, si c, b si t d e, e b Fid the gle o itesectio o the ollowig pi o cves si d si b d c, d d e d d 5 Fid the pedl eqtio o the ollowig cves b c sec d

3 m m l e m si m e Deivtive o c legth ds Fid d ds Fid d whe c h c whe ds Fid dt o the cve t t e si t, e t Fid 5 Fid ds ds ds,,, o the ccloid si, d d d ds ds, d d o the cve k Fo the cve show tht k Rdis o Cvte Fid the dis o cvte o the cve h Fid the dis o cvte o the cve ds is tt d t the poit, t the poit, o it Fid the dis o cvte o the cve -is t the poit whee the cve meets the Fid the dis o cvte o the cve t the poit whee it cts the lie pssig thogh the oigi mkig gle 5 with the -is 5 Fid the dis o cvte o the cve t the poits whee the tgets e pllel to the -is Fid the dis o cvte o the cve sec t t t, sec t 7 Fid the dis o cvte o the stoid, si t 8 Show tht the dis o cvte t poit o the ccloid si, is

4 9 I d be the dii o cvte t the ed poits o the ocl chod o the pbol, the show tht Show tht the dis o cvte t poit o the cdioide vies s Show tht o the cve vies s Pove tht t the poits i which the Achimedi spil itesects the hpebolic spil, thei cvtes e i the tio : Fo the cve show tht b I d be the dii o cvtes t the etimities o the pol chod o the cdioide, show tht 9 Wite the p eqtio o the pol cve si d id the dis o cvte to the cve 5 Show tht o the ellipse i the pedl om p b b the dis o cvte t the b poit p, is p Poblems o Idetemite oms : Modle Evlte the ollowig limits : e sih si lim 5 lim t lim lim cot t sec t lim 7 lim si b lim 8 lim t 9 lim t lim e

5 lim lim t lim t t b ì ü lim í ý î þ 5 lim sec t lim cot 7 lim cot 8 lim t 9 lim [ ]cot lim lim si lim t Fid the vles o the tts,b,c sch tht 5 lim b c lim lim lim e e lim b si Fid the tts d b so tht lim Tlo s d Mcli s seies limt t t lim sih lim lim t b ce si m be eql to Obti the Tlo s epsio o bot the poit = pto the tem cotiig oth degee tems d hece evlte e Obti the Tlo s epsio o bot the poit Epd si i powes o Epd t 5 Epd h i powes o - pto oth degee tems si i scedig powes o h pto the tem cotiig ppoimte vle o si 5 pto the oth degee tems h d hece dedce

6 Pove tht h h h h 7 Epd e si sig Mcli s theoem pto the tems cotiig 8 Epd i powes o b Mcli s theoem pto the tems cotiig 9 Epd t sig Mcli s theoem t pto d degee tems d hece id t Usig the Mcli s seies pove tht si Pove tht si Pove tht si Obti the Mcli s epsio o d hece dedce tht 5 5 Ptil Dieetitio I e show tht i t si si ep show tht t I t vei tht I e vei tht 5 I the pove tht I ep v pove tht v v v = 7 I ep b b the pove tht b 8 I t t the show tht t 9 I the show tht I t t t the show tht si si si

7 I the pove tht d hece dedce tht 9 I c b the show tht I si pove tht ïþ ï ý ü ïî ï í ì I tht show the whee 5 I si, d the pove tht I, the show tht e e Homogeeos ctios 7 Stte d pove Eles theoem d hece dedce tht 8 Vei Eles theoem o the ollowig ctios b c 9 I the show tht 5 I tht show the t I tht show the ec sec I, tht pove the I tht pove the I tht show the t si si 5 I ú û ù ê ë é sec t tht show the ec Jcobis Fid the Jcobi o v w with espect to give tht =++, v=+, w=

8 7 Fid the Jcobi o v w with espect to give tht, v, w, v, w 8 I, v, w id J,,,, 9 I, v, vw the id the vl e o J, v, w, v, w I, v w the id the vle o J,,, v, w I v w the id the vle o J,,,, t Pove tht the ctios t t d v e ctioll depedet sig the cocept o jcobis Also epess v items o Give tht, v w e these ctioll depedet, v I, v d, si the det e mi e the jcobi o,, v 5 i, v d, si the pove tht si, I, v, w id the ivese jcobi b ist coimig tht v w e ctioll idepedet 7 I e sec v e t v id the ivese jcobi b ist coimig tht d e ctioll idepedet Gdiet, Divegece, cl d Lplci Give id Ñ d Modle - : Vecto Clcls Ñ Give A i j k id div A, cl A d Fid the it vecto oml to the ollowig sces : t,, b t,, c t,, Fid the diectiol deivtives o the ollowig : t,, i j k Ñ A b t,, i j k c t,, i the diectio o A i j k

9 5 Fid the diectiol deivtive o the ctio the diectio o the oml to the sce t the poit,, Also id the eqtio o the tget ple d the oml lie to this sce I the diectiol deivtive o b c t,, hs mimm mgitde o its i the diectio pllel to -is id,b,c 7 Fid the gle betwee the sces 9 d t the poit,, 8 Fid the gle betwee the omls to the sce t,, d,, 9 Fid the vle o the tts d b sch tht the sces b d e othogol t the poit,, Fid div F d cl F whee F Ñ I F Ñ id div F d cl F t the poit,, I F i j k I A i j k id I F i j k Ñ A, id gddiv F t,, Ñ A d Ñ Ñ A t,, show tht F cl F = 5 Fid clcl A give tht A i j k I V w pove tht cl V w whee w is tt vecto 7 I A is tt vecto pove the ollowig : 8 I i j k i ii Ñ gd A A ì ü b cl í A ý A î þ ì ü ï ï c div í A A ý A ïî ïþ d div A e cl A = A Ñ d pove the ollowig : 9 Pove with sl meigs Ñ i j Show tht F is both soleoidl d iottiol iii Ñ iv Ñ

10 Show tht F i j k F Ñ Show tht F i j k potetil is iottiol Also id scl ctio sch tht Fid the vle o the tt sch tht F i j k hece id scl ctio sch tht F Ñ Fid the tts d b sch tht F i j b k id scl ctio sch tht F Ñ 5 I F i b j c k scl ctio sch tht F Ñ Pove the vecto idetities : clgd = b divcl A = c clcl A =gddiv A - Ñ A d e Redctio omle: Ñ A Ñ A Ñ A Ñ A Ñ A Ñ A div A B B cl A A cl B Evlte the ollowig itegls ò si d b ò d Modle - : INTEGRAL CALCULUS is evtive ield Fid its scl is iottiol d is iottiol d hece id, b, c sch tht cl F d the id c d ò ò 7 d si d e ò d

11 ò d, hece id the vle o ò si d g ò si d h ò t t 7 dt, i ò 9 / d j ò d si k ò / d l ò d Dieetil Eqtios: Solve the ollowig Dieetil Eqtios: d d b si d d c si d d e d, i d d e e d d g h i 5 j k 7 l 8

12 m 9 o p q s 5 t v w 7 A bod oigill t 8 C cools dow to C i mites, the tempete o i beig C Wht will be the tempete o the bod te mites om the oigil? Fid the k o the ollowig mtices: Modle - 5: LINEAR ALGBRA b Redce the ollowig mtices ito oml om d hece id its k b Solve +-=-5, +-=-, --= b Gsss-Elimitio method Solve -7++5µ=, µ =5, +++ µ=, µ=7 b Gss_Elimitio method 5 Solve ++=9, -+=, ++5= b Gss-Jod method Solve -7++5µ=, µ =5, +++ µ=, µ=7 b Gss_Jod method 7 Solve -+=-, ++5=5, -+= b Gss_Jod method 8 Show tht is egl Wite dow the ivese tsomtio 9 Pove tht the olowig mti is othopgol

13 I is Othogol, id,b, c d Fid the eig vles d eige vectos o the mti Fid the eig vles d eige vectos o the mti Fid the eig vles d eige vectos o the mti Redce the mti A= to the digol om 5 Fid the mti P which tsoms the mti A= to the digol om Hece clclte Redce the qdtic om to the coicl om Ad speci the mti o tsomtio 7 Redce the qdtic om ++ ito coicl om b othogol tsomtio ***************************************************************************

Engineering Mathematics I (10 MAT11)

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