DE55/DC55 ENGINEERING MATHEMATICS-II DEC 2013

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1 DE55/DC55 ENGINEERING MATHEMATICS-II DEC Q. a. Evaluae LT sin x x sin 5x + sin 6x sin x Here we have, L x sin x + sin 6x sin 5x sin x L x L x x + 6x x 6x sin + cos 5x + x 5x x cos.sin sin 4x cos( x) cos 4x.cos x L sin 4x x cos x x 4 4x sin x cos 4x 4() () () 4 b. If f is a real funcion defined by f (x) + f (x) f (x) + x f(x), f(x) + x + x + x + ( x ) + ( x + ) ( x + ) x f (x) hen prove ha x + f(x) - x x + x + x + x + f (x) + x + x + hen f (x) x + (Applying componendo & devidendo) f (x) + x, Now, f(x) f (x) x x + IETE

2 DE55/DC55 ENGINEERING MATHEMATICS-II DEC f (x) + f (x) f (x) f (x) + + f (x) f (x) + + f (x) f (x) + + f (x) f (x) + f (x) There proof f (x) + Q. a. Find he volume of he righ circular cone formed by he revoluion of a righ angled riangle abou a side which conains he righ angle. Le O B A be he righ angled wih OA r and OB h. When he riangle is revolved abou he side y-axis is abou he side OB. We ge a righ circular cone of radius r and heigh h. The curve is he line AB, whose eqn. is x r + y h r or x (h y) (i) h Required volume π π r (h y) h dy IETE

3 DE55/DC55 ENGINEERING MATHEMATICS-II DEC πr h (h y) h r π h πr h + h b. Find he lengh of he curve y x from origin o he poin (, ). The curve can easily be raced and is shape is shown in below figure The eqn. of he curve is y x (i) y x dy y x dx dy x x / x from(i) / dx y x or [ ] Now, S dy + dx dx 9x + dx 4 4 9x.dx + 9 ( 4 + 9x) / IETE

4 DE55/DC55 ENGINEERING MATHEMATICS-II DEC 7 7 [() ( ) ] / 4 / [ 8] n Q.4 a. If n is a posiive ineger hen show ha ( + i) + ( i) where i Le n n + nπ cos 6 + i r(cosα + isin α) π r +, α an 6 π π + i cos + isin 6 6 n ( + i ) n + ( i) n π π π π cos + isin + cos isin n nπ nπ n nπ nπ cos + isin + cos isin π n+ nπ cos 6 R.H.S. Hence proved b. A resisance of ohms and inducance of. H and a capaciance of µ F are conneced in series a cross Vol, 5cycle/sec main. Deermine: (i) impedance (ii) curren (iii) volage across L,R and C (iv) power in wa (v) power facor There, R Ω L. H C µf V V f 5 c/s (a) Impendence (z) R J C + J L j + j Lw wc IETE 4

5 DE55/DC55 ENGINEERING MATHEMATICS-II DEC j + j(.)π π J + Jπ j.8 + J 6.89 π + J So. Z Z 6.89 ohms. v (b) i 5. 4 z 6.89 (c) V L i L 5.4 π f. 5.4 π 5. V R ir vols V c i πf c vols 6 π 5 (d) Power i R (5.4) was. R (e) Power facor. 54 z 6.89 Q.5 a. A rigid body is spinning wih an angular velociy of 7 radian/second abou an axis parallel o i +j k passing hrough he poin i+j-k. Find he velociy of he poin whose posiion vecor is 4i+8j+k. Le w be he angular of he body roaing abou an axis parallel o he vecor i + j k. Then w i + j k 7 + i + ( ) w 8i + 9j 8k Le r OP P.V. of P P.V. of O IETE 5

6 DE55/DC55 ENGINEERING MATHEMATICS-II DEC r (4i +8j+k)-(i+j-k) r i +5j +k Le i u w r 8 j 9 5 k 8 u 8i 9j + 6k 9(i j +7k) b. Find he area of he riangle formed by he poin whose posiion vecors are i+j, 5i+j+k, i-j+k. Le ABC be a riangle and le a i + j, b 5i + j+ k, and c i j+ k be he posiion vecors of is verices A, B, and C respecively, hen AB P.V. of B P.V. of A (5i +j+k) (i+j) i + j + k and AC P.V. of B P.V. of A (i-j+k) (i + j) -i -j + k IETE 6

7 DE55/DC55 ENGINEERING MATHEMATICS-II DEC AB AC i j k 6i 8j 4k AB AC Reqd. Area AB AC 6 9 (formula, Area of ABC AB AC BC BA CA CB d y dy x x Q.6 a. Solve 6 + 9y 6e + 7e log dx dx d y dy x y 5e dx dx (D + 6D + 9) y 5e x Auxiliary equaion is D + 6D+9 or D -, -, C.F. (C +C x)e -x x x e.5.e 5 D + 6D + 9 () + 6() + 9 x 5e 6 d y b. Solve + 9y secx dx d y + 9y secx dx Auxiliary equaion D +9 or D ± i C.F C cosx + C sin x.secx 6i D i D i () + ix ix Now secx e e secx dx D i ix cosx isin x ix e dx e ( i an x)dx cosx IETE 7

8 DE55/DC55 ENGINEERING MATHEMATICS-II DEC e ix i x + og cosx Changing I o I, we have ix secx e x D + i i og cosx Puing hese values in (i), we ge ix i P.I. e x + og cosx e 6i ix x i log cosx ix x ix og cosx x ix e e + e + og cosx 6i 8 6i 8 ix ix ix x ix e og cosx xe e e + + og cosx 8 6i 8 ix ix ix ix x e e e + e +. og cosx i 9 ix ix ix ix x e e e + e. +. og cosx i 9 x sin x +.cos x. og cosx 9 x Hence, complee soluion is y C cos x + sin x +.cos x. og cosx 9 Q.7 a. Expand f(x) x e in a cosine series over (, ) Here F(x) x and C c x a f (x)dx e dx (e ) c a n e x nπx cos dx x e (nπsin nπx + cos nπx n π + IETE 8

9 DE55/DC55 ENGINEERING MATHEMATICS-II DEC e n π x n π + (nπsin nπ + cos nπ) + n [( ) e ] n π + a f (x) + a cos πx + a cos πx + a cosπx... + e x e e e e- + cos πx + cos πx + cosπx +... π + 4π + 9π + b. Find he Fourier Series of he funcion when < < f() K " < < " < < C 4 or C a c C f () d c c k c c k kd ( ) ( ) k an C c nπ f ()cos d C nπ k cos d C k c c n nπ sin π + k nπ nπ sin sin nπ k n sin π nπ b n C c c nπ f ()sin d IETE 9

10 DE55/DC55 ENGINEERING MATHEMATICS-II DEC c nπ k sin d c k nπ k nπ nπ cos nπ cos cos nπ Fourier series is a π π π f () + a cos + a cos + a cos... c c c + π π π + b sin + b sin + b sin +... c c c k k π π π π π f() + sin.cos + sin π.cos + sin.cos +... π k k π π + cos cos +... π Q.8 a. Evaluae L { e cosh } L{ e cosh } L e L{ + e } + s s + (e cosh ) S e d ds ( s + ) + e + s s + (s + ) + s s (s + ) s + s + s (s + ) IETE

11 DE55/DC55 ENGINEERING MATHEMATICS-II DEC b. Evaluae L e Here o find L We have L (sin ) s + sin L s + [ ] e sin sin d d an s π an s co s sin L e co (s ) f (s) sin L e d f (s) s co (s ) s S Q.9 a. Show ha L (cosh a.sin a + sinh a.cos a) 4 4 S + 4a a s L 4 4 s + 4a L L s 4 ( ) s + a 4a s s 4 ( ) s + a 4a s s 4 4a s + a 4a s s ( ) 4 s + a + as L Resolving ino parial fracion. IETE

12 DE55/DC55 ENGINEERING MATHEMATICS-II DEC 4a 4a L L s ( s a) ( s a) + a (s a) + a + a L L ( s + a) a ( ) s + a + a ( s + a) a ( ) s + a + a a s + a a (s a e.l e L 4a s + a s + a by firs shifing heorem. a s a a s e.l ae L e.l 4a s + a s + a s + a a a ( ) a a + ( ) + e e.l a e e.l 4a s a s + a (sinh a)cosa + a(cosh a).sin a 4a a [ cosh a.sin a + sinh a.cos a] Hence proved a b. Evaluae To evaluae s + L og s s + L log s Le L - s + og f () s Then L{ f ()} s + f (s) og s d s + L{ + ()} ( ) og ds s d ds { og(s + ) og(s ) } IETE

13 DE55/DC55 ENGINEERING MATHEMATICS-II DEC - s + s s s + L{ e e } L{ sinh } f () sinh or f() Hence L - sinh s + og s sinh Tex Books. Engineering mahemaics Dr. B.S.Grewal, h ediion 7, Khanna publishers, Delhi.. Engineering Mahemaics H.K.Dass, S. Chand and Company Ld, h Revised Ediion 7, New Delhi.. A Tex book of engineering Mahemaics N.P. Bali and Manish Goyal, 7h Ediion 7, Laxmi Publicaion(P) Ld. IETE