15. Numerical Methods
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- Horatio McLaughlin
- 5 years ago
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1 S K Modal' 5. Numrical Mhod. Th quaio i o b olvd uig h Nwo-Rapho mhod. If i ak a h iiial approimaio of h oluio, h h approimaio uig hi mhod will b [EC: GATE-7].(a (a (b 4 Nwo-Rapho iraio chm i f( +,,,... f' ( Hr, f( 8. f'( (c (d. Th rcurio rlaio o olv uig Nwo-Rapho mhod i [EC: GATE-8] (a X + (b X X + (c X (+ X + + (d X X (+ +. (c Nwo-Rapho iraio chm i f( +,f(. f' ( + ( + ( + ( W wih o olv by Nwo Rapho chiqu. L h iiial gu b. Subqu ima of (i.. will b: [ME: GATE-999] (a.44 (b.5 (c. (d No of h abov 4.(b. Pag 64 of 9
2 S K Modal' (, ( f f ( hr f.5 5. Th valu of a fucio f( ar abulad blow [ME: GATE-4] f( Uig Nwo' forward diffrc formula, h cubic polyomial ha ca b fid o h abov daa, i (a (b (c (d A. (d f( f( f( f( - 9 Uig Nwo forward irpolaio formula w g ( (( f( f( + f( + f( + f(... ( ( ( + ( + ( + (, ++(- +( , Sarig from X, o p of Nwo-Rapho mhod i olvig h quaio + -7 giv h valu ( a [ME: GATE-5] (a.5 (b.46 (c.5 (d 6. (c Nwo-Rapho iraio chm i f( +,,,... f' ( Giv f (.5 f' 6 ( Pag 65 of 9
3 S K Modal' 7. Th ordr of rror i h Simpo rul for umrical igraio wih a p iz h i [ME: GATE-997] (a h (b h (c h (d h 4 7. A. (b 8. Th accuracy of Simpo' rul quadraur for a p iz h i [ME: GATE-] (a O(h (b O(h (c O(h 4 (d O(h 8. A. (d 9. Wih a ui chag i b, wha i h chag i i h oluio of h ym of quaio + y, y b? [ME: GATE-5] (a Zro (b ui (c 5 ui (d ui 9.. A. (c Giv + y.. (i y b.. (ii Muliply.99 i quaio (i, ad ubrac from quaio (ii, w g (.-.99 b b-.98. Δ Δb Δ 5 ui.. Mach h im i colum I ad II. [ME: GATE-6] Colum I Colum II P. Gau-Sidl mhod. Irpolaio Q. Forward Nwo-Gau mhod. No-liar diffrial quaio R. Rug-Kua mhod. Numrical igraio S. Trapzoidal Rul 4. Liar algbraic quaio (a P-, Q-4, R-, S- (b P-, Q-4, R-, S- (c P-, Q-, R-, S-4 (d P-4, Q-, R-, S-. (d (P Gau Sidal mhod Liar algbraic quaio (Q Forward Nwo Gau mhod Irpolaio (R Rug Kua mhod No-liar diffrial quaio (S Trapzoidal Rul Numrical igraio. A calculaor ha accuracy up o 8 digi afr dcimal plac. Th valu of π i d wh valuad uig hi calculaor by rapzoidal mhod wih 8 qual irval, o 5 igifica digi i [ME: GATE-7] (a. (b. (c.5 (d.5. A. (a Pag 66 of 9
4 S K Modal' π π h 8 4 y i( π y i.77 4 π y i π y i.7 4 y i π 4 ( 5π y5 i π y6 i 4 y 7π 7 i π y8 i 4 Trapzoidal rul + h π h f(.d [(y + y + (y+ y y ] h i.d 8 [( + + ( ].. I h oluio of h followig of liar quaio by Gau limiaio uig parial pivoig 5 + y + z 4; 4y z ; ad y + z 4; h pivo for limiaio of ad y ar [CE: GATE 9] (a ad 4 (b ad (c 5 ad 4 (d 5 ad 4. A.(a Th quaio ar 5 + y + z 4 + 4y z ad y + z 4 Th augmd mari for gau-limiaio i Sic i h fir colum maimum lm i abolu valu i, w d o chag row wih row. Pag 67 of 9
5 S K Modal' R(, So h pivo for limiaig i a Now o limia y, w d o compa h limia i cod colum a ad blow h diagoal. Sic a 4 i alrady largr i abolu valu compar o a Th pivo lm for limiaig y i a 4 ilf. Th pivo for limiaig ad y ar rpcivly ad 4. CE Yar GATE Quio Q. Th quar roo of a umbr N i o b obaid by applyig h Nwo Rapho iraio o h quaio N. If i do h iraio id, h corrc iraiv chm will b N N (a i+ i+ (b i+ i+ i i N (b (c i+ i+ i A. (a Ep. f,i,,... ( i ( i+ i f' i i N i f ( N i N i (d i+ i [CE-] i i + N i i + N i N i + i Sam for Likd Awr Quio ad : Giv a >, w wih o calcula i rciprocal valu by uig Nwo Rapho Mhod for a f(.. Th Nwo Rapho algorihm for h fucio will b [CE: GATE 5] a a (a k + k + (b k + k + k k a (c k + k ak (d k + k k. (c Pag 68 of 9
6 S K Modal' a a L f( a Nwo Rapho iraio chm f( + f' ( a + a a. For a 7 ad arig wih X., h fir wo iraio will b (a.,.99 (b.,.9 (c.,.46 (d.,.48.(b a a Th followig quaio d o b umrically olvd uig h Nwo-Rapho mhod Th iraiv quaio for hi purpo i (k idica h iraio lvl [CE: GATE 7] k + 9 k + 4 (a k + (b k + k + 4 k + 9 4k + (c k + k k + 4 (d k k 4.(a Nwo Rapho iraio chm i f ( + f' ( Pag 69 of 9
7 S K Modal' A d dgr polyomial, f( ha valu of, 4 ad 5 a, ad, rpcivly. Th igral f( d i o b imad by applyig h rapzoidal rul o hi daa. Wha i h rror (dfid a ru valu approima valu i h ima? 4 (a (b [CE: GATE 6] (c (d 5. (a Giv ( f( 4 5 ( ( ( ( ( ( ( ( ( ( ( ( f( f ( + f ( +.f :5 4 + b Error f ( d y + y + y ( 4 + d ( 6. Th abl blow giv valu of a fucio F( obaid for valu of a irval of [CE: GATE ] F( Th valu of h igral of h fucio bw h limi o uig Simpo rul i (a.7854 (b.56 (c.46 (d (a h f ( d ( y + + ( + + y4 4 y y y ( Pag 7 of 9
8 S K Modal'.7854 EE All GATE Quio 7. Equaio - i rquird o b olvd uig Nwo mhod wih a iiial gu - Th afr o p of Nwo mhod, ima of h oluio will b giv by (a.788 (b.6784 (c.587 (d. [EE: GATE-8] 7. (a f( Nwo iraio chm f ( + f' (.. ( (a L f( 7 Nwo iraio chm i f ( + f' ( d 9. A diffrial quaio u( ha o b olvd uig rapzoidal rul of igraio d wih a p iz h.. Fucio u( idica a ui p fucio. If (-, h valu of a. will b giv by [EE: GATE-8] (a.99 (b.495 (c.99 (d A. (c Pag 7 of 9
9 S K Modal' d 49. Th diffrial quaio i dicrid uig Eulr umrical igraio d τ mhod wih a im p Δ T >.Wha i h maimum prmiibl valu of ΔT o ur abiliy of h oluio of h corrpodig dicr im quaio? [EE: GATE-7] (a (b τ / (c τ (d τ 49. A. (d IE All GATE Quio. For k,,, h p of Nwo-Rapho mhod for olvig a o-liar quaio i giv a [IE: GATE-6] 5 - k+ k + k Sarig from a uiabl iiial choic a k d o, h ira k d o (a.799 (b.6 (c.5 (d 5.. (a 5 k+ k+ k 5 k k + k 5 k+ k k+ k f ( k 5 k 5 k k f' ( k k f( 5 (by wo-rapho mdhod f( Idify h Nwo-Rapho iraio chm for fidig h quar roo of. (a + + (b + (c + (d + + [IE: GATE-7].(a f( N R chm i Pag 7 of 9
10 S K Modal' + ( ( f f' + + CS All GATE Quio 9. Coidr h ri + +,.5 obaid from h Nwo-Rapho mhod. 8 Th ri covrg o [CS: GATE-7] (a.5 (b (c.6 (d.4. (a ;.5 8 Th ri covrg wh α 9 4α + 9 α + 8α 8α 4α 9 α.5 + α R 4. Th Nwo-Rapho iraio + + ca b ud o compu h [CS: GATE-8] (a quar of R (b rciprocal of R (c quar roo of R (d logarihm of R 4.(c 5. Nwo-Rapho mhod i ud o compu a roo of h quaio wih.5 a h iiial valu. Th approimaio afr o iraio i [CS: GATE-] (a.575 (b.677 (c.667 (d (d N-R iraio chm i f ( + f' ( Pag 7 of 9
11 S K Modal' ( ( ( f.5.5 f' A picwi liar fucio f( i plod uig hick olid li i h figur blow (h plo i draw o cal. [CS: GATE-] f(. a (.55,.5 d (.5,. (.5, b c (.8,. If w u h Nwo-Rapho mhod o fid h roo of f( uig, ad rpcivly a iiial gu, h roo obaid would b (a.,.6 ad.6 rpcivly (b.6,.6 ad. rpcivly (c.,. ad.6 rpcivly (d.,.6 ad. rpcivly 6. A. (d Sarig from, lop of li a.5.5 y-ircp Eq, of a i y m + c + Thi li will cu ai (i.., y, a Sic i > ha.8, a prpdicular a will cu h li c ad o li b. roo will b. Sarig from, h prpdicular a i cuig li b ad roo will b.6. Sarig from,.5 Slop of li d.5.55 Equaio of d i y.5 (.55 i.. y.5 Thi li will cu ai a.5 Sic.5 i > ha.8, h prpdicular a.5 will cu h li c ad o li b. h roo will b hrfor qual o.. So arig from, ad h roo will b rpcivly.,.6 ad.. 7. Th miimum Numbr of qual lh ubirval dd o approimar d o a Pag 74 of 9
12 S K Modal' 6 accuracy of a la uig h rapzoidal rul i [CS: GATE-8] (a (b (c (d 7 A. (a Hr, h fucio big igrad i f( f( + ( + f ( + + ( + Trucaio Error for rapzoidal rul TE (boud h ma f ( ξ.ni Whr Ni i umbr of ubirval Ni b a h h b a T E ma f ( ξ. h h (b ama f ( ξ ξ h ( [ ( + ] h 6 6 h h Ni b a h Pag 75 of 9
13 S K Modal' 6. Traform Thory Prviou yar GATE Quio EC All GATE Quio. L ( u(, y( ( ad Y( jw b h Fourir raform of y(. Th Y( j i [EC: GATE-5] (a 4 (b (c 4 (d 4. A. (a ( y( y( y( 4 u( u( u( u( u( y( 4. Th igal ( i dcribd by [EC: GATE-8] for + (: ohrwi Two of h agular frquci a which i Fourir raform bcom zro ar (a π, π (b.5 π,.5 π (c, π (d π,.5 π. A. (a Giv : ( < + < ohrwi Fourir raform i i d [ ] i i Pag 76 of 9
14 S K Modal' i i [ ] [i ] i for π ad π. Coidr h fucio f( havig Laplac raform ω F( R[] > +ω Th fial valu of f( would b [EC: GATE-6] (a (b (c f( (d. A. (c f( L f( i w A i θ Thu, f( f( T T π ω ω 4. Giv ha F( i h o-idd Laplac raform of f(, h Laplac raform of i (a F( f( 4. A. (b 4 f( τ d τ f( (b F( (c (Lapalac formul f( τ d τ [EC: GATE-9] F( τ d τ (d [F( f(] +.Iflim f (, h h valu of K i 4 K (a (b (c (d 4 5. Giv f( L + + ( [EC: GATE-] 5. A. (d Pag 77 of 9
15 S K Modal' + K ( L ( ( L f( ( ( K f ( SF( f F lim lim ( + ( lim K K K (a z < (b z > (c 5 < z < 6 (d z 5 < < 6. Th rgio of covrgc of Z-raform of h quc 5 6 u( u( mu b A. (c 5 6 f( u( u( Now, or f( u( u( F(z z z.z Hc, rgio of covrgc, z < ad z <. 5 6 For wo rm 5 < z < [EC: GATE-5] 7. Coidr h z-raform X(z 5z + 4z - + ; < z <. Th ivr z-raform [] i [EC: GATE-] (a 5 δ[ + ] + δ[] + 4 δ[ ] (b 5 δ[ ] + δ[] + 4 δ[ + ] Pag 78 of 9
16 S K Modal' 7. A. (a (c 5 u[ + ] + u[] + 4 u[ ] (d 5 u[ ] + u[] + 4 u[ + ] (z 5z + 4z - + <IZl < [] 5 δ[ + ] + 4 δ[ ] + δ[] ME Yar GATE Quio 8. If f( i a fii ad coiuou fucio for, h Laplac raformaio i giv by F f( d. For f( co h m, h Laplac raformaio i..[me: GATE-994] 8. A. m w 9. Th Laplac raform of co ω i. δ +ω (a Tru (bfal [ME: GATE-995] 9. A. (bfal w Laplac raform of co ω i. δ ω. (+ - i h Laplac raform of [ME: GATE-998] (a (b (c - (d -. A.(d L( By fir hifig horm L(. ( +. Laplac raform of (a + b whr a ad b ar coa i giv by: [ME: GATE-999] a ab b a ab b (a (a+b (b (c + + (d + + (a+b. A.(c (a + b a + b + ab. L Laplac raform of Laplac raform of + a b ab L (a+b + + Pag 79 of 9
17 S K Modal'. Th Laplac raform of h fucio i i [ME: GATE-] (a (/-/[( +6] (b /( +6 (c (/-/( +4 (d /( +4. A.(a co4 i co4 L. + 6 ( + 6. Laplac raform of h fucio i ω [ME: GATE-] ω ω (a (b (c (d +ω +ω ω ω. A. (b ω L[i ω ] +ω, for <a 4. A dlayd ui p fucio i dfid a u(-a. I Laplac raform i, for a - a -a (a a. (b (c (d 4. A. (d a a L[U( a] U( ad,..d+..d,. d, a a 5. If F( i h Laplac raform of fucio f (, h Laplac raform of f( τ dτ i [ME: GATE-7] (a F( (b F( - f( (c F( f ( (d F( d 5. A. (a L f( d F( 6. Th Ivr Laplac raform of ( S + S i (a + (b - (b - - (d A. (c [ME: GATE-9] Pag 8 of 9
18 S K Modal' L? + + ( + + L L L [ ] Uig adard formula Sadard formula: a L L + a L a a 7. Th Laplac raform of a fucio f( i ( +. Th fucio f( i (a -+ - (b ++ - (c -+ - (d + [ME: GATE-] 7. A. (a L[ f( ] S ( S+ f( S ( S+ L L S + L d SS ( + + L S ( S+ CE Yar GATE Quio 8. If L dfi h Laplac Traform of a fucio, L [i (a] will b qual o a a (a (b [CE: GATE ] a + a (c + a 8. A. (b L [f (] L [i (a] a + a (d a f(d i(ad Pag 8 of 9
19 S K Modal' 9. Laplac raform for h fucio f( coh (a i [CE: GATE 9] a (a (b a a a (c + a 9. A. (b I i a adard rul ha a L (coh a. (d + a Q. Thr ar wo coair, wih o coaiig 4 Rd ad Gr ball ad h ohr coaiig Blu ad 4 Gr ball. O bal i draw a radom form ach coair. Th probabiliy ha o of h ball i Rd ad h ohr i Blu will b (a /7 (b 9/49 (c /49 (d /7 [CE- ] A. (c EE All GATE Quio Sam for Likd Awr Quio ( ad ( A a variabl ym X( ( X + u(, wih h iiial codiio X([- ] T ad h ui p ipu u( ha. Th a raiio quaio [EE: GATE-5] ( (a ( (c. A. (a (b ( ( (d Pag 8 of 9
20 S K Modal' ( I A ( L ( I A Adj( I A + I A + +. ( + + ( + + φ (. Th a raiio quaio [EE: GATE-5] (a X( (b X( (c X(. A. (c zro a rpo L φ BU S ( + L + L Sa raiio quaio ( φ ( X X( ( ( zro ipu rpo+zro a rpo. X + (d X( L ( rc (whr rc ( for ad zro ohrwi. Th if ic ( i( π, h Fourir Traform of (+(- will b giv by [EE: GATE-8] π ω (a ic π ω (b ic π Pag 8 of 9
21 S K Modal' ω (c ic π co ω ω (d ic π i ω. A. (c rc ( for Giv ( rc Simpliyig h hlp of quaio. ( wih ( Pag 84 of 9
22 S K Modal', (, hrwi ( ( jω Now, F a. jω a jω jω ( ( jω jω jω jω jω jω ω ω jω jω. ω j ω i F ( ω / (,, ohrwi ( (. jω jω ( jω jω ( jω jω jω jω jω jω jω F a jω a jω jω jω ω j ω i jω F ( ω / ω i jω jω ( ( F + + ω / ω i ω ω co / ω ω i π co Pag 85 of 9
23 S K Modal'. L ( b h p rpo of a liar ym wih zro iiial codiio; h h rpo of hi ym o a ipu u( i [EE: GATE-] (a ( ( τ uτ dτ (c ( τ u( τ dτ dτ. A. (b d τ u τ dτ d (b ( ( (d ( ( τ uτ dτ 4. L Y( b h Laplac raformaio of h fucio y (, h fial valu of h fucio i [EE: GATE-] LimY LimY (a ( (c Lim Y ( (b ( (d Lim Y ( 4. A. (c 5 5. Coidr h fucio, F( ( + + whr F( i h Laplac raform of h fucio f(.th iiial valu of f( i qual o [EE: GATE-4] (a 5 (b 5 (c 5 (d 5. A. (d Iiial valu 5 LimF( Lim S S Th Laplac raform of a fucio f( i F( A, f( approach ( + + (a (b 5 (c L f L F L A. (a ( ( (d [EE: GATE-5] 7. If u(, r( do h ui p a ui ramp fucio rpcivly ad u(* r( hir covoluio, h h fucio u(+* r(- i giv by [EE: GATE-7] (a (/(- (- (b (/(-(- (c (/(- u(- (d No of h 7. A. (c L u( + ( L r ( L L ( u( 8. A fucio y( aifi h followig diffrial quaio ( dy d ( δ ( + y Pag 86 of 9
24 S K Modal' Whr δ ( i h dla fucio. Aumig zro iiial codiio, ad doig h ui p fucio by u(, y( ca b of h form [EE: GATE-8] (a (b - (c u( (d - u( 8. A. (d dy ( + y( δ d Takig Laplac rafrom of boh id, w hav y( -y(o+y( ( ( + y y( + Takig ivr Laplac raform, w g - y( u ( 9. Th Laplac raform of g( i [EE: GATE-] 9. A. (c 5 5 (a ( (b ( 5 (c ( (d ( Commo Daa for Quio ad : Giv f( ad g( a how blow:. g ( ca b prd a [EE: GATE-]. A. (d (a g ( f( (b g ( f (c g ( f (d g ( f. Th Laplac raform of g( i [EE: GATE-]. A. (c 5 5 (a ( (b ( 5 (c ( (d (. If u( i h ui p ad δ ( i h ui impul fucio, h ivr z-raform of F(z for k> i [EE: GATE-5] z + Pag 87 of 9
25 S K Modal' (a ( k δ ( k (b δ ( k ( (c ( uk ( (d uk ( ( k. A. (b z+ z F( z z+ z+ z -. z ( ( 8( ( z F z z z a z a k. Th ruig igraor, giv by [EE: GATE-6] y ' d'. A. (b ( ( (a ha o fii igularii i i doubl idd Laplac Trafrom Y( (b Produc a boudd oupu for vry caual boudd ipu (c Produc a boudd oupu for vry aicaual boudd ipu (d ha o fii zro i i doubl idd Laplac Trafrom Y( 7. Th a raiio mari for h ym X AX wih iiial a X( i [EE: GATE-] (a (I-A - (b A X( (c Laplac ivr of [( I-A - ] (d Laplac ivr of [(I-A - X (] 7. A. (c 4. Coidr h mari P. Th valu of p i (a 5 (b [EC: GATE-8] (c (d A. (d P L [(I P ] ad P whr (I P + Pag 88 of 9
26 S K Modal' + ( + ( + + ( + ( + ( + ( + ( + ( + ( + ( + P L Q4. L h Laplac raform of a fucio f ( which i for > b F ( Laplac raform of i dlayd vrio f ( - τ b F (. L ( cojuga F ( wih h Laplac variabl a σ+ j ω. If ( ( ( G F.F * F ( h h ivr Laplac raform of G( i (a A idal impul δ ( (b a idal dlayd impul δ( - τ (c A idal p fucio u( (d a idal dlayd p fucio u( - τ A. [EE-] (b ad h F* b h compl, IE All GATE Quio. If h Fourir raform of [] i X( jω, h h Fourir raform of ( [] i [IE: GATE-4] (a ( j ω X( jω (b ( ω X( jω (c X( j(ω π d j (d (X( ω d ω. A. (c 4. If h wavform, how i h followig figur, corrpod o h cod drivaiv of a giv fucio f (, h h Fourir raform of f ( i (a + i ω ( co ω (c ω (b + co ω ( + co ω (d ω [IE: GATE-6] Pag 89 of 9
27 S K Modal' d f( d + 4. A. (c df( δ ( + δ ( + δ ( d Takig Laplac raform of boh id, w g F( + (jω F (jω j ω + j ω ( co ω F (jω ω 5. Th Fourir raform of a fucio g ( i giv a ω + G(ω ω +9 Th h fucio g ( i giv a (a δ ( + p ( (b co ω + p ( (c i ω + 7 co ω (d i ω + p ( 5. A. (a g( ( + p. ( Takig Laplac raform boh id, [IE: GATE-6] G (ω + p(p( jω.d+ p(.p( jω.d + p( j ω.d+ p( j ω.d + + j ω + j ω ω + G (ω ω Th Fourir raform of ( a u(, whr u( i h ui p fucio, [IE: GATE-8] (a Ei for ay ral valu of a (b Do o i for ay ral valu of a (c Ei if h ral valu of a i ricly gaiv (d Ei if h ral valu of a i ricly poiiv 6. A. (d Pag 9 of 9
28 S K Modal' 7. Th fudamal priod of ( i π + i π, wih prd i cod, i [IE: GATE-9] (a (b.67 (c (d 7. A. (d H.C.F. of π ad π i 6π. Th, fudamal frqucy 6π Priod, T 6 π π c 8. u( rpr h ui p fucio. Th Laplac raform of u( τ i (a τ τ (c 8. A. (c f( u( τ L{f(} L{u( τ} τ F( (b τ (d τ [IE: GATE-] 9. A maurm ym wih ipu ( ad oupu y( i dcribd by h diffr dy quaio +5y 8. Th aic iiviy of h ym i d [IE: GATE-] (a.6 (b.6 (c.67 (d A. (d dy 5y d + 8 Takig Laplac raform, w hav y( + 5y( 8X( y( [ + 5] 8X( y( ( For aic iiviy, Y( X( Th fudamal priod of h dicr-im igal [] (a 6 5π 4. A. (b (b 5 ω 5 π 6 (c 6 (d 5π j 6 i [IE: GATE-8] or π 5π T 6 or T 5 Pag 9 of 9
29 S K Modal' 4. A pla wih a rafr fucio i corolld by a PI corollr wih Kp ad Ki ( + i a uiy fdback cofiguraio. Th low valu of Ki ha ur zro ady a rror for a p chag i h rfrc ipu i [IE: GATE-9] (a (b (c 4. A. (b i G ( k kp + ( + R( lim R( + G ( lim ki + kp + ( + ( + lim ( + (k + k p i (d Low valu of ki for G o b zro. Pag 9 of 9
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