NET/JRF, GATE, IIT JAM, JEST, TIFR

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Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Mathmatical Physics JEST-6 Q. Giv th coditio φ, th solutio of th quatio ψ φ φ is giv by k.

1 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Mathmatical Physics JEST-6 Q. Giv th coditio φ, th solutio of th quatio ψ φ φ is giv by k. kφ kφ lφ kφ lφ (a) ψ (b) ψ kφ (c) ψ (d) ψ As: (a) Solutio: φ. φ. φ φ α + βy+ γz φ α + βy+ γz k φ. φ k α + β + γ kφ k If ψ ( α + βy+ γz) ψ ψ ψ y z ψ k α β γ ˆ ˆ ˆ ψ φ φ k. Q. Th ma valu of radom variabl with probability dsity As. : (a) p (a) ( + μ).p σ σ (b) μ is: (c) μ (d) σ Solutio: μ p d p d σ σ σ Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com

2 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Q. Giv a matri M M, which of th followig rprsts cos 6 (a) (b) (c) (d) As. : (b) Solutio: W hav λ λ λ λ+ λ or λ For λ y givs Thus + y y. Takig, th igvctor associatd with λ is For λ y y Takig, th igvctors associatd with λ is Thus i M 6 / / M / / i 6 / / / / i 6 Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com

3 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES i M 6 i 6 / / i / / 6 i i i i 6 6 i i i i i i i i i i M M + + cos i si i i + + i i M M cos i si i i Thus M cos 6 Q. Th sum of th ifiit sris is 5 7 (a) (b) (c) (d) As. : (d) Solutio: Th sris for ta for >, is giv by ta Puttig, w obtai ta Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com

4 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Q5. A smicircular pic of papr is foldd to mak a co with th ctr of th smicircl as th ap. Th half-agl of th rsultig co would b: o (a) 9 o (b) 6 o (c) 5 o (d) As. : (d) Solutio: Wh th smicircular pic of papr is foldd to mak a co, th circumfrc of bas is qual to th circumfrc of th origial smicircl. Lt r b th radius of th bas of th cor ad R b th radius of th smicircl. R Hc, r R r. Th stay hight of th com will also b R. R / Hc, siα R Thus, α α r R R JEST-5 Q6. Giv a aalytic fuctio f ( z) φ(. y) + iψ (, y), whr (, y) + y + y As. : (c) If C is a costat, which of th followig rlatios is tru? (a) ψ (, y) y+ y+ C (b) ψ, y y + C (c) ψ (. y) y+ y + C (d) Solutio: u Q, y + y + y ψ, y y + C φ. From C.R. quatio u v y u y v u + Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com

5 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES v + y v y+ y+ f (i) u y + y v + y v y+ + f y y+ y+ f y + f y (ii) f, f y y v y+ y + c Q7. If two idal dic ar rolld oc, what is th probability of gttig at last o 6? As: (a) 6 (a) (b) 6 Solutio: Numbr of poit i sampl spac ( S ) (c) 6 (,6 ),(,6 ),(,6 ),(,6 ),( 5,6 ),( 6, ),( 6, ),( 6, ),( 6, ),( 6,5 ),( 6,6) Numbr of poit i populatio ( P ) 6 6 S Probability that at last o si o fac of dic P 6 (d) 6 5 Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com 5

6 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Q8. What is th maimum umbr of trma of th fuctio f P (, ) ad P k is a arbitrary polyomial of dgr k? k + whr As: (a) k + (b) k + 6 (c) k + (d) k (c) Solutio: f P + + ( + ) f P P + f P P + is polyomial if ordr k + From th sig schm maimum umbr of trma k + s Q9. Th Broulli polyomials B () s ar dfid by, B () s! followig rlatios is tru? (a) ( s ) B +! () s ( ) (b) ( ). Which o of th s B +! ( s) ( ) (c) ( s ) B ( s)( )! (d) ( s ) ()( s ) B! As: (d) S Solutio: B ( S) Put S ( S ), ( ) ( ) ( ) B S B S S B ( S ) ( ) S B ( S) Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com 6

7 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Q. Cosidr th diffrtial quatio G + kg δ followig statmts is tru? (a) Both G ad (b) G is cotiuous at but G is ot. (c) G is discotiuous at. (d) Th cotiuity proprtis of G ad G ar cotiuous at. As: (c) Q. 99 Th sum is qual to m m + + m (a) 9 (b) As: (a) Solutio: 99 m m+ + m m+ m m+ m m+ m m ; whr k is a costat. Which G at dpds o th valu of k. 99 (c) ( 99 ) m JEST- f f + f? Q. What ar th solutios to (d) (a) c / (b) c + c / (c) c + c (d) As.: (d) Solutio: Auilary quatio D + D D +, + D Roots ar qual th f ( c+ c) + f c c c + c Q. Th valu of. Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / d by usig th o-sgmt trapzoidal rul is clos to (a).67 (b).87 (c).99 (d).9 As.: (c) h Solutio: h.. I y(.) + y(.).99 y Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com 7

8 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Q. Giv th fudamtal costats (Plack s costat), G (uivrsal gravitatio costat) ad c (spd of light), which of th followig has dimsio of lgth? G (a) c G (b) 5 c G (c) c (d) c 8G As.: (a) [ ][ ] Solutio: ML T M L T [ ] L L L T gr ML T, G [ M L T ] m Q5. Th Laplac trasformatio of t si t is (a) s + s + 5 (b) s + s + (c) s s + s + (d) s s + s + As.: (b) Solutio: L at si bt b s+ a + b t L si t s + + s + s+ Q6. Lt us writ dow th Lagragia of a systm as L( ) m k c dimsio of c? Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / ,, + +. What is th (a) MLT (b) MT (c) MT (d) ML T As.: (c) Solutio: Accordig to dimsio rul sam dimsio will b addd or subtractd th dimsio of M dimsio of C [ ] [ ] ML T C L LT [ ] [ ML T ] [ L M ] C [ MT ] Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com 8

9 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Q7. Th Dirac dlta fuctio δ satisfis th rlatio d f ( ) As.: (d) bhavd fuctio f. If has th dimsio of momtum th (a) δ has th dimsio of momtum (b) δ has th dimsio of ( momtum ) (c) δ is dimsiolss (d) δ has th dimsio of ( momtum) Solutio: f δ d f ( ) f δ d f ( ) [ f ] δ P f ( ) Sic, [ f ] [ f ( ) ] If α + β ( ) β Q8. Th valu of limit [ ] δ [ ] P F is forc [ M LT ] F is also [ M LT ] z + lim z i z 6 + is qual to (a) (b) (c) -/ (d) 5/ As.: (d) f δ for a wll Solutio: z + z lim i z 6 lim + z i 6z 9 z lim 5 z 5 z i 6 6 Q9. Th valu of itgral si z I dz c z with c a circl z, is (a) (b) i (c) i (d) i Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com 9

10 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES As.: (c) si z Solutio: I pol z z C z Rsidu at z z so it will b lis withi th cotour iz I mg R i C z Rs Now z iz z i / i z I i i (takig imagiary part) ; Rsidu JEST- Q. A bo cotais cois out of which 99 ar fair cois ad is a doubl-hadd coi. Suppos you choos a coi at radom ad toss it tims. It turs out that th rsults of all tosss ar hads. What is th probability that th coi you hav draw is th doublhadd o? (a).99 (b).95 (c).75 (d). As.: (c) R( z ) + Im( z ) Q. Comput lim z z (a) Th limit dos ot ist. (b) (c) i (d) - As.: (a) Solutio: R z + Im z y + y y + y lim lim lim z y + iy y + iy z z y Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com

11 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES y + y lim y y + iy ad + y y lim y y + iy i Q. Th vctor fild ziˆ + yˆj i cylidrical polar coordiats is ˆ (a) ρ( z cos φ + si φ) ρ + ρ siφ cosφ( z) φ ˆ (b) ρ( z cos φ + si φ) ρ + ρ siφ cosφ( + z) φ ˆ (c) ρ( z si φ + cos φ) ρ + ρ siφ cosφ( + z) φ ˆ (d) ρ( z si φ + cos φ) ρ + ρ siφ cosφ( z) φ As.: (a) Solutio: A ziˆ + yˆj A z, A y, A y z A A ˆ A ˆ ˆ + A yˆ ˆ + A z ˆ ( ˆ ) ρ ρ ρ y ρ z ρ Aρ ρcosφz( cosφ) + ρsiφ( siφ) + ρ ( ρcosφ + ρsi φ) A A ˆ A ˆ ˆ + A y ˆ ˆ + A z ˆ ( ˆ ) φ φ φ y φ z φ ˆ ˆ ˆ ˆ A z ˆ ρ A cos ( si ) z+ si cos A ρcosφ siφ( z) ˆ φ ρ φ φ ρ φ φ φ A Aˆ + Aˆ + Aˆ ρ φz+ φ ˆ + ρ φ φ z ˆ ( cos si ) cos si ( ) ρ ρ φ φ z z ρ φ Q. Thr ar o avrag buss pr hour at a poit, but at radom tims. Th probability that thr ar o buss i fiv miuts is closst to (a).7 (b).6 (c).6 (d).9 As.: (d) Q. Two druks start out togthr at th origi, ach havig qual probability of makig a stp simultaously to th lft or right alog th ais. Th probability that thy mt aftr stps is!! (a) (b) (c)! (d)!!! As.: (a) φ Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com

12 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Solutio: Ito probability of takig ' r ' stps out of N stps r N r N C r total stps N + for takig probability of stps out of N N N!!! P NC ( N ) ( )!!!!! Q5. What is th valu of th followig sris? As.: (d) !!! 5! (a) (b) (c) (d) Solutio: 5 θ θ θ θ cosθ +..., si θ θ +...!!! 5! cos + si si θ + cos θ!!! 5! / λ Q6. If th distributio fuctio of is f ovr th itrval < <, th ma valu of is (a) λ (b) λ (c) λ (d) As.: (b) Solutio: it is distributio fuctio so λ f d. d f d λ d λ λ d λ d Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com

13 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Q7. Th valu of th itgral (a) As. : (b) (b) l + ( ) d JEST- is (c) (d) Solutio: l l z d ( + ) ( z + ) dz Lt us cosidr w fuctio f ( z) Pol at z Rsidu at z l z z ± i is simpl pol of scod ordr. iis ( l z) ( ) ( + ) d ( z i ) dz z i z i ( l z) ( + ) d dz z i +, th I l z z + dz A B r R z ( z+ i) l ( z). ( l z).( z+ i) ( z+ i) l l. z ( z+ i) ( z) ( z) ( z+ i) l l i ( i) i ( i) ( i) Rs + i z i 6 i i 8i i + 8i Similarly at z i; Rs i z i Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / l z I dz i i i i + z i f z dz f z dz ; vaish RABr AB Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com AB

14 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Alog path A; z + iε ad alog path B; z iε Thus l ( ε) ( ε) l ( ε) ( ε) + i i i f z dz d d AB + i + i + i i + i + i + l ( + ε) ( ε) l ( ε) ( ε) i d d l l + i i i d d ε + + ; ( l + i) ( l i) l ( + ) ( + ) i d i l i i ( + ) Q8. If [] dots th gratst itgr ot cdig, th [] (a) As.: (a) (b) (c) Solutio: [ ] < [], < [], < [ ] ow [] d [] d + [] d + [] d + [] d +. d +. d +. d [ ] + ( ) + ( ) d (d) Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com

15 fiziks Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES r + Q As, th ifiit sris (a) divrgs (b) covrgs to uity (c) covrgs to / (d) o of th abov As.: (c) 5 7 Solutio: ta ta 5 7 Q. What is th valu of th followig sris? !!! 5! (a) (b) (c) (d) As.: (d) Solutio: ,!! +!...! cosh + +! +! +... sih ( ) ! 5! i. cos h si h Q. A ubiasd di is cast twic. Th probability that th positiv diffrc (biggr - smallr) btw th two umbrs is is (a) / 9 (b) / 9 (c) / 6 (d) / As.: (a) p Solutio: ( ) ( E) ( S ) Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com 5

16 Istitut for NET/JRF, GATE, IIT JAM, JEST, TIFR ad GRE i PHYSICAL SCIENCES Th umbr of ways to com positiv diffrc [(,), (, ),( 5, ), ( 6, ) ] p 6 9 Q. For a N N matri cosistig of all os, (a) all igvalus (b) all igvalus (c) th igvalus ar,,., N (d) o igvalu N, th othrs As.: (d) Solutio:, so far,, N N matri o ig valu is N ad aothr s ig valu is zro Nar IIT, Hauz Khas, Nw Dlhi 6 Pho: / Brach offic Aad Istitut of Mathmatics, 8 B/6, Jia Sarai, Nar IIT Hauz Khas, Nw Dlhi 6 fiziks.physics@gmail.com 6

1985 AP Calculus BC: Section I

1985 AP Calculus BC: Section I 985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b