JEE(Advnced) 08 TEST PAPE WITH SOUTION (HED ON SUNDAY 0 th MAY, 08) PAT- : JEE(Advnced) 08/Ppe-. The potentil enegy of pticle of mss m t distnce fom fixed point O is given by V () k /, whee k is positive constnt of ppopite dimensions. This pticle is moving in cicul obit of dius bout the point O. If v is the speed of the pticle nd is the mgnitude of its ngul momentum bout O, which of the following sttements is (e) tue? (A) Ans. (B,C) k V k n (B) m O F k (towds cente) At, mv k [Centipetl foce] k v m k m m k m v n k m (C) mk (D) é dvù êf - ë d ú û mk. Conside body of mss.0 kg t est t the oigin t time t 0. A foce F ( ti ˆ +bˆj) is pplied on the body, whee.0 Ns nd b.0 N. The toque cting on the body bout the oigin t time t.0 s is t. Which of the following sttements is (e) tue? (A) t Nm 3 (B) The toque t is in the diection of the unit vecto - (C) The velocity of the body t t s is ( ) ˆk + v i ˆ + j ˆ ms (D) The mgnitude of displcement of the body t t s is m 6 JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08
Ans. (A,C) F ( t)ˆi+bˆj, b F ti ˆ+ ˆj dv m ti ˆ ˆj dt + On integting [At t 0, v 0, 0 ] JEE(Advnced) 08/Ppe- t mv ˆi + tj ˆ d t ˆi + tj ˆ dt On integting 3 t ˆ t i+ ˆj 6 [m kg] [ 0 t t 0] At t sec, ( ) æ ˆ ˆ ö t F i+ j ( ˆ i+ ˆ j) t- ˆk 3 t v ˆi + tj ˆ ç è6 ø æ At t ˆ ˆ ö + ( ˆ + ˆ ) v ç i j i j m / sec è ø At t s - 0 é ù ê ë6 ú û ˆi+ ˆj -[ 0] ˆ s i+ ˆj 6 0 s æ ç ö + æ ç ö Þ m è6ø èø 6 3. A unifom cpilly tube of inne dius is dipped veticlly into beke filled with wte. The wte ises to height h in the cpilly tube bove the wte sufce in the beke. The sufce tension of wte is s. The ngle of contct between wte nd the wll of the cpilly tube is q. Ignoe the mss of wte in the meniscus. Which of the following sttements is (e) tue? (A) Fo given mteil of the cpilly tube, h deceses with incese in (B) Fo given mteil of the cpilly tube, h is independent of s. (C) If this expeiment is pefomed in lift going up with constnt cceletion, then h deceses., (D) h is popotionl to contct ngle q. Ans. (A,C) JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08
JEE(Advnced) 08/Ppe- s gh s h g cos q scosq h g [ dius of meniscus] (A) Fo given mteil, q constnt h µ (B) h depend on s (C) If lift is going up with constnt cceletion, g eff (g + ) scosq h g + It mens h deceses ( ) [ dius of cpilly; q contct ngle] (D) h is popotionl to cos q Not q 4. In the figue below, the switches S nd S e closed simultneously t t 0 nd cuent stts to flow in the cicuit. Both the btteies hve the sme mgnitude of the electomotive foce (emf) nd the polities e s indicted in the figue. Ignoe mutul inductnce between the inductos. The cuent I in the middle wie eches its mximum mgnitude I mx t time t t. Which of the following sttement(s) is (e) tue? I (A) mx Ans. (B,D) V S I S V V (B) Imx (C) 4 V t l n (D) t l n i i i i mx (i i ) mx Di (i i ) Vé æ ö æ ö -ç tù t Vé ç - ù ê è ø è ø ë-e úû- êë-e úû V é êë e æ ö æ ö -ç t -ç t è ø èø - e ù úû d( D i) Fo (Di) mx 0 dt JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08 3
JEE(Advnced) 08/Ppe- Vé æ ö æ ö -ç t t -ç è ø æ ö ù è ø ê- e -ç- e ú 0 ë è ø û e æö æ ö -ç t t -ç è ø è ø e e æ ö -ç t è ø æ ö ç t l n è ø t l n time when i is mximum. V é imx êëe -e i mx æ ö æ öæ ö - ç ln -ç ç ln è ø èøè ø V é ù V ê - ë4 úû 4 ù úû 5. Two infinitely long stight wies lie in the xy-plne long the lines x ±. The wie locted t x + cies constnt cuent I nd the wie locted t x cies constnt cuent I. A cicul loop of dius is suspended with its cente t (0, 0, 3 ) nd in plne pllel to the xy-plne. This loop cies constnt cuent I in the clockwise diection s seen fom bove the loop. The cuent in the wie is tken to be positive if it is in the + ĵ diection. Which of the following sttements egding the mgnetic field B is (e) tue? (A) If I I, then B cnnot be equl to zeo t the oigin (0, 0, 0) (B) If I > 0 nd I < 0, then B cn be equl to zeo t the oigin (0, 0, 0) (C) If I < 0 nd I > 0, then B cn be equl to zeo t the oigin (0, 0, 0) (D) If I I, then the z-component of the mgnetic field t the cente of the loop is Ans. (A,B,D) Z æ m I 0 ö ç - è ø l C (0,0, 3) Y x O x w Y 4 JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08
(A) At oigin, B 0 which is non-zeo. due to two wies if I I, hence ( B net ) JEE(Advnced) 08/Ppe- t oigin is equl to B due to ing, (B) If I > 0 nd I < 0, B t oigin due to wies will be long + ˆk diection nd B due to ing is long - ˆk diection nd hence B cn be zeo t oigin. (C) If I < 0 nd I > 0, B t oigin due to wies is long - ˆk nd lso long - ˆk due to ing, hence B cnnot be zeo. Z B B x (D) I B I X At cente of ing, B due to wies is long x-xis, m0i hence z-component is only becuse of ing which B (-kˆ) 6. One mole of montomic idel gs undegoes cyclic pocess s shown in the figue (whe V is the volume nd T is the tempetue). Which of the sttements below is (e) tue? (A) Pocess I is n isochoic pocess (C) In pocess IV, gs eleses het Ans. (B,C,D) T I II IV III V T I II IV III V (B) In pocess II, gs bsobs het (D) Pocesses I nd II e not isobic (A) Pocess-I is not isochoic, V is decesing. (B) Pocess-II is isotheml expnsion DU 0, W > 0 DQ > 0 (C) Pocess-IV is isotheml compession, DU 0, W < 0 DQ < 0 (D) Pocess-I nd III e NOT isobic becuse in isobic pocess T µ V hence isobic T V gph will be line. JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08 5
JEE(Advnced) 08/Ppe- 7. Two vectos A nd B e defined s A i ˆ nd B ( cosw ti ˆ + sinwtj ˆ), whe is constnt nd w p/ 6 d s. If A+ B 3 A-B t time t t fo the fist time, the vlue of t, in seconds, is. Ans..00 sec B wt A wt A + B cos wt A - B sin wt æ wtö So cos 3 ç sin è ø w t tn 3 w t p Þw t p 6 3 p p t t.00 sec 6 3 8. Two men e wlking long hoizontl stight line in the sme diection. The mn in font wlks t speed.0 ms nd the mn behind wlks t speed.0 ms. A thid mn is stnding t height m bove the sme hoizontl line such tht ll thee men e in veticl plne. The two wlking men e blowing identicl whistles which emit sound of fequency 430 Hz. The speed of sound in i is 330 ms. At the instnt, when the moving men e 0 m pt, the sttiony mn is equidistnt fom them. The fequency of bets in Hz, hed by the sttiony mn t this instnt, is Ans. 5.00 Hz A q m/s 3m 5m Obseve (sttiony) m 5m 3m q B m/s 5 cosq 3 é 330 ù é ù é cosqù fa 430 ê 430 430 + ë330 -cosqúû ê cosqú ê 330 ú ë û ê - ú ë 330 û (By bionomil expnsion) é 330 ù é cosqù fb 430 ê 430 - ë330 + cosqú û ê ë 330 ú û é3cosqù ë 330 û D f fa - fb 430ê ú 3 cos q æ 5 ö 3ç 5.00Hz è3 ø 6 JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08
JEE(Advnced) 08/Ppe- 9. A ing nd disc e initilly t est, side by side, t the top of n inclined plne which mkes n ngle 60 with the hoizontl. They stt to oll without slipping t the sme instnt of time long the shotest pth. If the time diffeence between thei eching the gound is ( - 3 ) / 0s, then the height of the top of the inclined plne, in metes, is. Tke g 0 ms. Ans. 0.75m c gsin q I + M C ing disc gsin q gsin q 3 h æ g sin q ö ç t 4h 6h Þ t sin q è ø gsin q 3g h æ g sin q ö ç t 3h 4h Þ t sinq è 3 ø gsin q g 6h 4h - 3 Þ - 3g g 0 é 4 ù h ê - - 3 3 ú ë û ( - ) ( ) 3 3 3 3 h Þ h 0.75m 4-3 4 q60 0. A sping-block system is esting on fictionless floo s shown in the figue. The sping constnt is.0 N m nd the mss of the block is.0 kg. Ignoe the mss of the sping. Initilly the sping is in n unstetched condition. Anothe block of mss.0 kg moving with speed of.0 m s collides elsticlly with the fist block. The collision is such tht the.0 kg block does not hit the wll. The distnce, in metes, between the two blocks when the sping etuns to its unstetched position fo the fist time fte the collision is. h kg ms kg Ans..09 m JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08 7
JEE(Advnced) 08/Ppe- m T p psec k block etuns to oiginl position in T sec p m/s 0 kg kg Just befoe collision /3m/s kg kg 4/3m/s Just fte collision d ( p ) ( 3.4).0933m 3 3 d.09 m. Thee identicl cpcitos C, C nd C 3 hve cpcitnce of.0 µf ech nd they e unchged initilly. They e connected in cicuit s shown in the figue nd C is then filled completely with dielectic mteil of eltive pemittivity Î. The cell electomotive foce (emf) V 0 8V. Fist the switch S is closed while the switch S is kept open. When the cpcito C 3 is fully chged, S is opened nd S is closed simultneously. When ll the cpcitos ech equilibium, the chge on C 3 is found to be 5µC. The vlue of Î. Ans..50 V 0 +8µC V 08V µf 8µC S S C C C 3 Î 3µC 3µC +5µC µf 5µC µf 3µC 3µC Applying loop ule 5 3 3 - - 0 Î Î.50 3 Î 8 JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08
JEE(Advnced) 08/Ppe-. In the x-y-plne, the egion y > 0 hs unifom mgnetic field ˆ Bk nd the egion y < 0 hs nothe unifom mgnetic field Bk. ˆ A positively chged pticle is pojected fom the oigin long the positive y-xis with speed v 0 pms t t 0, s shown in the figue. Neglect gvity in this poblem. et t T be the time when the pticle cosses the x-xis fom below fo the fist time. If B 4B, the vege speed of the pticle, in ms, long the x-xis in the time intevl T is. y B Ans..00 () Avege speed long x-xis v x v x ò ò dt dt () We hve, mv mv, qb qb Since B B 4 d t + d + t v 0 p ms B x \ 4 pm Time in B Þ t qb pm Time in B Þ t qb Totl distnce long x-xis d + d + ( + ) (5 ) Totl time T t + t 5t \ Avege speed 0 mv qb 5t qb pm JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08 9
JEE(Advnced) 08/Ppe- 3. Sunlight of intensity.3 kw m is incident nomlly on thin convex lens of focl length 0 cm. Ignoe the enegy loss of light due to the lens nd ssume tht the lens petue size is much smlle thn its focl length. The vege intensity of light, in kw m, t distnce cm fom the lens on the othe side is. Ans. 30 0 0 0cm \ tio of e 00 et enegy incident on lens be E. \ Given E.3 A So finl, E?? E A.30 Also A 00 cm \ Avege intensity of light t cm E A.3 00.3 30kW / m 4. Two conducting cylindes of equl length but diffeent dii e connected in seies between two het bths kept t tempetues T 300 K nd T 00 K, s shown in the figue. The dius of the bigge cylinde is twice tht of the smlle one nd the theml conductivities of the mteils of the smlle nd the lge cylindes e K nd K espectively. If the tempetue t the junction of the two cylindes in the stedy stte is 00 K, then K /K. 0 JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08
JEE(Advnced) 08/Ppe- Insulting mteil T K K T Ans. 4.00 300k k We hve in stedy stte, 00k æ00-300 ö æ00-00 ö ç 0 + ç ç k è p ø ç k ( ) p è ø p p k 00 00k 4 Þ k k Þ 4 k 00k PAAGAPH "X" In electomgnetic theoy, the electic nd mgnetic phenomen e elted to ech othe. Theefoe, the dimensions of electic nd mgnetic quntities must lso be elted to ech othe. In the questions below, [E] nd [B] stnd fo dimensions of electic nd mgnetic fields espectively, while [ Î 0 ] nd [µ 0 ] stnd fo dimensions of the pemittivity nd pemebility of fee spce espectively. [] nd [T] e dimensions of length nd time espectively. All the quntities e given in SI units. (Thee e two questions bsed on Pgph "X", the question given below is one of them) JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08
JEE(Advnced) 08/Ppe- 5. The eltion between [E] nd [B] is :- (A) [E] [B][][T] (B) [E] [B][] [T] (C) [E] [B][][T] (D) [E] [B][] [T] Ans. (C) We hve E B C \ [B] [ E ] - [ E] T [ C] Þ [E] [B] [][T ] PAAGAPH "X" In electomgnetic theoy, the electic nd mgnetic phenomen e elted to ech othe. Theefoe, the dimensions of electic nd mgnetic quntities must lso be elted to ech othe. In the questions below, [E] nd [B] stnd fo dimensions of electic nd mgnetic fields espectively, while [Î 0 ] nd [µ 0 ] stnd fo dimensions of the pemittivity nd pemebility of fee spce espectively. [] nd [T] e dimensions of length nd time espectively. All the quntities e given in SI units. (Thee e two questions bsed on Pgph "X", the question given below is one of them) 6. The eltion between [Î 0 ] nd [µ 0 ] is :- (A) [µ 0 ] [Î 0 ][] [T] (B) [µ 0 ] [Î 0 ][] [T] (C) [µ 0 ] [Î 0 ] [] [T] (D) [µ 0 ] [Î 0 ] [] [T] Ans. (D) We hve, C m Î 0 0 [ ] é ù \ C ê ú ëm0î0û Þ T - m 0 Î 0 [ ][ ] Þ [µ 0 ] [Î 0 ] [] [T] PAAGAPH "A" If the mesuement eos in ll the independent quntities e known, then it is possible to detemine the eo in ny dependent quntity. This is done by the use of seies expnsion nd tuncting the expnsion t the fist powe of the eo. Fo exmple, conside the eltion z x/y. If the eos in x, y nd z e Dx, Dy nd Dz, espectively, then JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08
JEE(Advnced) 08/Ppe- x±dx z±d z y ±D y xæ Dxöæ Dyö ç ± ± yè x ç y øè ø -. The seies expnsion fo æ Dy ö ç ± y è ø -, to fist powe in Dy/y, is m ( Dy/ y ) independent vibles e lwys dded. So the eo in z will be ædx Dyö D z zç + x y è ø.. The eltive eos in Dx Dy The bove deivtion mkes the ssumption tht <<, <<. Theefoe, the highe powes of x y these quntities e neglected. (Thee e two questions bsed on Pgph "A", the question given below is one of them) ( - ) 7. Conside the tio ( + ) to be detemined by mesuing dimensionless quntity. If the eo in the mesuement of is D(D/ << ), then wht is the eo D in detemining? D D (A) ( ) (B) + ( ) + Ans. (B) æ- ö ç è+ ø D D( -) D ( + ) + ( - ) ( + ) D (C) ( ) - (D) ( ) D - D D + ( - ) ( + ) D ( + + -) ( - )( + ) D ( -) D \D ( - )( + ) ( + ) ( + ) JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08 3
JEE(Advnced) 08/Ppe- PAAGAPH "A" If the mesuement eos in ll the independent quntities e known, then it is possible to detemine the eo in ny dependent quntity. This is done by the use of seies expnsion nd tuncting the expnsion t the fist powe of the eo. Fo exmple, conside the eltion z x/y. If the eos in x, y nd z e Dx, Dy nd Dz, espectively, then x±dx z±d z y ±D y xæ Dxöæ Dyö ç ± ± yè x ç y øè ø -. The seies expnsion fo æ Dy ö ç ± y è ø -, to fist powe in Dy/y, is m ( Dy/ y ) independent vibles e lwys dded. So the eo in z will be ædx Dyö D z zç + x y è ø.. The eltive eos in Dx Dy The bove deivtion mkes the ssumption tht <<, <<. Theefoe, the highe powes of x y these quntities e neglected. (Thee e two questions bsed on Pgph "A", the question given below is one of them) 8. In n expeiment the initil numbe of dioctive nuclei is 3000. It is found tht 000 ± 40 nuclei Ans. (C) decyed in the fist.0 s. Fo x <<, In ( + x) x up to fist powe in x. The eo Dl, in the detemintion of the decy constnt l, in s, is :- (A) 0.04 (B) 0.03 (C) 0.0 (D) 0.0 N N 0 e lt lnn lnn 0 lt dn d t N - l Conveting to eo, D N Dl t N 40 \Dl 0.0 (N is numbe of nuclei left undecyed) 000 4 JEE(Advnced) 08/Ppe-/Held on Sundy 0 th My, 08