Per -.(D).() Vdymndr lsses Solutons to evson est Seres - / EG / JEE - (Mthemtcs) Let nd re dmetrcl ends of crcle Let nd D re dmetrcl ends of crcle Hence mnmum dstnce s. y + 4 + 4 6 Let verte (h, k) then k, h 6 4 hk y 64 64 Locus of verte s 64y.() y f 4.() 5.() Let f () then ut y f f z 5 7, z nd (, ) s centre, z z centre, f or + + f. f f f f Let f () then ut y f f + + f f f 4 tn, tn 6 + + tn tn cot tn. tn tn tn + tn rs 84 uu vv u, v u v u+ v u+ v z z + u v + uv dy dy 6.() ( e e ) d + + d dy dy e e dy e d d d dy dy e d or e d 4 nd,, D (, ) f constnt functon. f D r. 4 4 s s 7 + z z s urely rel, hence m(z) dy dy e e d d y e + c or VM/JEE-/Solutons evson est Seres-/Per - y e + c
7.() Put ( sn α. sn t ) + we get, Vdymndr lsses 8.() + sn α sn α 4 t. dt sn α Let the dstnce from the orgn s oordnte of ont s ( cos θ, snθ ) 4 t t sn α + snα sn α 4 cot α. cosec α 9.() cos θ+ sn θ+ sn θ. cosθ 6 sn θ + 6 6 6 mn 4 snθ + + mn 4 6 sn θ + 8 9 8 8 9 8 + + + +.. d. d 9 9 8 9 8 8+ + + + + 8 8 8 8 8 +.. d +. d. d + 8+ 9 9 8 8 8.() Let f ( ) ( c) 9( b) ( d) f ( ) < nd f s ± one root le between b nd c nd one root le between (, ) `.(D) M 6 t t t t 6 r O t + t+ + 4 t t. b + c b. + c c. + b + b + c + b + c. + b + c + b + c +. b + b. c + c. 57.().() b c log, log, log b c re n.p. b c b c nd b c cos < 9 6 4 4.(D) ( [ ])( [ ]). d / / / ( )( ) ( ) ( ) ( )( ) ( ) ( ). d+. d+. d+. d / / / 7 5 9 + + + 54 6 7 7 7 y t, t VM/JEE-/Solutons evson est Seres-/Per - O
5-6. Vdymndr lsses 5.() y f nd f + f ( ) + nd f ( ) [, ) 6.() + f 9 + y 9 where y f + y + + y 9 7-8. 7.() f log ( ) + 4+ ( ) f log ( 4 ) ( 4 ) + 4+ ( 4 ) log + log 4 4log 8.() f cos ( cos ) 4 f cos cos + + 4 log log 4 f ( ) + 4+ ( ) cos cos( + ) cos ( cos ) f 9.(). f nd re tngents to the rbol, then the sloe of s of rbol wll be sme s sloe of L where L s md ont of L Snce length of tngents P nd P re equl Pont P les on the eteror rt of s of rbol Lne L s the s of rbol Snce ont of netersecton of erendculr tngents les on the drectr of the rbol nd the chord of contct of erendculr tngents s focl chord. (4, 4) s focus of rbol nd ltus rectum of rbol s equl to length. usng ths, we get (, 4) P(, 4) (4, 6) (4, 4) (4, ) L Equton of rbol s ( y 4) 4( ) y 4 8y+ 8 h, b, g, f 4, c 8.() Drectr of rbol nd drector crcle of crcle Must touch ech other.e. s tngent of + y r r VM/JEE-/Solutons evson est Seres-/Per -
.() Equton of Plne ssng through the lne y+ z + y z s y+ z + λ + y z Vdymndr lsses Snce t s erendculr to gve lne λ Equton of the lne of rojecton s 8y+ z+ 4 + y+ z ts drecton rto re (, 9, 5) nd ont (,, ).() Norml to rbol les on the lne. + y z s lso the equton of lne of rojecton. 4 5 t ssng through (5, 6) t t t,, my be tken s y.. t t( t ) y 4.. Hence normls re y 9 y+ 6.(D) he bove equton hs only one rel root α such tht < α< 4.() Let α where, q, q α + α + α ( q q ) q + + m q where m mq where m nd q re ntegers m whch s rtonl for m k q where k s nteger q q k whch s nteger nd there ests no. nteger vlue of k between & q α s rrtonl Smlrly see tht α s rrtonl 7 7 7 7 f g + g f 4 4 f + g 4 4 5.() he vector ˆ + λ ˆ j+ ˆ k s rotted We hve λ ˆ ( λ ) ˆ+ ˆj + k 4ˆ 4 ˆj k ˆ ( λ ) + λ + 9 6+ 4 + 4 λ, f + g 4 + 5 VM/JEE-/Solutons 4 evson est Seres-/Per -
Vdymndr lsses Solutons to evson est Seres - / EG / JEE - (Physcs) Per - 6.(D) ml ml 5l + + m 5ml µ µ φ ln dφ µ ln d εnd d t d t 7.() φ ( d ) 8.() ( ) µ ω ω ln cos t ε nd µ ln n cosωt L L ω g g g s Puttng vlues. m 9.(D) UV / constnt g g l O d s G l 5l snωt V / constnt.() PV / constnt + + γ n 5 / ml V L ω sn θ L snθ mvl snθ λ.(d). cm λ 6. 6cm 6. 6 V f λ ( 5 ). m / s m / s γ γ 8. V γ. 4 M 8. V. 7 J / mol k γ. 4.() Equton of refrcted ry s y tnθ + b b y + b f b f θ Y X VM/JEE-/Solutons 5 evson est Seres-/Per -
.() V V l V 4.() P Vdymndr lsses V V l V ; V V ( l V + l V ) From energy conservton 5.() o lft M, K Mg Mg k For from OE mg k k M m g 6.() V V. 6. 8 m / s P L L 7.() Equtng the horzontl comonent of velocty, u cosθ cos () lso sn u snθ gt u snθ tnθ θ 6 () 8.() E 4 + / 8 V / µ µ 8 ( + ) E µ V V 9. () q q qq qq qq U + + 4 (. 4) 4 (. ) 4 (. 4) 4 (. 5) q q 8q q 4 4 [ ] K 8q 7 5g sn7 4-4. 4.() 4.() 4.() For : + 5 ().5g For : 4 () For : 5. 5 () nd + (v) Solvng, m / s, 7/4 m/s, 4 86 4 m / s VM/JEE-/Solutons 6 evson est Seres-/Per -
Vdymndr lsses 4-45. 4.() 44.() 45.() θ r r (usng r + r nd geometry) sn 45 µ sn µ δ 45 r + 8 r + 45 5 ( θ 45, usng Snell s Lw t ) f s not olshed sn r sn e e 45 45 6 r r θ 6 6 6 θ 46.(D) 47.(D) 48.(D) r r r G m M r r E E r G m M r > r k < k r otl energy ( > )( > ) 56 8 Knetc energy r r r 4 ε V + 5 V 5 5V lso ε ( ) ( ) 5 Ω Ω Ω 5 5W Power n V Pcell ε W 49.() 5.(D) V t ˆ + t ˆj ˆ + t ˆj. V t + t t t V t VM/JEE-/Solutons 7 evson est Seres-/Per -