SAFE HANDS & IIT-ian's PACE EDT-10 (JEE) SOLUTIONS
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1 . If their mea positios coicide with each other, maimum separatio will be A. Now from phasor diagram, we ca clearly see the phase differece. SAFE HANDS & IIT-ia's PACE ad Aswer : Optio (4) 5. Aswer : Optio () 6. ( ) Aswer : Optio () 7. Aswer : Optio (3) 8. Phasors: A As all three side are of same legth, it is equilateral triagle ad phase differece is π/3. Aswer : Optio (). Total eergy = ka PE at ( = A/3) = k (A/3) = k KE at ( = A/3) = TE PE = k As velocity is tripled, KE will become 9 times. New KE at ( = A/3) = k (5A ) New TE at ( = A/3) = k Now, this will also be TE at ew amplitude. Hece, ka ew = k A ew = Aswer : Optio (4) 3. As particle starts from rest, let the equatio of SHM be - = A cos(ωt) I time T it covers distace a. So its coordiate will be A a. A a = A cos(ωt) a = A ( cosωt) Similarly, A 3a = A cos(ωt) 3a = A ( cosωt) Takig ratio, cos ωt = 3 3cosωT cos ωt = / Puttig this i first equatio, a = A/ A = a. Aswer : Optio () 4. ad ad Whe sprigs are coected i series, = + B Aswer : Optio (3) 9. Equilibrium displacemet of block will be y = So amplitude will also be zero. Also, ω = as iitial elogatio is As block starts from etreme positio, we will use cos. The mea positio is. Aswer : Optio (3) 0. T L ( ) ϕ π/3 Aswer : Optio (4). Effective legth will decrease the CoM will be elevated. Aswer : Optio (). As legth of each part is /3 rd of origial legth, sprig costat of each part will be 3k. For two sprigs i parallel, sprig costat will be k + k = 6k. Now this is i series with sprig of costat 3k. ( )( ) = k Aswer : Optio () 3. Aswer : Optio (3) 4. y = A si(ωt + π/6) At t = 0, y = = A si (π/6) A = 0.6 m At t =, y = = 0.6 si (ω + π/6) si (ω + π/6) = ½ ω = π/3 Hece, y = 0.6 si (πt/3 + π/6) A
2 At t = 3s, y = 0.6 si (π + π/6) = 0.3m Aswer : Optio (3) 5. Aω = ad Aω = Hece, A = 0.5m ad ω = rad/s v = ω = 0.6 ms - Aswer : Optio () 6. Aswer : Optio (3) 7. v = ω (A y ) ad v = ω (A y ) Solvig these two, Aswer : Optio (4) 8. k = k (A ) = A/ Aswer : Optio () ω = 9. E X = k ad E Y = ky E X+Y = k ( + y) = ( E X + E Y ) Aswer : Optio () 0. T = π ad T = π Solvig these two eq. we get a = 3g/5. Aswer : Optio (). All the three sprigs are i parallel ad so effective force costat is k eff = k + k + k = 3k Aswer : Optio (). mg = k k = T = π = π Aswer : Optio () 3. Whe block is displaced by distace i the dowward directio, sprig will be stretched by. SAFE HANDS & IIT-ia's PACE Force o the pulley is T. So restorig force o the block = F = T = (k) = 4k. Aswer : Optio (3) 4. Aswer : Optio (4) 5. Aswer : Optio () 6. As eergy becomes oe-fourth i time t, half-life period is t/. So eergy will become oe-eighth i three half lives i.e. i 3t/. Aswer : Optio () 7. E = ka e -bt/m Hece, ( ) ( ) = e-bt/m Aswer : Optio (3) 8. At distace from the ceter of the earth, depth is d = R. g d = g ( ) = g( ) = (g/r) F = mg d = (mg/r) = m = m Aswer : Optio () 9. K = mv = ma ω cos ωt = ka cos ωt Time average of Kietic Eergy = = = ka Aswer : Optio (4) 30. Aswer : Optio (3)
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8 SAFE HANDS & IIT-ia's PACE 6. A log B log 00 = C A log 5 + B log = C log 00 = C log(5 3 ) = C log C log hece, A = C ad B = 3C for o commo factor greater tha, C = A = ; B = 3 A + B + C = m 5 3 m m 5m 4.8 5m , N Whe 7 divides 3 3 remaider =4 3 3m, m N 63. Let, S S Subtract above equatios, S S coefficiet of m m Cm m k pk 50 i S = coefficiet of 00 m! p! p! m p! k! p k!... 00] i C 50 8
9 m m mk m! Cm C pk m k pk k! m k! SAFE HANDS & IIT-ia's PACE m m m m Cm Cm3 Cm m m C Ck m mk m m m k r 0 r Tr 0 Cr This is ratioal, if 0 r ad 5 r are itegers. There are oly two ratioal terms Namely C ad C0 0 sum = = Give C C c... C 0 Itegratig both sides with respect to, we get 3 C C C C Puttig 0, k We get k 3 C0 C C C... 3 Multiplyig with both sides 3 4 C C C C
10 Differetiatig with respect to SAFE HANDS & IIT-ia's PACE 3 C 0 3C 4C C... 3 Now puttig both sides, we get C 0 3C 4C C... 3 C 3C 4C 3 C Give t ro r r0 r r r Cr C r C C r C C r0 r r0 r S... 0 C C C S r0 t r C r t S t t S t S 68. D k k k k,3, 4 3 values 69. Let,, 3 required part is 3 7 0
11 3 0 & SAFE HANDS & IIT-ia's PACE 70. First pait ay colour o ay face. Now the opposite face ca be paited i 5 ways (with ayoe of the remaiig 5 colours). Now, the remaiig 4 faces ca be paited with the remaiig 4 colours i (4-)! ways. (circular permutatios) As = 5 (4 )! 30 ways. 7. (By hypothesis) Number of participats = me + wome =3. 7. Let A(7 3, 5, 3 ), B(, 4, 3 ) D.R s of AB are i : : ,, A(,,), B(3,3,) 73. Cosider eactly three persos betwee the brothers as referece ow secod brother ca be placed i two ways (left or right), rest 8 i Ways. 74. Similarly we have arragemets for eactly 4 persos or 5 persos or 6 persos or 7 persos or 8 persos i betwee the brothers. For eactly 9 persos i betwee them we have oly 8! Ways. These will also iclude 0 or or or 3 or 4, 5 persos i betwee the brothers. So total umber of arragemets Alterate If oe of the brothers is made referece poit the remaiig 8 persos (ecludig the secod brother) ca be seated i 8! Ways. For the secod brother we have oly places to sit. Total umber of ways r m d, A a
12 SAFE HANDS & IIT-ia's PACE distace from Aa to the plae r m d is d a m m 9 If A a i j 3k the d 78 If A a 3i 3j 3k the d = Give plae passes through a & b cotaiig the lie is AP AB c 0 r a b a c r b c c a a b c 0 legth of r from the origi a b c b c c a 76. BC is the - ais ad AB BC Area of ABC Volume DE DE gives p = p 3 a c b c a b c a b c a b c a b a b a b , a. b 0 a, b, ab are mutually perpedicular c 79. Give OA a, OB b where O is the origi of vectors, also give AC 3AB OC OA 3OB OA OC 3OB. OA 3b a Agai OD. OA OB a b BD BA OD OB OA OB
13 SAFE HANDS & IIT-ia's PACE 80. Give a ad b are o-colliear. The l a m b, la m b, l3a m3 b are colliear if 8. Give 3i j 5k l i j k m i 3j k i j 3k l m 3, l 3m, l m 3 5 Solvig these equatios: l 3, m,. Hece 8. Let l l l 3 m m m l m m l m m l m m l m m are i A.P l m i l m j l m k m, l /, c a yb, c Also give that Where c a a ybi j k 0 a i j k & b i j k y y y 0 3 y c i j k y i j k y y y c i j k i j k i j k Let d d i d j d k. Now d a d d d () 3 d c d d d () 3
14 SAFE HANDS & IIT-ia's PACE Solvig () ad () d 0 d d3 d d j dk d j k d j k d d j k d d d 83. Let be the first term ad y be the commo ratio the a y, b y, c y p q r log a log p log y; log b log q log y,log c log r log y Let a log a i log b j log c k ad b q ri r p j p qk The a. b log a q r log b r p log c p q q r log p log y r p log q log y p q log r log y ab. 0 agle ab,. 84. Give B. Sice O, the give poits are ot colliear. If A,B,C are the vertices, the AB BC CA Hece the give triagle is equilateral. 85. Let the corer meetig the edge ad the face be at origi. Here a b = c k (say) ad a.b b.c c.a = k cos60 = k (90 ) would be the agle betwee the edge (say a ) ad the ormal (bc ) to the face 4
15 SAFE HANDS & IIT-ia's PACE a (bc) The si (90 ) = = a b c (a.c) b (a.b)c k.k si 60 k b c cos = = 3 3 k. k k k 86. a b c b c c a a b a b c a b c [ ] [ ] [ ] 87. Observe that the lies L, L, L3 are parallel to the vector (,-,-) Also, ad The three plaes itersect i a lie 88. AC PR ad AC PR So, ABPC is a parallelogram comparig the coordiates of mid-poit of diagoals, we get P a,b,c ad Qa, b,c ad Ra,b, c Also, AP is media of ABC ad PQR so cetroids are Coicidig. The perpedicular bisector of PR is also perpedicular to AC. Therefore circumcetre of ABC PQR is orthoceter of arpqr 4arABC 4 OAB OBC OAC Where OAB is the area of the projectio of ABC o the plae XOZ etc. 89. [ ] 0 [] + 0 [] should ot belog to [, ) Domai of f is (, ) [, ). 90. gf g f. f o puttig = 0 g f 0. f 0 g. g 5
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