MODEL SOLUTIONS TO IIT JEE 2009

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MDEL SLUTINS T IIT JEE 9 Pper II PART I D A D B 5 7 8 9 A, B, D A, D A A, B, C B, C A p, s B q, s C r, t D q, t A p, q B p, s, t C r, s D p 5 7 8 9 9 8 8 9. log K log A Give log K A s E. R Sectio I E.

1 MDEL SLUTINS T IIT JEE 9 Pper II PART I D A D B A, B, D A, D A A, B, C B, C A p, s B q, s C r, t D q, t A p, q B p, s, t C r, s D p log K log A Give log K A s E. R Sectio I E. RT T E. 8. kj mol 8. kj mol. C is strog legd. Cofigurtio of Cr i Cr(C) is t g e g. All electros re pired. So dimgetic, BM.. If H t C migrtes, positive chrge will develop t C cotiig H group which c be stbilized b resoce ivolvig the loe pir o oge.. Betwee structures I & III, I is most stble s more electro egtive itroge crries egtive chrge th less electro egtive crbo. Betwee II & IV, II is more stble due to the sme reso. I I & III ll the toms/ios hve completel filled octet. Sectio II 5. Metls with lesser reductio potetil th the N ios get oidised ie. V, Fe d Hg re oidised i combitio with N (q). Amog the give optios, the stte fuctios re iterl eerg d molr ethlp Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.), 95B, Siddmsett Comple, Prk Le, Secuderbd 5. All rights reserved. No prt of this mteril m be reproduced, i form or b mes, without permissio i writig. This course mteril is ol for the use of bofide studets of Triumpht Istitute of Mgemet Eductio Pvt. Ltd. d its licesees/frchisees d is ot for sle. ( pges) ( ) SLJEE9/

2 7. Ammoi forms the ioic dduct [BH (NH ) ] + [BH ] Amies form the simple dduct RH N BH 8. N N, N N N N N, 9. X is o reducig sugr becuse the ldehde d keto group of the two mooscchride uits re ivolved i glcosidic likge. But Y is reducig sugr becuse the two mooscchride uits re joied ol b, glcosidic likge. Sectio III N. RT M RT M M M w 8. ph pk pk log C 8 5. Totl umber of - prticles Totl umber of prticles Aswer + 8. fusig M with KH K M is formed.. (A) Cu + 8dil.HN Cu(N ) + H + N (B) Cu + HN (coc) Cu(N ) + H + N (C) Z + HN (dil) Z(N ) + N + 5H (D) Z + HN (coc) Z(N ) + N + H. The compoud (A) is bromo compoud which udergoes ucleophilic substitutio d dehdrohlogetio (elimitio). It cot udergo dehdrogetio. (B) c udergo ucleophilic substitutio, esterifictio d dehdrogetio. (C) c udergo ucleophilic dditio t crbol group d esterifictio t H group. (D) c udergo ucleophilic substitutio of bromie tom becuse of the presece of electro withdrwig N group t the ortho positio.. Z.5 kj K.5K M 8g mol m.5 g Qv Z m M Sectio IV 9 kj mol () Cclopete () methl cclobute () ethl ccloprope Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE8/ 7. H H As + Cu H H H S - 8. I the crstllie AlCl, the C.N of Al is d it chges to four coordited dimer t its meltig poit. 9. Cclic Structures 5 Cclo pete, methl cclobute, ethlccloprope,, dimethl ccloprope d,- dimethlccloprope. Stereoisomers Cis d trs ( isomers) CH CH CH CH CH CH CH CH CH H H H H cis - isomer trs - isomer trs-, dimethl ccloprope is opticll ctive d eists i (+) d () forms. The questio is mbiguous. If it is iterpreted s totl umber of cclic structurl s well s cclic stereo isomers possible there re ol 7 isomers.

3 (), -dimethl ccloprope (5) Cis-,-dimethlccloprope (meso) () d-trs-,-dimethl ccloprope (7) l-trs-,-dimethl ccloprope If it is iterpreted s totl umber of stereo isomers d cclic structures there re 9 isomers i.e., i dditio to the bove 7 isomers there re (8) Cis--petee d (9) trs-- petee. PART II C C C D A, B, C A, B B, C, D A, D C, D 9 A q, s B p, r, st C t D r A p B q, s C q, r, s, t D r Sectio I. Give tht d c d d c d d c d c The AP is c, c, 5c, 7c Sum of the squres of the terms of the A.P c + (c) + (5c) + ( ) c c ( ) c r r c r r c r r r r r r c c c c c c. Equtig the lie is k The poit o the lie is (k +, k, k + ) sice lies o + + z 9, we hve k, Q is (,, ) PQ Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE8/

4 . (cos, si) ' 8 ' 8. cos si cot Slope of orml t si t ( t) si + t 8si t si si cos t Q(cos, ) P(cos, si) 7 cos Midpoit of PQ, si 7 h cos ; k si h cos ;k si 7 h k 9 Locus of M 9, b b ( e ) ( e ) e e Equtio of the ltus rectum is A(p, ) L L C(pq, ( + p) ( + q) D L B(q, ) p( + q) pq pq( + q) (q ( + p) p( + q)] pq [ + p ( + q)] (q p) pq[p q] pq p ( + p) pq + p( + p) p( + p) (q + ) ( + p) ( + q) pq Equtio of the AD q p q q pq q q p q pq q pq, pq + Stright lie. I Sectio II p q( + p) pq pq ( + p) Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE8/ si() d si si() d si si d si + si d si si d si (b replcig with i the first itegrl) si si I + d si( ) d si si( ) cos d si si cos cos si si si si si d si si d si si d I + cos( ) d I cos( ) si d si cos d cos cos d cos cos d

5 I I + I I I m m m m si d si si si si ( I I 5...I ) d si d cos d d cos d I I + I I I si d cos d si 5. Eccetricit of the ellipse Cosider the ellipse + b ( _ e ) ( e ) e e Solvig the two equtios, + Product of the slopes of the curves d + t poit of itersectio (, ) Foci of the ellipse + re t (, ). f() cos, f () si si cos si cos si cos cos si lim f' () lim lim + lim cos + Agi f(), sice f( + ) ( + ) f( + ) f() > cos si f" () si cos si si cos < ' () f is decresig i [, ) Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE8/5

6 7. Let P be (t, t) Tget t P is t () t Norml t P is + t t + t () Put i () T is (t, ) Put i () N is ( + t, ) Let (, ) deote the cetroid of PTN t t t t t + t ( ) t 9 Prbol whose verte is t, Ltus rectum, Focus is t (, ) 8. S (m ) m m si.si S si A m si A m m si si A m m si A si But A m A m-. m cot A cot m m A m further simplifictio, cot A cot A sia A si A si A We hve A d A ; Simplifig we get, si ( ) d the correct optios 5 re d. Sectio III 9. (A) - Give equtio is si si si + si cos si + si ( si ) si si + (si ) (si ) si, si, A (p, s), (B) We ote tht [] is discotiuous t ll iteger vlues of. Give fuctio is discotiuous wherever or is iteger. Hece the vlues re,, d. (C) - V (C) t (D) - b c b c. c + +. b..b cos D (r). (A) f() e si cos si f' () e. cos si f' () i, f() is icresig i, ; Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE8/

7 f() < d f f() hs ectl oe root i, (A) (p) (B) - The ples itersect i the lie correspodig homogeeous sstem hs o-trivil solutios k (C) - k k(k ) ( ) + (8 k) k 7 k + 8 k k 7 k + 8 (k ) (k ) k, (B) (q, s). 8 Sectio IV + 9 D 8 8 Roots re 5 d + 9 whe [, 5] Cosider f() f () + Let f () 5 +, so the grph of f() will be s give bove I the itervl [, 5] f() is icresig d the mimum vlue is t 5 f(5) (5) 5(5) + (5) Sstem hs ifiite solutios f() k k (C) q, r, s, t. Let z k k,, z k But,, z re iteger vlues k, ; z k But + + z k k k 7 iteger vlues re there. B B d (D) - d d d log( + ) + C () log C log e + e e (log) (D) (r) The trigles re ABC d A B C with ABC A B C AB A B, AC A C (Note tht it mes A + C 5 A + C ) Usig sie lw, for the two s we hve si A si sic Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE8/7 Ad A C A C

8 si A' si sic' si A,siC d si A',si C' As the trigles re o cogruet, si C sic C, C' or C, C' We cosider C BAC 5 Are of ABC cos5 C A 5 Are of A B C si5 ABC A' si 5 si 5 B' C'. Note tht p() lim p() p () d p () If p() + b + c + d + e p () + b + c + d p () + b + c p() gives e p () gives d p () gives e p() + b + p () + b + p (), p () + b + () + b + () Solvig, b b p() 8 5. f() f (t)dt Differetitig w.r.t. f' () f() d d d d Ce Sice must be zero whe, (becuse f() f (t)dt ) C must be zero f() f( 5) Smllest vlue of k Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE8/8. Let, d be the cetres of the circles C, C d C respectivel. The P P uits Let T T be the commo tget T T uit; Let T r uits. The C T r T P Now beig isosceles trigle d P is the mid poit of, therefore P is the ltitude Let P h uits. The, from P, ( + r) + h () from T P, h r () From () d () we get ( + r) r r r 8 uits C 7. 8k + (k k + ) Roots re rel d distict D > k (k k + ) > k k k + > k > () f() k + (k k + ) k + k k k 8k + k k + k must lie beod d () Usig () d (), k () Agi, the miimum poit k must be > k > P h T C

9 8. g() f () f(g()) Differetitig w.r.t f' (g())g' () g () () f' (g()) To fid g() As g f, g() is the vlue of for which f() i.e., + e () + e is icresig fuctio ( () + e ) () hs tmost oe zero We observe tht mkes () g() g () f' () But, f'() + e so tht f () g () PART III 9 D B A C 5 7 B, D D B, D A B, C 8 9 A P, Q. T B Q C S D S A P, S B Q C T D R, S, T k eff. kk k k KK F K K Sectio I Let Amplitude A K A K A A A K K K Slope k g kg k K f m M Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE8/9.. ev: p q.5 r. Sturtio currets will be the rtio :: for p : q : r Agulr SHM : Torque Kr.r r m

10 d. sme i dt Sectio II 5. Sectio IV F F i m m h A > h B A B > Bm m A B A >. Amplitude of vibrtio much less th cm Itesit (frequec) sme progs i verticl ple. Closed pipe with ed correctio Air colum < bviousl (B) d (C) cot be mtch (kowledge bsed). Now optio (A) is lso ruled out becuse; Cosider the coditio for forced resoce, t sted stte eerg iput to the ir colum b the tuig fork is equl to eerg output t resoce whtever be the hrmoics. Itesit received b the studet is the sme. 5. AB is ot shped like rectgulr hperbol ot isotherml BCD : W : egtive, T decreses u : egtive W ABC (Are) W ccle >, (Are). Similr to Kepler s lw Torque is zero 7. o Vcm Vcm Sectio III 8. Kowledge bsed 9. Itroducig film will shift cetrl frige up workig out for ech A, B, C, D bsed o cos, where is phse differece : Mimum i tesit ( ) p (5 H) (p ). H H mm T 5. p, p p + p r v r, pv costt B A m m 5. v m.9 pm m m m m Fil velocit m s m 5. R P Q s i si(7) i B p (si 7 si5) 5 I.k k 7 8 m m 55. g, (m.) m g g s m m(m) g T g m.g; w mg. 8 J m 5. E r kr. r dr r E r + R E t E t R 8. ( ) 9 T v 57. v,, 5cm Triumpht Istitute of Mgemet Eductio Pvt. Ltd. (T.I.M.E.) H: 95B, d Floor, Siddmsett Comple, Secuderbd 5. Tel : 78989/95 F : 787 emil : ifo@timeeductio.com website : SLJEE9/

Time: 2 hours IIT-JEE 2006-MA-1. Section A (Single Option Correct) + is (A) 0 (B) 1 (C) 1 (D) 2. lim (sin x) + x 0. = 1 (using L Hospital s rule).

Time: 2 hours IIT-JEE 2006-MA-1. Section A (Single Option Correct) + is (A) 0 (B) 1 (C) 1 (D) 2. lim (sin x) + x 0. = 1 (using L Hospital s rule). IIT-JEE 6-MA- FIITJEE Solutios to IITJEE 6 Mthemtics Time: hours Note: Questio umber to crries (, -) mrks ech, to crries (5, -) mrks ech, to crries (5, -) mrks ech d to crries (6, ) mrks ech.. For >, lim