iitjee 2006

Section I:Multiple Choice Questions with onecorrect answer.IIT-JEE (2006)(Memory Based Question Paper)PHYSICSQ.1 A student performs experiment to determine g24πlusing formula g = 2T; here l (length of thewire) is approximately equal to 1m, ∆lrepresents error in measurement of length l,∆T represents error in measurement of time, nis total number of oscillation. Then for which ofthe data of the measurement of g will be mostaccurate ?∆l ∆T n Amplitude ofoscillation(a) 5 mm 0.2sec 10 5mm(b) 5 mm 0.2 sec 20 5mm(c) 5 mm 0.1 sec 20 1 mm(d) 1 mm 0.1 sec 50 1mmAns. (d)Q.2 Focal length of the shown plano-convex lens is15 cm. Plane surface of the lens is silvered.An object is kept on the principal axis of thelens at a distance 20 cm. Image of the objectwill form.(a) πr 2 ∝ f(b) πr 2 ∝ f 2(c) lens is covered half so half image is formed(d) on increasing focal length, brightness of theimage increasesAns. (b)Q.5 Time constant for the given circuits are.1Ω2Ω2µf 4µf1Ω 2Ω 4µf(a) 18 µs,2µf1Ω2µf 4µf2Ω8 8 µs, 4 µs (b) 18 µs, 4 µs, µs99object20 cm(a) 60 cm, left(c) 12 cm, left(b) 60 cm, right(d) 30 cm, rightAns. (c)Q.3 Half life of a radioactive sample is 4 days. Findthe probability that a particular nucleus of theradioactive material decays after 2 half life is(a) 11(b) 23(c) 23(d) 4Ans. (d)Q.4 The image of sun is formed on focal plane oflens which is a circular in shape of radius r andarea πr 2 .(c) 4 µs. 98 µs, 18 µs (d) 98 µs, 18 µs, 4 µsAns. (a)Q.6 In the system shown if the inextensible stringconnecting 2m and m is cut, the accelerationsof mass m and 2m are(a)g g ,2 2\\\ \\\ \\ \\\ \\\\\ \\\ \\\\\ \\\ \\ \\\ \\\2mmstring(b) g, 2g(c)2g , g (d) g, gAns. (b)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 25004921PAPER - IIT-JEE-2006

Q.7 In the figure shown length of each wire is l /2and their radii are 2r and r. ThenP (2r) Q(r) RIIl/2 l/2(a) current density in both wires is same(b) power dissipated in QR is 4 times that inthe PQ(c) ratio of potential drops on PQ & QR is 4(d) resistance of PQ is 4 times that of QRAns. (b)Q.8 Graph of position of image vs position of a pointobject from a convex lens is shown in the figure.Then focal length of the lens isu(cm) –31–30–10v (cm)313010(a) (0.50 ± 0.05) cm(b) (0.50 ± 0.10) cm(c) (5.00 ± 0..05) cm(d) (5.00 ± 0.10) cm Ans. (c)Q.9 Moment of inertia of solid sphere of mass mand radius R about axis passing through centerof mass is Ι as shown in figure 1. The sphere ismoulded in the form of disc of radius 'r' andthickness 't'. The moment of inertia of disc aboutthe axis shown in figure 2 is Ι.Q.10 Wire 1 is vibrating in first harmonic: Wire 2 isvibrating in second harmonic : What is theposition x of mass m4L(a) 5(c) 4L//////////////////////////////////////////////////////1 2xm(b) 5L3L(d) 4Ans. (b)Q.11 A double star system having two stars ofmasses m 1and m 2, is rotating about commoncenter of mass in radius r 1and r 2, with periodsT 1and T 2, then3⎛ T1⎞ ⎛ r1⎞(a) ⎜ ⎟ ⎜ ⎟T=2 r⎝ ⎠ ⎝ 2 ⎠(b) if T 1> T 2then R 1> R 2(c) if T 1> T 2then m 1> m 2(d) T 1= T 2Ans. (d)Q.12 The circular divisions of shown screw gauge are50 It moves 0.5 mm on main scale in onerotation. The diameter of the ball is -01050rtFig.1Fig.20302520The radius of disc is(a)2R15(b)2R5(c)R15(d)R5Ans. (a)(a) 2.25 mm(c) 1.20 mm(b) 2.20 mm(d) 1.25 mmAns. (c)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 25004922PAPER - IIT-JEE-2006

Section II: Multiple Choice Questions with one ormore correct answer(s).Q.13 An infinite wire carrying current Ι passes throughpoint O perpendicular to the plane containinga current carrying loop ABCD as shown in thefigure.CB(a) friction force is dissipative(b) on decreasing θ, frictional force decreases(c) friction force does help in rotation andopposes translation(d) friction force is necessarily µmgcosθAns. (b, c)Q.16 The given figure shows lines of force of aparticular field. Out of the following option, thefield line can not represent.IOAD(a) Net force on the loop is zero(b) Net torque on the loop is zero(c) The loop rotates in anticlockwise directionas seen from O(d) The loop rotates in clockwise direction asseen from O Ans. (a, d)(a) An electrostatic field(b) A magneto static field(c) A gravitational field(d) An induced electric fieldAns. (a, c)Q.14 A ball is rolling on the track as shown in thefigure. AB is rough surface and BC is smooth.Ball reaches to the height C. K A, K Band K Care the kinetic energies at A, B and C.Q.17 The adjacent figure shows variation of electricpotential with distance for a spherical symmetriccharge distribution system and given asqφr=4πε0r(r ≥ R0)Ah Ah CCqφr= (r ≤ R0)4πε0R0Which of the following option is/are correct ?φ rB(a) h A> h C; K B> K C(b) h A < h C ; K B > K C(c) h A= h C; K B= K C(d) h A> h C; K A< K CAns. (a, d)Q.15 A solid cylinder is rolling over an inclined planeas shown in the figure.θR 0r(a) For spherical region r ≤ R 0 total electrostaticenergy stored is zero(b) Within r = 2R 0, total charge is q.(c) There will be no charge anywhere except atr = R 0(d) Electric field is discontinuous at r = R 0Ans. (a, b, c, d)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 25004923PAPER - IIT-JEE-2006

Q.18 f(t) = Asin 2 ωt + Bcos 2 ωt + CsinωtcosωtThe above function represents SHM(a) for all values of A and B, with C ≠ 0(b) A = –B, C = 2B; with amplitude √2 B(c) A = B, C = 2B; amplitude |B|(d) A = B, C = 0 Ans. (b, c)Section III:Questions based on comprehensionswith one correct answer.PASSAGE - 1A tank has a cylindrical opening as shown ofdiameter 2r. A cylinder of diameter 4r and densityρ/3 is placed on the top of the opening as shown infigure. Liquid of density ρ is to be filled in the tank.Q.19 The graph between1 and stopping potentialλh 1h h 24rdisc(V) of three metal having work functions φ 1, φ 2and φ 3in an experiment of photo electric effectis plotted as shown in the figure which of thefollowing statement(s) is/are correct ?(λ represents wavelength of the incident ray.)vmetal 1 metal 2 metal 3θ0.001 0.002 0.0041 –1nmλ(a) Ratio of work functions φ 1: φ 2:φ 3= 1: 2 : 4(b) Ratio of work functions φ 1: φ 2: φ 3= 4:2 : 1(c) tan θ is directly proportional to hc/e, whereh is plank's constant and c is the speed oflight(d) the violet color can eject photoelectrons frommetal 2 & 3. Ans. (a, c)Q.20 A black body at temperature T is kept in a darkroom with surrounding temperature T 0. Sunrays are allowed to fall on the black body througha hole in the roof of the room while T and T 0aremaintained. Assuming that there is no changein the surrounding temperature of the room,select the correct statement(s).(a) The quantity of radiation absorbed by theblack body per unit time will increase(b) The quantity of radiation emitted by blackbody per unit time will increase(c) Black body radiates more energy per unittime in the visible spectrum(d) The reflected energy per unit time by theblack body remains same Ans. (d)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 25004924Q.21 What is the height h 1of the liquid level for themass to be lifted.(a)(c)5hh 1 =42hh 1 =32r(b)(d)5hh 1 =35hh 1 =2Ans. (b)Q.22 The disc is kept pressed and water is loweredin the tank upto the height 'h 2' as shown, whatis the height h 2for the disc to be again lifted.(a)(c)h32h34h(b) 9(d) h Ans. (b)Q.23 Water is still lowered in the tank, what is theanother height h' 2, for the disc to be lifted ?hh(a) (b) 3 4h(c) 5(d) no height h' 2at which disc will be liftedAns. (d)PASSAGE-2Equations of two waves are given asy 1= A cos[0.5πx - 100 πt)y 2 = A cos[0.46πx - 92 πt)Q.24 How many maximum will a stationary observerhear ?(a) 4 (b) 5(c) 6 (d) 7Ans. (a)PAPER - IIT-JEE-2006

Q.25 Resultant wave is given as y R= y 1+ y 2at x = 0.How many times is the resultant y Rzero in 1sec.(a) 100 (b) 192(c) 46 (d) 96Ans. (d)Q.26 Maximum velocity of the wave having maximumintensity.(a) 200 m/s(c) 192 m/s(b) 180 m/s(d) 100 m/sAns. (a)PASSAGE-3The capacitor of capacitance C can be charged(with the help of resistance R) by a battery ofe.m.f. V while keeping switch S 2open. Thecapacitor can be connected to an inductor ofinductance L by closing switch S 2and openingS 1.VQ.29 If maximum charge stored on the capacitor inLC circuit is Q 0then for t ≥ 0.(a) the charge on the capacitor⎛ π 1Q = Q 0cos⎟ ⎞⎜ − t⎝ 2 LC ⎠2d Q(b) the charge on the capacitor is Q = –LCdt 2(c) the charge on the capacitor is⎛ π 1Q = Q 0cos⎟ ⎞⎜ + t⎝ 2 LC ⎠(d) the charge on the capacitor is21 d QQ = –2LC dtAns. (b)RLQ.27 Initially the capacitor was uncharged, Nowswitch S 1is closed and S 2is kept open. If timeconstant of the given circuit is τ then(a) After 1 time constant, charge on thecapacitor is CV/2(b) After time interval 2τ, charge on the capacitoris CV(1 – e –2 )(c) The work done by the voltage source will behalf of the heat dissipated, when thecapacitor is fully charged(d) After 2 time constant, charge on thecapacitor is CV(1 – e –1 )Ans. (b)Q.28 After the capacitor gets fully charged, S 1isopened, and S 2is closed so that the inductoris connected in series with the capacitor, then(a) at t = 0, energy stored in the circuit is purelyin the form of magnetic energy(b)direction of current is same for any time t(c) at t > 0, there is no exchange of energy takesplace between the inductor and capacitorC(d) maximum current in the circuit is vLAns. (d)CS 1S 2PASSAGE - 4A number of designs have been developed formagnetic leviation. In one of the designselectrodynamic system (EDS) is used. This modelis conceptually simple because it depends only onthe attractive force between magnets andferromagnetic materials. In Japan the magneticlevitated trains run based on the EDS model whichuses Lenz's Law in the simplest form. The traincarries a magnet. As the magnet passes over ametal plate or coils of wire currents are induced inthe plates or coils that tend to oppose the originalchange this results in repulsive force which lifts thetrain. This model has a stabilizing feature that if thetrain drops the repulsion becomes stronger andpushes the train back up. If the train rises the foredecreases and the train drops down. In Maglev Trainthe dissipative forces against friction are absent. Thedisadvantage of this modal is that induced currentresults in drag force as well as lift force. This requiresmore power for propulsion The drag force is largerthan the lift force for small speed, but the drag forcemaximizes at some speed and then begins todecrease. The lift force continues to increase as thespeed increases. Thus it is advantageous to travelat high speed, but the significant drag force at lowspeed must be overcome as the train starts up.Q.30 What causes the train to lift up -(a) Electrostatic force(b) Time varying electric field(c) Magnetic force(d) Induced electric fieldAns. (c)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 25004925PAPER - IIT-JEE-2006

Ans.A → SB → P, RC → RD → Q, SQ.39 Column-I Column II(A) A stationary uniformly (P)Time independentcharged dielectric ring electrostatic fieldout of system(B) Dielectric ring (Q) Magnetic fielduniformly chargedrotating with angularvelocity ω.(C) Constant current in (R) Induced electricring i 0field(D) i = i 0 cosωt(S) Magnetic momentAns. A → PB → P, Q, SC → Q, SD → Q, R, SQ.40 Focal length of objective & eyepiece of atelescope are f O& fe respectively.Column - IColumn - II(A) Intensity of the image (P) Radius ofcurvature (R)(B) Angular magnification (Q) Dispersion of lens(C) Length of telescope (R) Focal length f O& f e(D) Sharpness of image (S) SphericalaberrationAns. A → PB → RC → RD → P, Q, SCAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 25004927PAPER - IIT-JEE-2006

IIT–JEE (2006)(Memory Based Question Paper)CHEMISTRYSection I : Multiple Choice Questions with onecorrect answer.Q.1(I)NO 2OHSO 3 HNaHCO 3⎯⎯⎯⎯→NaHCO 3⎯⎯⎯→(B)(II)Gases released in reaction (I) and (II) are :(a) CO 2, CO 2(b) SO 2, NO 2(c) SO 2, CO 2(d) SO 2, NOAns. (a)OQ.2 IUPAC name of C H C ||6 5 − − Cl(a) Benzoylchloride(b) Benzenecarbonylchloride(C ) Chlorophenyl ketone ( d )phenylchloroketoneAns. (b)Q.3 CH 3- CH = CH 2 ⎯⎯NOCl ⎯→ProductProduct is :(a) CH3 − CH − CH2− NO|Cl(b) CH3 − CH − CH2− Cl|NOQ.4 CH 3 NH 2 + CHCl 3 ⎯⎯KOH ⎯→Product,Product is :(a)⊕ΘCH 3 –N C: (b) .. ⊕CH 3 –N .. C:(c ) CH 3- NH - CH 3(d) CH 3- C ≡ NAns. (a)Q.5 Order of boiling point for the following compoundsis :OH(II) () IOHOH((III))OH(a) (I) < (II) < (III) < (IV)(b) (I) < (II) < (IV) < (III)(c) (IV) < (I) < (II) < (III)(d) (II) < (I) < (III) < (IV)OHOH((IV)III)Ans.OH((I)IV)Q.6 On dilution with H 2 O and after boiling results inthe formation of white ppt. The formation of whitegelatenous ppt. take place OH treating it withexcess mixture of NH 4 OH & NH 4 Cl. Identify theppt. which dissolves in NH 4 OH/NH 4 Cl :(a) Zn(OH) 2 (b) Al(OH) 3(c) Ca(OH) 2 (d) Mg(OH) 2Ans. (a)Q.7 Express molar specific heat in terms of R, giventhat P/ V at any instant is constant and is equalto 1.3 5(a) RT (b) RT22(c)(c ) CH 3- CH 2- CH 2- Cl(d) NO - CH 2- CH 2- CH 2-ClAns.(a)(c)24 RT (d) 0Ans. (c)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 25004928PAPER - IIT-JEE-2006

Q.8 B(OH) 3+ NaOH NaBO 2+ Na[B(OH) 4]+ H 2OThe reaction can be made to proceed in theforward direction by adding :(a) cis 1,2 diol (b) borax(c) trans 1, 2 diol (d) Na 2HPO 4Ans. (a)Q.9 The conversion A to B is carried out by thefollowing path :C DGiven : ∆S (A →C) = 50 e.u. , ∆S (C →A BD) = 30 e.u. , ∆S (B → D)= 20 e.u.Where e.u. is entropy unit then ∆S (A → B)is :(a) +100 e.u.(b) + 60 e.u.(c) –100 e.u.(d) – 60 e.u.Ans. (b)(c) The catalyst will increase the rate of forwardreaction by α and that of backward reaction β.(d) If the reaction is carried out at 400 K thenforward rate constant remains unaffectedwhile the backward rate constant isdecreased.Ans. (a)Section II : Multiple Choice Questions with one ormore correct answer(s).Q.13 The bond length of CO is 1.128Å. The bondlength of CO in Fe(CO) 5is (in Armgostrons)(a) 1.76 (b) 1.128(c) 1.15 (d) 1.118Ans. (d)OQ.14 CH – C – CH3 3+ next homologueQ.10 CuSO 4decolorizes on addition of KCN theproduct is :(a) [Cu(CN) 4] 2–(b) Cu 2+ get reduced to form [Cu(CN) 4)] 3–(c) CuCN(d) Cu(CN) 2Ans. (b)Q.11 Ag + + NH 3[Ag(NH 3) + ] ; K 1= 1.6 × 10 3[Ag(NH 3 )] + + NH 3 [Ag(NH 3 ) 2 ] +K 2 = 6.8 × 10 3The formation constant of [Ag(NH 3 ) 2 ] + is(a) 6.08 × 10 6 (b) 1.08 × 10 7(c) 1.08 × 10 3 (d) 3.8 × 10 5Ans.Q.12 In the Haber process which of the following iscorrect if the reaction is carried out in a vesselof constant volumeN 2 + 3H 2 2NH 3(a) If N 2 is added the equilibrium will shift toright. Because according to II law ofthermodynamics entropy must increase inthe direction of spontaneous reaction.(b) The condition for equilibrium isG N + 3G2 H = 2G2 NH where G is Gibbs3free energy per mole of the gaseous speciesmeasured at that partial pressure. Thecondition of equilibrium is unaffected by theuse of catalyst which increases the rate ofboth the forward reaction by α and that ofbackward reaction by β.(b)Q.15NH OH 2mixtureFollowing statements is/are correct aboutmixture :(a) mixture is 3-types of oximes(b) mixture is 2-types of oximes(c) all are optically active(d) one is optically active.Ans. (a)P(a)(b)CH –CH –CH –Cl3 2 2(i) O 2/ ∆+(ii) H 3O& Q are :AlC l 3P Q ++ PhenolOand CH – CH – C – H3 2and CH 3 – C – CH3(c) and CH 3 – C – CH3(d)OOOand CH – CH – C – H3 2Ans.(c)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 25004929PAPER - IIT-JEE-2006

Q.16Q.17CH 3CH 3–CH–CH 2–CH3Cl /hv 2N (no. of Isomers)Fractional distillati on (F),⎯⎯⎯⎯⎯⎯⎯⎯→(N) and (F) are :(a) 6,4 (b) 4,4 (c) 6,6 (d) 3,3Ans. (a)OCOThe above reagent can besynthesis from the following compounds inalkaline medium following by acidification :(a)(c)CHOCHOCOOCH 3CH 3(b)(d)Ans.COOHCHOCOOCH 3COOHCAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 2500492(a)Q.18 Which of the following is the only incorrectstatement ?zABCIdeal gasP(a) For the gas (A) ; a = 0 and it varies linearlywith P(b) For the gas (B), b = 0 & it varies linearly withpressure.(c) C represent a typical real gas for whichneither a nor b is equal to 0. If minima & thepoint of intersection is known with Z = 1, a &b can be calculated.(d) C will represent a typical vander Waals gaswhose a & b not equal to zero. If minima ofthe curve is known then value a & b can bedetermined.Ans. (d)Q.19 When CO 2 is passed through water which onethe species present :(a) CO 2 ,H 2 CO 3 ,HCO 3 – , CO 32–(b) HCO 3 – , CO 32–(c) CO 2 ,H 2 CO 3(d) H 2 CO 3 ,HCO 3–Ans.(a)10Q.20 The reaction of MgSO 4and NH 4OH + Na 2HPO 4results in the formation of white ppt. The ppt.are of :(a) MgNH 4PO 4(b) Mg 3(PO 4) 2(c) MgSO 4.MgCl 2(d) MgSO 4. 2MgCl 2Ans. (a)Section III : Questions based on comprehensionswith one correct answer.Passage : 1The reaction sequence given below representsHoffmann bromamide reaction in which RCONH 2is converted into R - NH 2:OC – NH 2ClΙNH 2ClVIOC – NH–BrClΙΙHN – C – O –ClVOClOΘC – N–Br ..ΙΙΙN = C = OElectron donating groups attached to phenylactivates the reaction. The above reactionsequence is interamolecular reaction.Q.21 The reagent used to convert (I) to (II) is :(a) Br 2/NaOH(b) KBr/NaOH(c) NBS(d) KBrAns. (a)Q.22 Rate determining step of reaction is :(a) formation of (II) (b) formation of (III)(c) formation of (IV) (d) formation of (VI)Ans. (c)ClΙVPAPER - IIT-JEE-2006

Passage : 4Carbon-14 is used to determine the age oforganic material. The procedure is based on theformation of 14 C by neutron capture in the upperatmosphere 14 11417 N + 0 n ⎯⎯→6C + 1 p , 14 C isabsorbed by living organisms duringphotosynthesis. The 14 C content is constant inliving organism once the plant or animal dies,the uptake of carbon dioxide by it ceases andthe level of 14 C in the dead being, falls due tothe decay which 14 C undergoes14−C ⎯⎯→N+ β . The half life period of614 C is5770 years. The decay constant (λ) can becalculated by using the following formula0.693λ =t. The comparison of the β – activity1/ 2of the dead matter with that of the carbon still incirculation enables measurement of the periodof the isolation of the material from the livingcycle. The method hoever, ceases to beaccurated over periods longer than 30,000 years.The proportion of 14 C to 12 C in living matter is 1: 10 12 .Q.30 Which of the following is correct ?(a) In living organism, circulation of 14 C fromatomosphere is high so the carbon contentis constant in organism(b) Carbon dating can be used to find out theage of earth crust and rocks(c) Radioactive absorption due to cosmicradiation is equal to the rate of radioactivedecay, hence the carbon content remainsconstant in living organism.(d) Carbon dating can not be used to determineconcentration of 14 C in dead beings.Ans. (c)Q.31 What should be the age of fossil for meaningfuldetermination of its age ?(a) 6 years(b) 6000 years(c) 60,000 years(d) It can be used to calculate any ageAns. (b)Q.32 A nuclear explosion has taken place leading toincrease in concentration of 14 C in nearby areas.14 C concentration is C 1in nearby areas and C 2in areas far away. If the age of the fossil isdetermined to be T 1and T 2at the placesrespectively then :(a) The age of the fossil will increase at the placewhere explosion has taken place and1 CT11 − T2= lnλ C2(b) The age of the fossil will decrease at theplace where explosion has taken place and1 CT11 − T2= lnλ C2(c) The age of fossil will be determined to besame.T 1 C1(d) =T2C2Ans. (b)Section IV : This section consists of 4 questions.Answers are to be given in between0000 to 9999 in the form of nearestinteger.Q.33 For a cubical system the following informationare available.Edge length = 5Å ; density = 2 gm/cm 3 , Atomicwt. = 75Determine the radius of the atom in pm ?Ans. 0217Q.34 75.2 g phenol is added in (1kg) of the solvent .The depression in freezing point is 7. Calculatethe percentage association if phenol undergoesdimerisation (k f= 14).Ans. 0075Q.35 2 moles of CO are mixed with 1 mole of O 2in acontainer of 1l. They completely to form 2 molesof CO 2. If the pressure in the vessel changesfrom 70 atm to 40 atm, what will be the value of∆U in kJ at 500 K (The gases deviateappreciably from ideal behaviour.)2 CO + O 2⎯→ 2CO 2; ∆H = -560 kJ/mol,1 L atm = 0.1 kJAns. ∆U = 0557CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049212PAPER - IIT-JEE-2006

Q.36 K spof AgBr = 12 × 10 –14 . To the saturatedsolution of AgBr, AgNO 3is added at the conc.of 10 –7 (M) λ ∞of Ag + = 6 × 10 –3 ohm –1 m 2 mol –1 , λ ∞of NO 3–= 7 × 10 –3 ohm –1 m 2 mol –1 , λ ∞of Br – = 8 × 10 –3 ohm –1 m 2 mol –1ohm –1 m –1 .Ans. 55 × 10 7Section V : This section consists of 4 questions. Eachhaving two columns with 4 entries ineach column. Entries of column I areto be matched with column II. Oneentry of column I may have more thanone matching in column II.Q.37 (a) SiO4–4 ⎯→ Si 2 O4–7 (P) Heat(b) B 4 O2–7 ⎯→ B (OH) 3 (Q) Hydrolysis(c) Bi 3 + ⎯→ [BiO] + (R) Acidification(d) AlO-2 ⎯→ Al(OH) 3 (S) Dilution bywaterAns. (a) → P ; (b) → Q & R ; (c) → Q ; (d) → RQ.38 Match the extraction process listed in ColumnI with metals listed in Column II :Column IColumn II(a) Self reduction(P) lead(b) Carbon reduction(Q) Silver(c) Complex formation and (R) Copperdisplacement by metal(d) Decomposition of iodide (S) BoronAns. a - P , R ; b - P, R ; c - Q ; d - SQ.40 Match the following(A)(B)(C)(D)Ph–CH 2–CH 2–Br& Ph–CD 2–CH 2–BrRea cts with thesame ratePh–CH–CH 3Brfaster thanPh–CH–CD 3BrPh–CH –CH –Br2 2Ph–CD = CH 2BrCH –CH–CD3 3alc. KOHCH =CH–CD2 3(Major Product)reacts–CHOD/CHO2 5 2 51(P) E reaction2(Q) E reaction1(R) ECBReactionst(S) 1 orderreactionAns. (a) → (P) & (S) ; (b) → (Q) ; (c) → (R) ;(d) → (Q)Q.39 According to Bohr's theory ,E n= Total energy ; K n= Kinetic energyV n= Potential energy r n= Radius of n th orbitMatch the following :Column IColumn II(a) V n/K n= ? (P) 0(b) If radius of nth orbital (Q) –1∝ Exn, x = ?(c) Angular momentum in (R) –2lowest orbital1 y(d) ∝ Z ,y = ?nr(S) 1Ans. a→ R ; b→ Q; c→ P; d→ SCAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049213PAPER - IIT-JEE-2006

IIT-JEE (2006)(Memory Based Question Paper)MathematicsSection I:Multiple Choice Questions with onecorrect answer.Q.1 If t 1 = (tanθ) tanθ , t 2 = (tanθ) cotθ ,t 3= (cotθ) tanθ , t 4= (cotθ) cotθ and letθ ∈ (0, 4π ) then(a) t 4< t 2< t 1< t 3(b) t 4< t 1< t 3< t 2(c) t 4< t 3< t 2< t 1(d) t 2< t 1< t 3< t 4Ans. [d]Q.2 The axis of parabola is along the line y =xand the distance of vertex from origin is 2and that from its focus is 2 2 . If vertex andfocus both lie in the first quadrant, so theequation of parabola is –(a) (x–y) 2 = 8 (x+ y– 2)(b) (x–y) 2 = 4 (x+ y– 2)(c) (x –y) 2 = (x + y– 2)(d) (x– y) 2 = (x – y– 2)Ans. [a]Q.3x⎛⎜⎜ sin x0 ⎜⎝Lim→( )1xsin x ⎞⎛ 1 ⎞ ⎟+ ⎜ ⎟ ⎟⎝ x ⎠ , for x > 0⎟⎠(a) 0 (b) –1(c) 2 (d) 1 Ans. [d]Q.4 Given an isoscels triangle, whose one angleis 120° and inradius istriangle is–3 . So the area ofQ.5 Let a, b, c be sides of a triangle and any twoof them are not equal and λ∈R. If the rootsof the equation. tx 2 + 2 (a+b+c)x+3λ(ab+bc+ca) = 0 are real, then(a) 34 < λ < 35(c) λ > 35(b)(d) λ < 341 5 < λ 0. When 0< θ < 2π⎛13π ⎞(a) ⎜ , 2π ⎟⎝ 48 ⎠(c)Q.7∫ 3x(a)(c)⎛ π 5π⎞⎜ , ⎟⎝ 8 6 ⎠⎛ π ⎞ ⎛ π 5π⎞(b) ⎜0,⎟ ∪ ⎜ , ⎟⎝ 8 ⎠ ⎝ 6 6 ⎠⎛ π ⎞ ⎛ 5π ⎞(d) ⎜0,⎟⎝ 6∪ ⎜ , 2π ⎟⎠ ⎝ 6 ⎠2x − 1dx is equal to4 22x − 2x + 14 22 x − 2x + 122x(b)4 22 x − 2x + 12x+ c4 22 x − 2x + 1+ c3x+ cAns. [d](a) 4π (b) 12 + 7 3(c) 7+12 3 (d) 12 – 7 3Ans. [b](d)Ans. [a]4 22x − 2x + 1x+ cCAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049214PAPER - IIT-JEE-2006

Q.8 If ω = α+iβ where β ≠ 0 and z ≠ 1 satisfies⎛ ω − ωz⎞the condition that ⎜ ⎟⎝ 1−z ⎠is purely real,then set of z is –(a) {z : z = z } (b) {z : |z| = 1, z ≠1}(c) {z : z ≠ 1} (d) {z : |z| = 1}Ans. [b]Q.9 A plane passes through (1, –2, 1) and isperpendicular to two planes 2x – 2y + z = 0and x – y + 2z = 4. The distance of theplane from point (1, 2, 2) is –(a) 2 2 (b) 0(c) 1 (d) 2 Ans. [a]Q.10 If f ′′ (x) = –f(x) and g(x) = f′(x) and22⎛ ⎛ x ⎞⎞⎛ ⎛ x ⎞⎞F(x) =⎜f⎜⎟2⎟ +⎜g⎜⎟⎝ ⎝ ⎠⎠2⎟⎝ ⎝ ⎠⎠and given thatF(5) = 5, then F (10) is(a) 15 (b) 0(c) 5 (d) 10 Ans. [c]Q.11 Let a = î + 2 ĵ +kˆ , b = î – ĵ +kˆand c = î + ĵ – kˆ . A vector in the planeof a and b whose projection on c is of1length unit is –3(a) 4 î + ĵ – 4 kˆ(c) 2 î + ĵ – 2 kˆ(b) 4 î –ĵ + 4 kˆ(d) 3 î +ĵ – 3 kˆAns. [b]Q.12 If r, s, t are prime numbers and p, q are thepostive integers such that the LCM of p,q isr 2 t 4 s 2 , then the number of ordered pair (p,q)is –(a) 224 (b) 225(c) 252 (d) 256Ans.[b]Section II:Multiple Choice Questions with one ormore correct answer(s).Q.13 If f(x) = min{1, x 2 , x 3 }, then(a) f ′(x) > 0 ∀ x ∈ R(b) f(x) is continuous ∀ x ∈ R(c) f(x) is not differentiable for two values of x(d) f(x) is not differentiable but continuous∀ x ∈ R Ans. [b, d]Q.14 If a hyperbola passes through the focus of2 2x ythe + = 1 and its transverse and25 16conjugate axes coincide with the major andminor axis of ellipse, and product ofeccentricities is 1, then(a) Focus of hyperbola is (5, 0)(b) Focus of hyperbola is (5 3 , 0)2 2x y(c) The equation of hyperbola is − = 19 25(d) The equation of hyperbola is2 2x y− = 19 16Ans. [a, d]Q.15 The equation(s) of common tangent(s) to theparabola y = x 2 and y =–(x–2) 2(a) y = – 4 (x–1) (b) y = 0(c) y = 4(x–1) (d) y = – 30x–50Ans. [b, c]Q.16 Internal angle bisector of ∠A of triangle ABC,meets side BC at D. A line drawn through Dperpendicular to AD intersects the side ACat P and the side AB at Q. If a, b, c representthe sides of ∆ABC then.(a) AD =(b) PQ =2bcb + c4bcb + ccos 2Asin 2A(c) The triangle APQ is isosceles(d) AP is HM of b and cAns.[a, b, c, d]CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049215PAPER - IIT-JEE-2006

Q.17 A tangent at P(x,y) point on the curve y =f(x) intersects to the axes at A and B pointsrespectively such that AP : BP ≡ 1 : 3,given that f(1) = 1, then–(a) normal at (1, 1) is x + 3y = 4(b) equation of curve is 3y + xy′ = 0(c) curve passes through (2, 1/8)(d) equation of curve is xy′ – 3y = 0Ans. [b,c]Q. 18 f(x) is a cubic polynomial such that f(3) = 18,f(–1) = 2 and f(x) has local maximum atx = –1. If f ′(x) has local maximum at x =0, then(a) f(x) is increasing for x ∈ [1, 2, 5 ](b) the distance between (–1, 2) and (a, f(a))where x = a is the point of local minimumSection III :Questions based on comprehensionswith one correct answer.Passage – INow we define the definite integral using theformula ∫ b b − af (x) dx = (f(a) + f(b)), for morea2accurate result for c ∈ (a,b) ,b − a⎢⎣ 2b − c⎢⎣ 2⎡⎤ ⎡⎤F (c) = ( f(a) + f(c) ) ⎥⎦ + ( f(b) + f(c) ) ⎥⎦when c =a + b,2∫ b ⎛ b − a ⎞f (x) dx = ⎜ ⎟a ⎝ 4(f(a)+ f(b) + 2f(c))⎠is 2 5(c) f(x) has local minima at x = 1(d) the value of f(0) = 5Ans. [b c]Q.19 Let A is a vector parallel to line ofintersection of planes P 1and P 2throughorigin. P 1is parallel to the vectors 2 ĵ +3 kˆand 4 ĵ – 3 kˆ and P 2is parallel to ĵ – kˆπQ.21 Evaluate 2 sin x dx∫0π(a) (1 + 2 )8 2π(c) (1 2)4 +(b)π(18 + 2)(d)π(1 24 + 2)Ans [b]and 3 ĵ +3 kˆ than angle between vectorsA and 2 î + ĵ –2 kˆ is–π(a) 2(c)π6⎧ xe⎪Q.20 Let f(x) = ⎨ 2−ex−1⎪⎩ x −e(b) 4π3π(d)4Ans [b, d]0 ≤ x < 11

Passage – IITotal n urns each containing (n+1) balls suchthat the i th urn contains i white balls and(n + 1 – i) red balls.Now u ibe the event of selecting i th urn,i = 1, 2, 3 .... n and w denotes the event ofgetting a white ball.Q.24 If P (u i) ∝i, where i = 1, 2, 3 , ...... n, thenlimn → ∞P(w) is equal to(a) 32(b) 1Q.28 A variable circle touches to the line L andcircle C 1 externally such that both circlesare on the same side of the line, then thelocus of center of variable circle is –(a) ellipse(b) circle(c) Hyperbola (d) parabolaAns. [d]Q.29 A line M through A is drawn parallel to BD.Locus of point R, which moves such that itsdistances from the line BD and the vertex Aare equal, cuts to line M at T 2and T 3andAC at T 1, then area of triangle T 1T 2T 3is(c)34(d) 41Ans [a]Q.25 If P (u i) = c, where c is a constant thenP (u n/w) is equal to(a) 21 (sq.units)(c) 1 (sq.units)(b) 2(sq.units)(d) 34 (sq.units)Ans. [c]2(a)n + 1(c)1n + 1(b)nn + 11(d)2( n + 1)Ans [a]Passage – IVIf A =⎡10 1 ⎤⎢01 −1⎥, if U⎢⎣1 1 0 ⎥ 1, U 2and U 3are⎦column matrices satisfyingQ.26 If n is even and E denotes the event ofchoosing even numbered urn and alsoP(u i ) = n1 , then find the value of P (w/E)n + 2(a)2( n + 1)(b)n + 22n + 1n1(c)(d)n + 1n + 1Ans [a]Passage – IIILet C 1is a circle touching to all the sides ofsquare ABCD of side length 2 units internallyand C 2circle is passing through the verticesof square. A line L is drawn through A.Q.27 Let P is a point on C 1 and Q is on C 2 , then2 2 2 2PA + PB + PC + PD2 2 2 2QA + QB + QC + QD=(a) 0.75 (b) 0.5(c) 1.25 (d) 1Ans. [a]CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049217⎡1⎤⎡2⎤⎡3⎤A U 1 = ⎢0⎥, AU⎢⎣0⎥2 = ⎢3⎥, AU⎦ ⎢⎣0 ⎥ 3 = ⎢2⎥, and⎦ ⎢⎣1⎥⎦U is 3×3 matix whose columns are U 1, U 2,U 3then answer the following questions.Q.30 The value of |U| is –(a) 2 (b) 3(c) 6 (d) 12Q.31 Sum of the elements of U –1 is –(a) 1/12 (b) 1/6(c) 1/3 (d) 1/4Q.32 Find the value of –⎡3⎤[3 2 0] U ⎢2⎥⎢⎣0⎥⎦is -(a) 13 (b) 26(c) 12 (d) 24Ans. [b]Ans. [c]Ans. [a]PAPER - IIT-JEE-2006

Section IV :This section consists of 4 questions.Answers are to be given in between0000 to 9999 in the form of nearestinteger.Q.33 If roots of x 2 –10cx –11d = 0 are a, b and theroots of x 2 –10ax –11b = 0 are c, d, then thevalue of a+b+c+d is equal to (a,b,c,d aredifferent numbers) ………………………….Ans. [1210]Q.34 The value 50501∫∫01050 100( 1−x ) dx50101( 1−x ) dxis equal to…………………..... Ans [5051]2 33 ⎛ 3 ⎞ ⎛ 3 ⎞Q.35 Let a n = – ⎜ ⎟ + ⎜ ⎟ – ………………4 ⎝ 4 ⎠ ⎝ 4 ⎠n(–1) n–1 ⎛ 3 ⎞⎜ ⎟⎠ and b⎝ 4n= 1–a nthen find thenatural number n 0such that b n>a n, n > n 0.is………. Ans [0005]Q.36 If f(x) is twice differentiable and f(a) = 0,f(b) = 2, f(c) = –1, f(d) = 2, f(e) = 0,where a < b < c < d < e then the minimumnumber of zeroes of g(x)= {f ′(x)}2 + {f ′′(x)f(x)} in [a,e] is ?Ans.[0006]Section V: This section consists of 4 questions.Each having two columns with 4entries in each column. Entries ofcolumn I are to be matched withcolumn II. One entry of column I mayhave more than one matching incolumn II.Ans.–(a) → (p)(b) → (s)(c) → (r)(d) → (q)Q.38 Column I(a) Area bounded by (p) 0–4y 2 = x and x–1= – 5y 2(b) cosine of the angle (q) 6 ln 2of intersection of curvesy= x x –1 and y = 3 x–1 log xAns.–(c) / cos xsin x( sin x) ( cos x.cot x – log(sin x) )∫ π 20(r) 4/3,(d) Not available (s) 1,(a) → (r)(b) → (s)(c) → (s)Q.39 (a) –1⎛1 ⎞∑ ∞ tan ⎜ ⎟ = t , (p) 02i=1 ⎝ 2i ⎠then tan t =(b) A line perpendicular to (q) 35x + 2y + 2z = 0 and passesthrough (0, 1, 0) then theperpendicular distance ofthe line from origin is(c) Not available (r) 1(d) Not available (s) 2/3dxQ.37 Three normals drawn at P,Q and R on theparabola y 2 = 4x intersect at (3, 0). ThenColumn I Column II(a) Radius of circumcircle (p) 5/2of ∆ PQR(b) Area of ∆ PQR (q) (5/2, 0)(c) Centroid of ∆ PQR (r) (2/3, 0)(d) circumcentre of ∆ PQR (s) 2Ans.–(a) → (r)(b) → (s)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049218PAPER - IIT-JEE-2006

Column IQ.40(a) Two rays in the Ist quadrant ax–y = 1and x + y = |a| cut each other in theInterval a ∈ (a 0 , ∞) the value of a 0 is(b) sinA sinB sinC + cosAcosB = 1then sinC =(c)10(1 − y2 )dy + (y2∫−1)dy01∫(d) Let a = α î + β ĵ + γ kˆ , kˆ ×(kˆ × a ) = 0and a point (α,β,γ) lies on planex + y + z = 2, then γ =Column II(p) 4/3(q)11∫− xdx + 1∫+00–1x dx(r) 1(s) 2Ans.–(a) → (r)(b) → (r)(c) → (p) & (q)(d) → (s)CAREER POINT, 112, SHAKTI NAGAR , KOTA (RAJ) PH. 250049219PAPER - IIT-JEE-2006