PAGE www.aieeepage.com PAGE AGE 1 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 1. The reaction of toluene with 2 Cl in presence of 3 FeCl gives predominantly : 1) benzyl chloride 2) o -and p -chlorotoluence 3) m -chlorotoluence 4) benzoyl chloride Ans. (2) Sol. 3 CH − group is ortho para directing 2. Presence of a nitro group in a benzene ring 1) renders the ring basic 2) deactivates the ring towards nucleophilic substituion 3) deactivates the ring towards electrophilic substitution 4) activates the ring towards electrophilic substitution. Ans. (3) Sol. 2 NO − group is both -I and -m. 3. In which of the following ionization processes, the bond order has increased and the magnetic behaviour has changed ? a) NO NO+ → b) 2 2 O O+ → c) 2 2 N N+ → d) 2 2 C C+ → Ans. (1) AIEEE 07 CHEMISTRY SOLUTIONSPAGE www.aieeepage.com PAGE AGE 2 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 4. The actinoids exhibit more number of oxidation state in general than the lanthnoids. This is because 1) there is a similarity between 4f and 5f orbitals in their angular part of the wave function 2) the actinoids are more reactive than the lanthnoids 3) the 5f orbitals extend further from the nucleus than the 4f orbitals. 4) the 5f orbitals are more buried than the 4f orbitals. Ans. (3) 5. Equal masses of methane and oxygen are mixed in an empty container at o 25 C . The fraction of the total pressure exerted by oxygen is 1) 1 273 3 298 × 2) 13 3) 12 4) 23 Ans. (2) Sol. Partial Pressure = Total pressure x mole fraction x 32 P x x 16 32 = × + 1 P 3 = × 6. A 5.25% solution of a substance is isotonic with a 1.5% solution of urea (molar mass = 60 g 1 mol− ) in the same solvent. If the densities of both the solutions are assumed to be equal to 3 1.0 gcm , − molar mass of the substance will be 1) 1 115.0 g mol− 2) 1 105.0 g mol− c) 1 210.0 g mol− d) 1 90.0 g mol− Ans. (3) Sol. Since the solutions are isotonic their conc. must be equal Molar mass of unknown solute 1 5.25 60 210.0 g mol 1.5 − = × =PAGE www.aieeepage.com PAGE AGE 3 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 7. Assuming that water vapour is an ideal gas, the internal energy change ( ) U ∆ when 1 mol of water is vapourised at 1 bar pressure and o 100 C, (Given : Molar enthalpy of vapourisation of water at 1 bar and 373 K = 41 kJ 1 mol− and R = 8.3 J 1 1 mol K − − ) will be : 1) 1 3.7904 kJ mol− 2) 1 37.904 kJ mol− 3) 1 41.00 kJ mol− 4) 1 4.100 kJ mol− Ans. (2) Sol. H U nRT ∆ =∆ + ( ) ( ) 2 2 g H O H O → l n 1 ∆ = 3 41 U 8.3 10 373 − = ∆ + × ×1 U 37.904 kjmol− ∆ = 8. In a saturated solution of the sparingly soluble strong electrolyte 3 AglO (Molecular mass = 283) the equilibrium which sets in is ( ) ( ) ( ) S3 3 aq aq AglO Ag IO + − + ! If the solubility product constant sp K 3 AglO at a given temperature is 8 1.0 10 , − × what is the mass of 3 AgIO contained in 100 ml of its saturated solution 1) 3 2.83 10 g − × 2) 7 1.0 10 g − × 3) 4 1.0 10 g − × 4) 2 28.3 10 g − × Ans. (1) Sol. ( ) ( ) ( ) S3 3 aq aq AgIO Ag IO + − + ! sp 3 K Ag IO + − = 3 Ag IO + − = 8 4 sp Ag K 10 10 M + − − ∴ = = = 4 w 1000 10 283 100 −= × 3 w 2.83 10 g − = ×PAGE www.aieeepage.com PAGE AGE 4 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 9. A radioactive element gets spilled over the floor of a room. Its half-life period of 30 days. If the initial activity is ten times the permissible value, after how many days will it be safe to enter the room ? 1) 300 days 2) 10 days 3) 100 days 4) 1000 days Ans. (3) Sol. 0 N 2.303 K log t N = 0 12 N 0.693 2.303 log t t N = 0.693 2.303 log10 30 t = t 100 days = 10. Which one of the following conformations of cyclohexane is chiral ? 1) Rigid 2) Chair 3) Boat 4) Twist boat Ans. (4) Sol. First boat structure lacks plane of symmetry. 11. Which of the following is the correct order of decreasing 2 SN reactivity ? 1) 2 2 3 RCH R CH R C > > X X X 2) 3 2 2 R CH R CH RCH > > X X X 3) 2 3 2 R CH R C RCH > > X X X 4) 2 3 2 RCH R C R CH > > X X X Ans. (1) Sol. The order of reactivity of alkyl halides towards 2 SN is 0 0 0 1 2 3 > >PAGE www.aieeepage.com PAGE AGE 5 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 12. In the following sequence of reactions, 2 2 P I H O Mg HCHO 3 2 ether CH CH OH A B C D + → → → → the compound ‘D’ is : 1) n-butyl alcohol 2) n-propyl alcohol 3) propanal 4) butanal Ans. (2) Sol. 3 2 H O P I Mg HCHO 3 2 2 5 2 5 dry ether CH CH OH C H I C H MgI H C H + + → → → − − → 2 5 C H OMgI 2 5 C H H C H − − OH or 2 5 2 C H CH OH 13. Which of the following sets of quantum numbers represents the highest energy of an atom ? 1) 1 n 3, 2,m 1,s 2 = = = =± l 2) 1 n 4, 0,m 0,s 2 = = = =+ l 3) 1 n 3, 0,m 0,s 2 = = = =+ l 4) 1 n 3, 1,m 1,s 2 = = = =+ l Ans. (1) Sol. The Q.no. corresponds to ‘3d’ 1) 3d 2) 4s 3) 3s 4) 3p decreasing order of energy 3d 4s 3p 3s > > >PAGE www.aieeepage.com PAGE AGE 6 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 14. Which of the following hydrogen bonds is the strongest ? 1) F H.....F − 2) O H.....O − 3) O H.....F − 4) O H.....N − Ans. (1) Sol. Due to higher electronegativity of fluosene 15. In the reaction, ( ) ( ) ( ) ( ) ( ) 3 s aq aq aq 2 g 2A 6HC 2A 6C 3H , + − + → + + l l l l 1) 33.6 L ( ) 2 H g is produced regardless of temperature and pressure for every mole Al that reacts 2) 67.2 L ( ) 2 H g at STP is produced for every mole Al that reacts 3) 11.2 L ( ) 2 H g at STP is produced for every mole ( ) aq HCl consumed 4) 6 L ( ) aq HCl is consumed for every 3L ( ) 2 H g produced. Ans. (3) Sol. 1 equvalent of acid HCl liberates 1 equvalent of hydrogen gas ( 11.2 th at STP) 16. Regular use of which of the following fertilizers increases the acidity of soil ? 1) Urea 2) Superphosphate of lime 3) Ammonium sulphate 4) Potassium nitrate Ans. (3) Sol. Salt of weak base and strong acid 17. Identify the correct statement regarding a spontaneous process : 1) Endothermic processes are never spontaneous 2) Exothermic processes are always spontaneous 3) Lowering of energy in the reaction process is the only criterion for spontaneity 4) For a spontaneous process in an isolated system, the change in entropy is positive Ans. (4)PAGE www.aieeepage.com PAGE AGE 7 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 18. Which of the following nuclear reactions will generate an isotope ? 1) positron emission 2) α− particle emission 3) β− particle emission 4) neutron particle emission Ans. (4) 19. The equivalent conductances of two strong electrolytes at infinite dilution in 2 H O (where ions move freely through a solution) at o 25 C are given below : 3 o 2 CH COONa 91.0 S cm /equiv ∧ = o 2 HC 426.2 S cm /equiv ∧ =l What additional information/quantity one needs to calculate o ∧ of an aqueous solution of acetic acid ?1) o ∧ of 3 CH COOK 2) The limiting equivalent conductance of ( ) oH H λ + + 3) o ∧ of chloroacetic acid ( ) 2 C CH COOH l 4) o of NaC ∧ l Ans. (4) Sol. o o o o 3 3 CH COOH CH COONa HCl NaCl ∧ =∧ + ∧ − ∧ 20. Which one of the following is the strongest base in aqueous solution ? 1) Aniline 2) Dimethylamine 3) Methylamine 4) Trimethylamine Ans. (2) Sol. Aliphate amines are more basic than Aromatic amines Among aliphatic amines, decreasing order of basic nature of amines is 0 0 0 2 3 1 > >PAGE www.aieeepage.com PAGE AGE 8 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 21. The compound formed as a result of oxidation of ethyl benezene by 4 KMnO is : 1) acetophenone 2) benzoic acid 3) benzyl alcohol 4) benzophenone Ans. (2) Sol. Alkyl benzenes containing α− Hydrogen are oxidised to benzoic acid 22. The IUPAC name of is 1) 4, 4 -dimethyl-5, 5-diethylpentane 2) 5, 5-diethyl-4, 4-dimethypentane 3) 3-ethyl-4, 4-dimethylheptane 4) 1, 1-diethyl-2, 2-dimethylpentane Ans.(3) Sol. 7 6 5 4 3 1 2 23. Which of the following species exhibits the diamagnetic behaviour ? 1) 2 O+ 2) 2 O 3) NO 4) 22 O − Ans. (4) 24. The stability of dihalides of Si, Ge, Sn and Pb increases steadily in the sequence 1) 2 2 2 2 Si Ge Pb Sn << << << X X X X 2) 2 2 2 2 Si Ge Sn Pb << << << X X X X 3) 2 2 2 2 Pb Sn Ge Si << << << X X X X 4) 2 2 2 2 Ge Si Sn Pb << << << X X X X Ans. (2) Sol. The stability of +II state increases on moving down the group in carbon familyPAGE www.aieeepage.com PAGE AGE 9 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 25. Identify the incorrect statement among the following : 1) Silicon reacts with ( ) aq NaOH in the presence of air to give 2 3 Na SiO and 2 H O 2) 2 Cl reacts with excess of 3 NH to give 2 N and HCl 3) 2 Br reacts with hot and strong NaOH solution to give NaBr, 4 NaBrO and 2 H O 4) Ozone reacts with 2 SO to give 3 SO Ans. (2) Sol. With excess of 3 2 NH , Cl forms 4 NH Cl and 2 N gas. 26. The charge/size ratio of a cation determines its polarizing power, Which one of the following sequences represents the increasing order of the polarizing power of the cationic species, 2 2 2 K ,Ca ,Mg ,Be ? + + + + 1) 2 2 2 Be K Ca Mg + + + < < < 2) 2 2 2 K Ca Mg Be + + + + < < < 3) 2 2 2 Ca Mg Be K + + + + < < < 4) 2 2 2 Mg Be K Ca + + + + < < < Ans.(2) Sol. More is the charge density greater is the polarising power. 27. The density (in g 1 mL− ) of a 3.60 M sulphuric acid solution that is 29% 2 4 H SO (Molar mass = 98 1 g mol− ) by mass will be 1) 1.88 2) 1.22 3) 1.45 4) 1.64 Ans. (2) Sol. % d 10 M M.wt. × × =98 3.6 d 1.22 10 29 × = = ×PAGE www.aieeepage.com PAGE AGE 10 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 28. The first and second dissociation constants of an acid 2 H A are 5 1.0 10− × and 10 5.0 10− × respecitively. The overall dissociation constant of the acid will be 1) 15 5.0 10 × 2) 15 5.0 10− × 3) 5 0.2 10 × 4) 5 5.0 10− × Ans.(2) Sol. 1 2 K K K = ×5 10 1.0 10 5 10 − − = × × × 15 5 10− = × 29. A mixture of ethyl alcohol and propyl alcohol has a vapour pressure of 290 mm at 300 K. The vapour pressure of propyl alcohol is 200 mm. If the mole fraction of ethyl alcohol is 0.6, its vapour pressure (in mm) at the same temperature will be 1) 300 2) 700 3) 360 4) 350 Ans. (4) Sol. o o A A B B P P x P x = + 0 B 290 200 0.4 P 0.6 = × + × o B 290 80 0.6 P = + o B P 350. =PAGE www.aieeepage.com PAGE AGE 11 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 30. In conversion of lime-stone to lime, ( ) ( ) ( ) 3 2 CaCO s CaO s CO g → + the values of o H ∆ and o S ∆ are + 179.1 kJ 1 mol− and 160.2 J/K respectively at 298 K and 1 bar. Assuming the o H ∆ and o S ∆ do not change with temperature, temperature above which conversion of limestone to lime will be spontaneous is 1) 1200 K 2) 845 K 3) 1118 K 4) 1008 K Ans. (3) Sol. G H 7 S ∆ =∆ − ∆ ; T should correspond to G 0 ∆ = H T S ∆ = ∆ 3 179.1 T.160.2 10− = × T = 1118 K ∴ Above 1118K the reaction becomes spontaneous 31. The energies of activation for forward and reverse reactions for 2 2 A B 2AB + ! are 180 kJ 1 mol− respectively. The presence of a catalyst lowers the activation energy of both (forward and reverse) reactions by 100 kJ 1 mol− . The enthalpy change of the reaction ( ) 2 2 A B 2AB + → in the presence of catalyst will be (in kJ 1 mol− ) 1) 120 2) 280 3) 20 4) 300 Ans. (3) Sol. Enthalpy is a state function F.R B.R E. E. H − =∆ a a2 2 Zn Cu Zn Cu + + + → +PAGE www.aieeepage.com PAGE AGE 12 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 32. The cell, ( ) ( ) ( ) 2 2 0cell Zn Zn 1M Cu 1M Cu E 1.10V + + = was allowed to be completely discharged at 298 K. The relative concentration of 2 Zn + to 2 Cu + 22 Zn Cu ++ is 1) 37.3 2) 37.3 10 3) 4 9.65 10 × 4) antilog (24.08) Ans. (2) Sol. 2 0 2 Zn 0.059 E log 2 Cu++ = = 2 37.3 2 Zn 10 Cu++ = 33. The pKa of a weak acid (HA) is 4.5. The POH of an aqueous buffered solution of HA in which 50% of the acid is ionized is 1) 2.5 2) 9.5 3) 7.0 4) 4.5 Ans. (2) 34. Consider the reaction 2A B + → Products. When concentration of B alone was doubled, the half life did not change. When the concentration of A alone was doubled, the rate increased by two times. The unit of rate constant for this reaction is 1) no unit 2) mol 1 1 L S − − 3) 1 S− 4) 1 1 Lmol S − − Ans. (4) Sol. The reaction with respect to A and B is first order reaction ∴ Overall order is 2.PAGE www.aieeepage.com PAGE AGE 13 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 35. Identify the incorrect statement amont the following : 1) La and Lu have partially filled d oribital and no ther partially filled orbitals 2) The chemistry of various lanthanoids is very similar 3) 4f and 5f oribitals are equally shielded 4) d-Block elements show irregular and erratic chemical properties among themselves Ans. (4) 36. Which one of the following has a square planar gementry ? 1) [ ]2 4 FeC − l 2) [ ]2 4 NiC − l 3) [ ]2 4 PtC − l 4) [ ]2 4 CoC − l (At. nos. Co = 27, Ni = 28, Fe = 26, Pt = 78) Ans. (3) 37. Which of the following molecules is expected to rotate the plane of plane-polarised light ? 1) SH 2) 2 H N 2 NH Ph Ph H H 3) 2 H N H H COOH 4) H CHO HO 2 CH OH Ans. (4) Sol. Molecule is chiral [has one stereogenic or chirality centre]PAGE www.aieeepage.com PAGE AGE 14 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 38. The secondary structure of a protein refers to : 1) hydrophobic interactions 2) sequence of α− amino acids 3) fixed configuration of the polypeptide backbone 4) α− helical backbone Ans. (3) Sol. Secondary structure has α− Helic & β− Flat sheet (or) β− pdeated structure. If the sige of the group R is large intra molecula hydrogen bonds are formed between C=O of one molecule and NH of the fourth amino acid where as in β− flat sheet. If the sige of the group R is small intermolecular hydrogen bonding are formed between C=O of one polypeptide chain with alter of the other chain. 39. Which of the following reactions will yield 2, 2-dibromopropnae ? 1) 3 CH CH CHBr HBr = + → 2) 3 CH CH 2HBr ≡ + → 3) 3 2 CH CH CH HBr − = + → 4) 3 CH C CH 2HBr − ≡ + → Ans. (4) Sol. Markonikov's rule HBr 3 3 2 3 3 CH C CH HB CH C CH CH C CH − ≡ + → − = → − − Br Br BrPAGE www.aieeepage.com PAGE AGE 15 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 40. In the chemical reaction, 3 2 2 3 2 CH CH NH CHC 3KOH (A) (B_ 3H O, + + → + + l the compounds (A) and (B) are respectively 1) 3 2 2 CH CH CONH and 3KCl 2) 2 5 2 3 C H NCandK CO 3) 2 5 C H NC and 3KCl 4) 2 5 C H CNand3KCl Ans. (3) Sol. 2 5 2 3 2 5 2 C H NH CHCl 3KOH C H NC 3KCl 3H O + + → + +PAGE www.aieeepage.com PAGE AGE 16 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com AIEEE 07 MATHEMATICS SOLUTIONS 41. If p and q are positive real numbers such that 2 2 p q 1, + = then the maximum value of (p + q) is 1) 12 2) 12 3) 2 4) 2 Ans : (3) Sol : 2 2 p q 1 + = 2 p q 2p 1 = ⇒ = 1 p 2 ⇒ = Maximum value of p + q = 1 1 2 2 2 2 2 + = = 42. A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB (=a) subtends an angle of 60° at the foot of the tower, and the angle of elevation of the top of the tower from A or B is 30°. The height of the tower is 1) 2a 3 2) a /3 3) a 3 4) 2a /3 Ans : (2) Sol : A B O a a h ° 60 30° 30° OAB ∆ is equilateral ∴ OB = AB = a h Tan 30 a ° = ; 1 ha 3 ⇒ = a h 3 ⇒ =PAGE www.aieeepage.com PAGE AGE 17 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 43. The sum of the series 20 20 20 20 20 0 1 2 3 10 C C C C ... C − + − + + is 1) 20 10 1 C 2 2) 0 3) 20 10 C 4) 20 10 C − Ans : (1) Sol : 20 20 20 20 20 0 1 2 3 10 S C C C C .... C = − + − + + 20 20 20 20 20 20 19 18 17 10 S C C C C .... C = − + − + + _____________________________ 20 20 20 20 0 1 20 10 2S ( C C .... C ) C = − + + + 20 20 10 10 1 2S 0 C S C 2 = + ⇒ = 44. The normal to a cruve at P(x, y) meets the x-axis at G. If the distance of G from the origin is twice the abscissa of P, then the curve is a 1) parabola 2) circle 3) hyperbola 4) ellipse Ans : (3) Sol : D.E. is dy y x dx = y. dy x. dx = ∫ ∫ 2 2 1 y x C 2 2 = + 2 2 x y C ⇒ − = 45. If | z 4 | 3, + ≤ then the maximum value of | z 1| + is 1) 10 2) 6 3) 0 4) 4 Ans : (2) Sol : |z 1| |(z 4) 3| |z 4| | 3| 3 3 + = + − ≤ + + − ≤ + | z 1| 6 ⇒ + ≤PAGE www.aieeepage.com PAGE AGE 18 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 46. The resultant of two forces P N and 3N is a force of 7N. If the direction of the 3 N force were reversed, the resultant would be 19 N. The value of P is 1) 6N 2) 3N 3) 4N 4) 5N Ans : (4) Sol : 2 2 3 p 2(3) p cos 49 + + α= 2 2 3 p 2(3) p cos 19 + − α= ______________________ 2 2(p 9) 68 + = 2 p p 34 ⇒ + = 2 p 25 p 5N ⇒ = ⇒ = 47. Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is 1) 0.14 2) 0.2 3) 0.7 4) 0.06 Ans : (1) Sol : P(target hit by second) = 0.7 0.2 × = 0.14 48. If 1 1 1 D 1 1 x 1 forx 0,y 0 1 1 1 y = + ≠ ≠ + then D is 1) divisible by both x and y 2) divisible by x but not y 3) divisible by y but not x 4) divisible by neither x nor y Ans : (1) Sol : 1 1 1 0 x 0 xy 0 0 y =PAGE www.aieeepage.com PAGE AGE 19 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 49. For the Hyperbola 2 2 2 2 x y 1, cos sin − = α α which of the following remains constant when α varies ? 1) Directrix 2) Abscissae of vertices 3) Abscissae of foci 4) Eccentricity Ans : (3) Sol : 2 2 2 cos sin e sec cos α+ α = = α α ( ) cos sec ,0 ( 1, 0) ± α α = ± 50. If a line makes an angle of 4π with the positive directions of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is 1) 3 π 2) 4 π 3) 2 π 4) 6 π Ans : (3) Sol : 2 2 2 1 1 cos 1 2 2 + + γ = 2 cos 0 2π ⇒ γ= ⇒γ = 51. A value of C for which the conclusion of Mean value Theorem holds for the function e (x) log x = f on the interval [1, 3] is 1) e 1 log 3 2 2) 3 log e 3) e log 3 4) 3 2log e Ans : (4) Sol : 1 e log 3 f (3) f (1) 1 f (x) 2 x 2 − = ⇒ = 3 x 2log e ⇒ =PAGE www.aieeepage.com PAGE AGE 20 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 52. The function 1 (x) tan (sin x cos x) − = + f is an increasing function in 1)( ) , 2 4 π π − 2) ( ) 0, 2 π 3)( ) , 2 2 π π − 4) ( ) , 4 2 π π Ans : (1) Sol : Tan y sin x cos x = +2 dy cos x sin x dx sec y − = cos x sin x ⇒ > 53. Let 5 5 A 0 5 0 0 5 α α = α α If 2 | A | 25, = then | | α equals 1) 1 2) 15 3) 5 4) 2 5 Ans : (2) Sol : 2 | A | 25 , | A | 25 = α = 2 625 25 ⇒ α = 2 1 25 ⇒α = 1 | | 5 ⇒ α =PAGE www.aieeepage.com PAGE AGE 21 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 54. The sum of the series 1 1 1 .... 2! 3! 4! − + − upto infinity is 1) 1 e− 2) 12 e− 3) 12 e+ 4) 2 e− Ans : (1) Sol : 2 3 x x x x e 1 .... 1! 2! 3! − = − + − + 1 1 1 1 e 1 .... 1! 2! 3! − = − + − + 55. If ˆ ˆ u and v are unit vectors and θ is the acute angle between them, then ˆ ˆ 2u 3v × is a unit vector for 1) More than two values of θ 2) No value of θ 3) Exactly one value of θ 4) Exactly two values of θ Ans : (3) Sol : 2u 3v x 6| u v | | u | × = × = " " " " " " 6.1.1.sin 1 ⇒ θ= 1 sin 6 ⇒ θ=PAGE www.aieeepage.com PAGE AGE 22 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 56. A particle just clears a wall of height b at a distance a and strikes the ground at a distance c from the point of projection. The angle of projection is 1) 45° 2) 1 bc tan a(c a) − − 3) 1 bc tan a − 4) 1 b tan ac − Ans : (2) Sol : φ 1 θ a a c 2 θ b1 2 Tan Tan Tan φ= θ+ θ b b a c a = + − bc Tan a(c a) φ = − 1 bc Tan a(c a) − ⇒ φ = − 57. The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is 1) 20 2) 80 3) 60 4) 40 Ans : (2) Sol : 1 2 1 2 52x 42x 50 x x + = +1 2 1 2 2x 8x x 4x ⇒ = ⇒ =PAGE www.aieeepage.com PAGE AGE 23 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 58. The equation of a tangent to the parabola 2 y 8x = is y = x + 2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is 1) (0, 2) 2) (2, 4) 3) (-2, 0) 4) (-1, 1) Ans : (3) Sol : Slope of required tangent, m = -1 Equation of tangent is y = mx + a/m y x 2 ⇒ =− − Solving y = x + 2 and y = -x -2 , we get (-2, 0) 59. If (2, 3, 5) is one end of a diameter of the sphere 2 2 2 x y z 6x 12y 2z 20 0, + + − − − + = then the coordinates of the other end of the diameter are 1) (4, -3, 3) 2) (4, 3, 5) 3) (4, 3, -3) 4) (4, 9, -3) Ans : (4) Sol : (2, 3, 5) (3, 6, 1) (4, 9, 3) -60. Let ˆ ˆ ˆ ˆ ˆ ˆ a i j k,b i j 2k = + + = − + and ˆ ˆ ˆ c xi (x 2)j k. = + − − If the vector c lies in the plane of a and b, then x equals 1) 1 2) -4 3) -2 4) 0 Ans : (3) Sol : 1 1 1 1 1 2 0 x (x 2) 1 − = − − 1 0 0 1 2 1 0 x 2 1 x ⇒ − = − − − 2(1 x) 2 0 + + = x 2 ⇒ =−PAGE www.aieeepage.com PAGE AGE 24 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 61. Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypoitenuse. If the area of the triangle is 1, then the set of values which ‘k’ can take is given by 1) {0, 2} 2) {-1, 3} 3) {-3, -2} 4) {1, 3} Ans : (2) Sol : A(1, K) B(1,1) C(2,1) 1 (k 1) 1 1 2 − × = k 1 2 − =± k = 3, -1 62. Let P = (-1, 0) Q = (0, 0) and R = (3,3 3) be three points. The equation of the bisector of the angle PQR is 1) 3 x y 0 2 + = 2) 3x y 0 2 + = 3) x 3y 0 + = 4) 3 x y 0 + = Ans : (4) Sol : R ) 0 , 1 (− Q ° 60 ° 60 ) 0 , 0 ( ) 3 3 , 3 ( P Slope of QX = tan 120° 3 = − Equation of QX is y 3x = − 3x y 0 = + =PAGE www.aieeepage.com PAGE AGE 25 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 63. If one of the lines of 2 2 2 my (1 m )xy mx 0 + − − = is a bisector of the angle between the lines xy = 0, then m is 1) -2 2) 1 3) 2 4) 12 − Ans : (2) Sol : 2 m(1) (1 m )(1) m 0 + − − = //m 1 = ± 64. Let 1 F(x) (x) , x = + f f where x1 log t (x) dt. 1 t = + ∫ f Then F(e) equals 1) 0 2) 1 3) 2 4) 12 Ans : (4) Sol : log x '(x) (1 x) = + f 2 1 1 F'(x) f '(x) f ' x x ⇒ = − log x log x log x (1 x) (1 x)x x = + = + + log x F(x) dx x = ∫ 2 1 F(x) (log x) 2 = 1 F(e) 2 =PAGE www.aieeepage.com PAGE AGE 26 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 65. Let : → # # f be a function defined by (x) Min{x 1,| x | 1}. = + + f Then which of the following is true ? 1) (x) f is not differentiable at x = 1. 2) (x) f is differentiable everywhere 3) (x) f is not differentiable at x = 0 4) (x) 1 for all x ≥ ∈# f Ans : (2) Sol : f (x) x 1, x 0 = + < = x + 1, x > 0 = 1, x = 0 f '(x) 1, x 0 = < = 1, x > 0 = 1, x = 0 66. The function : \{0}→ # # f given by 2x 1 2 f (x) x e 1 = − − can be made continuous at x = 0 by defining f(0) as 1) -1 2) 0 3) 1 4) 2 Ans : (3) Sol : 2x 2x x 0e 1 2x f (0) Lt x(e 1) → − − = −2 x 0 2x (2x) 1 12x 1! 2! Lt 1 x(2x) → + + − − =PAGE www.aieeepage.com PAGE AGE 27 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 67. The solution for x of the equation x 2 2 dt 2 t t 1 π = − ∫ is 1) π 2) 3 /2 3) 2 2 4) 2 Ans : no option 1 x2 [sec t] /2 − = π 1 1 sec x sec 2 /2 − − − =π 1 sec x /4 /2 − −π = π 1 3 sec x 4 − π = x sec( /2 /4) 2 = π +π =− 68. dx cos x 3 sin x + ∫ equals 1) 12 x log tan C 2 12π − + 2) x log tan C 2 12π + + 3) x log tan C 2 12π − + 4) 12 x log tan C 2 12π + + Ans : (4) Sol : 1 dx 2 cos x 3π − ∫1 x log tan C 2 212π = + + PAGE www.aieeepage.com PAGE AGE 28 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 69. The area enclosed between the curves 2 y x = and y |x| = is 1) 1 2) 16 3) 13 4) 23 Ans : (2) Sol : 10 1 x (x) dx 6 − = ∫ 70. If the difference between the roots of the equation 2 x ax 1 0 + + = is less than 5, then the set of possible values of a is 1) ( 3, ) − ∞ 2) (3, ) ∞ 3) ( , 3) −∞− 4) ( 3,3) − Ans : 4 Sol : 5 α−β < 2 ( ) 4 5 α+β − αβ < 2 a 4 5 − < 2 a 4 5 − < 2 a 9 < | a | 3 <3 a 3 ⇒− < < a ( 3,3) ⇒ ∈ −PAGE www.aieeepage.com PAGE AGE 29 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 71. In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals 1) 1 5 2 2) 5 3) ( ) 1 5 1 2 − 4) ( ) 1 1 5 2 − Ans : (3) Sol : Let 2 a, ar, ar be progression 2 a ar ar ⇒ = + 2r r 1 0 + − = 1 1 4 r 2 − ± + =1 5 2 − ± = 5 1 1 5 , 2 2 − − − = 72. If 1 1 x 5 sin cos ec 5 4 2 − − π + = then a value of x is 1) 3 2) 4 3) 5 4) 1 Ans : (1) Sol : Given 1 1 x 5 sin cos ec 2 5 4 − − π + = 1 1 x 3 sin cos 5 5 2 − − π ⇒ + = x 3 ∴ =PAGE www.aieeepage.com PAGE AGE 30 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 73. In the binomial expansion of n (a b) ,n 5, − ≥ the sum of th th 5 and6 terms is zero, then ab equals 1) 6 n 5 − 2) n 5 6− 3) n 4 5− 4) 5 n 4 − Ans : (3) Sol : n 4 4 5 44 C4 T T n a b − + = = n 5 5 6 51 C5 T T n a b − + = =− Given 5 6 T T 0 + = n 4 4 n 5 5 C C 4 5 n a b n a b − − = CC54 n a n 4 b n 5− = = 74. The set S : = {1, 2, 3, ...,12} is to be partitioned into three sets A, B, C of equal size. Thus, A B C S,A B B C A C . ∪ ∪ = ∩ = ∩ = ∩ =φ The number of ways to partition S is 1) 4 12!3! (3!) 2) 3 12!(4!) 3) 4 12!(3!) 4) 3 12!3!(4!) Ans : (2) Sol : No.of ways of dividing (m + n + p) things into three groups containing m things, n things, p things is (m n p)! 12! m!n!p! 4! 4! 4! + + =PAGE www.aieeepage.com PAGE AGE 31 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 75. The largest interval lying in , 2 2 −π π for which the function x 1 2 x (x) 4 cos 1 log(cos x) 2 − − = + − + f is defined, is 1) , 2 2 π π − 2) , 4 2 π π − 3) 0, 2 π 4) [ ] 0, π Ans : (3) Sol : Domain of x2 4− is R ........(1) Domain of 1 x cos 1 2 − − is x 1 [ 1,1] 2− ∈ − x [0, 2] 2 ∈ x [0,4] ∈ ....(2) Domain of log (cos x) cos x > 0 x , 2 2 −π π ∈ .......(3) So the function is defined in intersection position of (1), (2) and (3) i.e. 0 2 π PAGE www.aieeepage.com PAGE AGE 32 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 76. A body weighing 13 kg is suspended by two strings 5 m and 12 m long, their other ends being fastened to the extremities of a rod 13 m long. If the rod be so held that the body hands immediately below the middle point. The tensions in the strings are 1) 5 kg and 5 kg 2) 5 kg and 12 kg 3) 5 kg and 13 kg 4) 12 kg and 13 kg Ans : (2) Sol : r θ 2 θ 12 m 13 2 T 1 T ° 90 β M 13r θ5 Lam’s theorem 1 2 T T w sin sin sin 90 = = α β ° 2 1 2 T T Q sin (90 ) sin(90 ) = = +θ +θ 2 1 2 T T W sin (90 ) sin(90 ) = = +θ + θ 1 2 1 2 T T 13 cos cos = = θ θ 1 1 2 2 T 13cos ;T 13cos = θ = θ 1 2 5 12 T 13 T 13 13 13 = × = × 1 T 5 = 2 T 12 =PAGE www.aieeepage.com PAGE AGE 33 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 77. A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is 1) 8/9 2) 8/729 3) 8/243 4) 1/729 Ans : (3) Sol : n = no of times of experiment conducted = 3 Experiment : pair of dice rolled Getting score ‘9’ when experiment conducted one time 4 4 1 P 6 6 369 9 = = = × 8 Q 9 = Now we need two times success 1 2 3 2 8 1 p(x 2) C 9 9 = = 38 8 8 3 9 9 3 243 9 = × = = × ×PAGE www.aieeepage.com PAGE AGE 34 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 78. Consider a family of circles which are passing through the point (-1, 1) and are tangent to x-axis. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval 1) 12 k ≥ 2) 1 1 2 2 k − ≤ ≤ 3) 12 k ≤ 4) 12 0 k < < Ans : (1) Sol : ) 1 , 1 (− axis x − ) k , h ( | K | radius = 2 2 | K | (h 1) (k 1) = + + − 2 2 2 2 K h 1 2h(h 1) k 1 2k = + + + + + − 2 2K (h 1) 1 = + + 2K [0, ) 1 = ∞ + [1, ) = ∞1 K ,2 = ∞ PAGE www.aieeepage.com PAGE AGE 35 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 79. Let L be the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2z = 2. If L makes an angle α with the positive x-axis, then cos α equals. 1) 12 2) 1 3) 1 2 4) 1 3 Ans : (4) Sol : Dr’s of line n ⊥ to first plane 2, 3, 1 Dr’s of line n ⊥ to first plane 1, 3, 2 Dr’s of line L = i j k 2 3 1 1 3 2 = 3i -3j + 3k = 3, -3, 3 = 1, -1, 1 1 1 cos 1 1 1 3 α = = + +PAGE www.aieeepage.com PAGE AGE 36 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 80 The differential equation of all circles passing through the origin and having their centres on the x-axis is 1) 2 2 dy x y 3xy dx = + 2) 2 2 dy y x 2xydx = + 3) 2 2 dy y x 2xydx = − 4) 2 2 dy x y xydx = + Ans : (2) Sol : (a, 0) (0, 0) Equation of circles 2 2 2 (x a) y a − + = 2 2 x y 2ax 0 ⇒ + − = dy 2 (x a) 2 y 0 dx − + = 2 2 x y a 2+ = ..........(1) From ...(1) 2 2 x y dy x y 0 2x dx + − + = 2 2 2 dy 2x x y 2xy 0 dx − − + = 2 2 dy x y 2xy 0 dx − + = 2 2 dy y x 2xydx = +PAGE www.aieeepage.com PAGE AGE 37 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 81. Carbon, silicon and germanium have four valence electrons each. At room temperature which one of the following statements is most appropriate? 1) The number of free conduction electrons is negligibly small in all the three. 2) The number of free electrons for conduction is significant in all the three. 3) The number of free electrons for conduction is significant in all the three. 4) The number of free conduction electrons is significant in C but small in Si and Ge. And : (1) Sol : All are semiconductors 82. A charged particle with charge q enters a region of constant, uniform and mutually orthogonal fields E and B " " with a velocity v " perpendicular to both E and B, " " and comes out without any change in magnitude or direction of v. " Then 1) 2 v B E/B = × " " " 2) 2 v E B/E = × " " " 3) 2 v B E/E = × " " " 4) 2 v E B/B = × " " " And : (4) Sol. The particle gets underheated Hence E V B = 2 E B V B× = " " " AIEEE 07 PHYSICS SOLUTIONSPAGE www.aieeepage.com PAGE AGE 38 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 83. The potential at a point x (measured in m) µ due to some charges situated on the x-axis is given by 2 V(x) 20 /(x 4) Volts = − The electric field E at x 4 m = µ is given by 1) 5/3 Volt /m µ and in the +ve x direction 2) 10/9 Volt /m µ and in the -ve x direction 3) 10/9 Volt /m µ and in the +ve x direction 4) 5/3 Volt /m µ and in the -ve x direction And : (3) Sol. ( ) ( ) 220 V x v x 4 = − dv E dx = − 2 d 20 dx x 4 = − − ( )1 2 d 20 x 4 dx − = − − ( )2 2 20 1 x 4 2x − = − ×− − × 84. Which of the following transitions in hydrogen atoms emit photons of highest frequency ? 1) n 6ton 2 = = 2) n 2ton 1 = = 3) n 1ton 2 = = 4) n 2ton 6 = = And : (2) Sol. 2 2 1 2 1 1 1 R n n λ = − k -lines lies in u.v spectran 85. A block of mass ‘m’ is connected to another block of mass ‘M’ by a spring (massless) of spring constant ‘k’. The blocks are kept on a smooth horizontal plane. Initially the bolcks are at rest and the spring is unstretched. Then a constant force ‘F’ starts acting on the block of mass ‘M’ to pull it. Find the force on the block of mass ‘m’. 1) (M m) F m + 2) mF (m M) + 3) MF (m M) + 4) mF M And : (2) Sol. ( ) F a M m = + ( ) 1 mF F ma M m ∴ = = +PAGE www.aieeepage.com PAGE AGE 39 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 86. Two lenses of power -15D and +5D are in contact with each other. The focal length of the combination is 1) 10 cm − 2) 20 cm + 3) 10 cm + 4) 20 cm − And : (1) Sol. 1 2 P P P 10 = + =− 1 100 P f fcm = = 100 fcm 10 cm 10 = =− −PAGE www.aieeepage.com PAGE AGE 40 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 87. One end of a thermally insulated rod is kept at a temperature 1 T and the other at 2 T . The rod is composed of two sections of lenghts 1 2 and l l and thermal conductivities 1 2 k andk respectively. The temperature at the interface of the two sections is 1 l 2 l 1 T 2 T 1 k 2 k 1) 2 1 1 1 2 2 2 1 1 2 (k T k T ) /(k k ) + + l l l l 2) 1 2 1 2 1 2 1 2 2 1 (k T k T ) /(k k ) + + l l l l 2) 1 1 1 2 2 2 1 1 2 2 (k T k T ) /(k k ) + + l l l l 3) 2 2 1 1 1 2 1 1 2 2 (k T k T ) /(k k ) + + l l l l And : (2) Sol : 1 l 2 l 1 T 2 T 1 k 2 k 1 1 2 2 1 2 k A(T ) k A( T ) −θ θ− = l l 1 2 1 2 1 2 T k T k −θ = θ− l l 1 2 1 1 1 2 1 2 1 2 k T k k k T − θ= θ− l l l l 1 2 2 1 1 2 1 2 1 2 (k k ) k T k T θ + = + l l l l 1 2 1 2 1 2 1 2 2 1 k T k T k k + θ = + l l l lPAGE www.aieeepage.com PAGE AGE 41 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 88. A sound absorber attenuates the sound level by 20 dB. The intensity decreases by a factor of 1) 10000 2) 10 3) 100 4) 1000 And : (3) Sol : 1 1 o L 10log(I /I ) =2 2 o L 10log(I /I ) = 2 (L 4) 20 − =2 1 10 log (I /I ) 20 = 2 1 I /I 100 = . 89. If p v C and C denote the specific heats of introgen per unit mass at constant pressure and constant volume respectively, then 1) p v C C R/14 − = 2) p v C C R − = 3) p v C C 28R − = 4) p v C C R/28 − = And : (4) Sol : P V C C r − = RM = P V C C R − = P V MC MC R − = R = M r Nitrogen M = 28 P V R R C C 28 M ∴ − = = 90. A charged particle moves through a magnetic field perpendicular to its direction. Then 1) both momentum and kinetic energu of the particle are not constant 2) both, momentum and kinetic energy of the particle are constant 3) kinetic energy changes but the momentum is constant 4) the momentum changes but the kinetic energy is constant And : (4) Sol : Particle moves in Uniform circular pathsPAGE www.aieeepage.com PAGE AGE 42 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 91. Two identical conduicting wires AOB and COD are placed at right angles to each other. The wire AOB carries an electric current 1 I and COD carries a current 2 I . The magnetic field on a point lying at a distance ‘d’ from O, in a direction perpendicular to the plane of the wires AOB and COD, will be given by 1) 1 2 2 0 2 1 2 (I I ) 2 d µ + π 2) 0 1 2 (I I ) 2 d µ + π 3) 2 2 0 1 2 (I I ) 2 d µ + π 4) 12 0 1 2 I I 2 d µ + π And : (1) Sol : O 1 I B A DC 2 I If the lines are infinitely lens 1 2 B B B = + " " "( ) 1 2 1/2 2 2 0 1 B B B B I I 2 d µπ = + = + " " "PAGE www.aieeepage.com PAGE AGE 43 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 92. The resistance of a wire is 5 ohm at 50 °C and 6 ohm at 100 °C. The resistance of the wire at 0 °C will be 1) 1 ohm 2) 4 ohm 3) 3 ohm 4) 2 ohm And : (2) Sol : 1 t 0 t 0 1 0 0 R R R R R t R t α − − = = 5 x 6 x x 50 x 100 − − = × × 10 2x 6 x = − = − x 4 = Ω 93. A parallele plate condenser with a dielectric of dielectric constant K between the plates has a capacity C and is charged to a potential V volts. The dielectric slab is slowly removed from between the plates and then reinserted. The net work done by the system in this process is 1) 2 CV (K 1) /K − 2) 2 (K 1) CV − 3) zero 4) 2 12 (K 1) CV − And : (3) Sol : Workdone is zeroPAGE www.aieeepage.com PAGE AGE 44 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 94. If E m g andg are the accelerations due to gravity on the surfaces of the earth and the moon respectively and if Millikan’s oil drop experiment cound be perfoumed on the two surfaces, one will find the ratio electronic charge on the moon electronic charge on the earth to be 1) 0 2) E M g /g 3) M E g /g 4) 1 And : (4) Sol : Electonic charge is independent of acceleration acceleration due to gravity 95. A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the circumferences of the discs coincide. The centre of mass of the new disc is Rα from the centre of the bigger disc. The value of α is 1) 12 2) 16 3) 14 4) 13 And : (4) Sol : 2RR R 1 1 2 2 M X M X = 2 2 2 R td(R) 3 R td x π =π 2 R x 3 = x 13 R ∴ =PAGE www.aieeepage.com PAGE AGE 45 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 96. A round uniform body of radius R, mass M and moment of inertia ‘I’, rolls down (without slipping) an inclined plane making an angle θ with the horizontal. Then its acceleration is 1) 2 g sin 1 MR /I θ + 2) 2 g sin 1 I/MRθ − 3) 2 g sin 1 MR /I θ − 4) 2 g sin 1 I/MRθ + And : (4) Sol. 22 gsink 1 Rθ = + a 2 I Mk = 2 I k M = 2 gsinI 1 MRθ = + a 97. Angular momentum of the particle rotating with a central force is constant due to 1) Constant linear momentum. 2) Zero Torque 3) Constant Torque 4) Constant Force And : (2) Sol. Acc to LCAM dL T dt = L = Constant when Text = 0PAGE www.aieeepage.com PAGE AGE 46 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 98. A 2 kg block slides on a horizontal floor with a speed of 4 m/s. It strikes a uncompressed spring, and compresses it till the block is motionless. The kinetic friction force is 15 M and spring constant is 10,000 N/m. The spring compresses by 1) 2.5 cm 2) 11.0 cm 3) 8.5 cm 4) 5.5 cm And : (4) Sol : 2 2 1 1 v gx kx 2 2 µ = + 2 1 16 15x 5000x 2× = + 2 5000x 15x 8 0 + − = 15 225 16000 15 1625 x 10,000 10,000 − ± + − + = = 99. A particle is projected at 60° to the horizontal with a kinetic energy K. The kinetic energy at the highest point is 1) Zero 2) K/4 3) K/2 4) K And : (2) Sol. KE at height Point 1 2 2 1 E m cos 2 µ θ = 2 K cos 60 = 1 K E 4 =PAGE www.aieeepage.com PAGE AGE 47 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 100. In a Young’s double slit experiment the intensity at a point where the path-difference is 6λ (λ being the wavelength of the light used) is I. If 0 I denotes the maximum intensity, 0 I I is equal to 1) 3 2 2) 12 3) 34 4) 12 And : (3) Sol. 2 x π φ λ ∆ = ∆ 2 6 3 π λ π λ = = R 2acos 2φ = 3 R 2a 2 = max R 2a = 2 2 0 max I R 3 3 I R 2 4 = = = PAGE www.aieeepage.com PAGE AGE 48 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 101. Two springs, of force constants 1 2 k andk , are connected to a mass m as shown. The frequency of oscillation of the mass is f. If both 1 2 k andk are made four times their original values, the frequency of oscillation becomes 2 k 1 k 1) f /4 2) 4f 3) 2f 4) f /2 And : (3) Sol. 1 2 k k 1 f 2 m π + = 1 1 2 4k 4k 1 f 2 m π + = 1f 2f =PAGE www.aieeepage.com PAGE AGE 49 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 102. When a system is taken from state i to state f along the path iaf, it is found that Q = 50 cal and W = 20 cal. Along the path ibf Q = 36 cl. W along the path ibf is i fb 1) 16 cal 2) 66 cal 3) 14 cal 4) 6 cal And : (4) Sol. U ∆ = constant 1 1 2 2 Q W Q W − = − 2 50 20 36 W − = −2 30 36 W = − 2 W 6Cal = 103. A particle of mass m executes simple harmonic motion with amplitude ‘a’ and frequency ‘v’. The average kinetic energy during its motion from the position of equilibrium to the end is 1) 2 2 1 m a v 4 2) 2 2 2 4 ma v π 3) 2 2 2 2 ma v π 4) 2 2 2 m a v π And : (4) Sol : 2 2 av 1 1 K mwa 2 2 = 2 2 2 1m4 a 4 π ν = 2 2 2 ma π ν =PAGE www.aieeepage.com PAGE AGE 50 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 104. The displacement of an object attached to a spring and executing simple harmonic motion is given by 2 x 2 10− = × cos t π metres. The time at which the maximum speed first occurs is 1) 0.75 s 2) 0.125 s 3) 0.25 s 4) 0.5 s And : (4) Sol : 2 x 2 10 Cos t π − = × Particle starts from extreme position ∴ After a time (t) = T4 ; maximum speed first occur w π = 2Tπ π = T 2s ∴ =T 1 t s. 4 2 ∴ = = 105. In an a.c. circuit the voltage applied is 0 E E sin t. = ω The resulting current in the circuit is 0 I I sin( t ). 2π = ω− The power consumption in the circuit is given by 1) P = zero 2) o o E I P 2 = 3) o o P 2E I = 4) o o E I P 2 = And : (1) Sol : rms rms P E I cos 0 = θ=PAGE www.aieeepage.com PAGE AGE 51 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 106. An electric charge 3 10 C − µ is places at the origin (0, 0) of X Y − co-ordinate system. Two points A and B are situated at ( 2, 2)and(2,0) respectively. he potential difference between the points A and B will be 1) zero 2) 2 volt 3) 4.5 volt 4) 9 volt And : (1) Sol : A r B r ( 2, 2) (2, 0) c 10 I 3µ = − O Ar 2 = Br 2 = A B V V ∴ = A B V V 0 ∴ − = 107. A battery is used to charge a parallel plate capactior till the potential difference between the electromotive force of the battery. The ratio of the energy stored in the capacitor and the work done by the battery will be 1) 2 2) 14 3) 12 4) 1 And : (3) Sol : 1 U qv 2 = W qv = U 1 W 2 =PAGE www.aieeepage.com PAGE AGE 52 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 108. An ideal coil of 10H is connected in series with a resistance of 5 Ω and a battery of 5 V.. 2 seconds after the connection is made, the current flowing in amperes in the circuit is 1) e 2) 1 e− 3) 1 (1 e ) − − 4) (1 e) − And : (3) Sol : 2 t o I I 1 e −τ = − 1 2 L 2 r R − τ = = o E 5 I 1A R 5 = = =1 I 1(1 e ) − = −PAGE www.aieeepage.com PAGE AGE 53 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 109. A long straight wire of radius ‘a’ carries a steady current i. The current is uniformly distributed across its cross section. The ratio of the magnetic field at a2 and 2a is 1) 4 2) 1 3) 12 4) 14 And : (2) Sol. ( ) 0 1 2 I a B 2 2 a µ π = ( ) 0 2 I B 2 2a µ π = 12 B 1 B 1 = 110. A current I flows along the length of an infinitely long, straight, thin walled pipe. Then 1) the magnetic field is different at different points inside the pipe 2) the magnetic field at any point inside the pipe is zero 3) the magnetic field at all points inside the pipe is the same, but not zero 4) the magnetic field is zero only on the axis of the pipe And : (2)PAGE www.aieeepage.com PAGE AGE 54 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 111. If o M is the mass of an oxygen isotope 17 8 P N O ,M andM are the masses of a proton and a neutron respectively, the nuclear binding energy of the isotope is 1) 2 O P N (M 8M 9M ) C − − 2) 2 O M C 3) 2 O N (M 17 M ) C − 4) 2 O P (M 8M ) C − And : (1) Sol. Nuclear binding energy 2 mc ∆ = ( )2 0 P N M 8M 9M C = − − 112. In gamma ray emission from a nucleus 1) there is no change in the proton number and the neutron number. 2) only the neutron number changes 3) only the proton number changes 4) both the neutron number and the proton number change And : (1)PAGE www.aieeepage.com PAGE AGE 55 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 113. If in a p-n junction diode, a square input signal of 10V is applied as shown 5V 5V -L R Then the output signal across L R will be 1) 10V -2) 5V -3) 5V + 4) 10V And : (4) 114. Photon of frequency v has a momentum associated with it. If c is the velocity of light, the momentum is 1) hvc 2) 2 h v/c 3) h v/c 4) v/c And : (3) Sol. h P mc Cν = =PAGE www.aieeepage.com PAGE AGE 56 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 115. The velocity of a particle is 2 0 v v gt ft . = + + If its position is x = 0 at t = 0, then its displacement after unit time (t 1) is = 1) 0 v g/2 f/3 + + 2) 0 v g f + + 3) 0 v g/2 f + + 4) 0 v 2g 3f + + And : (1) Sol : 2 0 V V gt ft = + + 2 0 ds V gt 8t dt= + + s 1 1 120 0 0 0 0 ds Vdt gtdt ftdt = + + ∫ ∫ ∫ ∫ 1 1 2 3 1 0 0 0 0 gt ft S Vt 2 3 = + + 0 g f V 2 3 = + +PAGE www.aieeepage.com PAGE AGE 57 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 116. For the given uniform square lamina ABCD, whose centre is O, A E O B F D C 1) AD EF I 3I = 2) AC EF I I = 3) AC EF I 2I = 4) AC EF 2 I I = And : (2) Sol : ADG F HBC E From perpendicular axis theorem EF GH AC DB I I I I + = + EF AC 2 I 2 I = EF AC I I =PAGE www.aieeepage.com PAGE AGE 58 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 117. A point mass oscillates along the x-axis according to the law 0 x x cos( t /4). = ω−π If the acceleration of the particle is written as a Acos( t ),then = ω+δ 1) 2 0 A x , 4 π = ω δ= 2) 2 0 A x , 4 π = ω δ=− 3) 2 0 A x , 3 4 π = ω δ= 4) 0 A x , 4 π = δ=− And : (3) Sol : o x x cos( t /4) = ω−π velocity 2 o dx v x cos t dt 4π = =− ω ω− acc, 2 o dv a x cos t dt 4π = =− ω ω− compaing with a Acos( t ) = ω+δ We get 2 2 o 3 a x cos 4π = ω ω + 2 o 3 A x and 4π ∴ = ω δ=PAGE www.aieeepage.com PAGE AGE 59 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 118. Charges are places on the vertices of a square as shown. Let E " be the electric field and V the potential at the centre. If the charges on A and B are interchanged with those on D and C respectively, then A BC D 1) Both E " and V change 2) E " and V remain unchanged 3) E " changes, V remains 4) E " remains unchanged, V changes And : (3) Sol : E AD BC centre V 0= E at centre is lr ⊥ to CD and towards CD E AD BC centre V 0= E and centre is lr ⊥ to AB and towards AB.PAGE www.aieeepage.com PAGE AGE 60 www.aieeepage.com www.iitjeepage.com www.bitsatpage.com 119. The half-life perod of a radio-active element X is same as the mean life time of another radio-active elementY. Initially they have the same number of atoms. Then 1) Y will decay faster than X 2) X and Y have same decay rate initially 3) X and Y decay at same rate always 4) X will decay faster than Y. And : (1) Sol : Mean life 1 = λ Way life 0.693 = λ 1 av(B) (A) 2T T= 0.693 1 , = λ λ 0.693 λ λ =y decreasing faster then x ∴ 120. A Carnot engine, having an effeiciency of 110 η = as heat engine, is used as a refrigerator. If the work done on the system is 10 J, the amount of energy absorbed from the reservoir at lower temperature is 1) 90 J 2) 1 J 3) 100 J 4) 99 J And : (3) Sol. Efficiency of absorbed w Q = absorbed 1 10 10 Q = absorbed Q 100 J =