4th AIEEE 2011 Test Series 1 www.vidyadrishti.com An education portal for future IITians Name Reg. No. AIEEE 2011 4th Test Series 12th Dec 2010 to 27th Dec 2010 from www.vidyadrishti.com General Instructions: Time 3 hours Maximum marks 432 The Test Booklet consists of 90 questions. The maximum marks are 432. There are three parts in the question paper. The distribution of marks subject-wise in each part is as under for each correct response. Part A – Physics (144 Marks) –Questions No. 1 to 2 and 9 to 30 consist FOUR (4) marks each and Question No. 3 to 8 consist EIGHT (8) marks each for each correct response. Part B – Chemistry (144 Marks) – Questions No. 1 to 24 consist FOUR (4) marks each and Question No. 25 to 30 consist EIGHT (8) marks each for each correct response. Part C – Mathematics (144 Marks) – Questions No.1 to 21 and 28 to 30 consist FOUR (4)marks each and Question No. 22 to 27 consist EIGHT (8) marks each for each correct response. 4th AIEEE 2011 Test Series 2 www.vidyadrishti.com An education portal for future IITians PHYSICS 1. The angle of dip at a place is 40.6o and the intensity of the vertical component of the earth’s magnetic field is V = 6 x 10-5 T. The total intensity of the earth’s magnetic field at this place is (a) 7 x 10-5 T (b) 6 x 10-5 T (c) 5 x 10-5 T (d) 9.2 x 10-5 T 2. A magnet of magnetic moment 50i Am2 is placed along the x-axis in a magnetic field ()0.53.0.BijT=+ The torque acting on the magnet is (a) 175k N-m (b) 150k N-m (c) 75k N-m (d) 2537k N-m 3. A deflection magnetometer is adjusted in the usual way. When a magnet is introduced, the deflection observed is θ, and the period of oscillation of the needle in the magnetometer is T. When the magnet is removed the period of oscillation is T0. The relation between T and T0, is (a) 220cosTTθ= (b) 0cosTTθ= (c) 0cosTTθ= (d) 220cosTTθ= SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 3 www.vidyadrishti.com An education portal for future IITians 4. In the shown network, current flow through 20 Ω resistor is (a) 32 A (b) 92 A (c) 1 A (d) 23 A 5. If the number of free electrons is 5 x 1028 m-3 then the drift velocity of electron in a conductor of area of cross-section 10-4 m2 for a current of 1.2 A is (a) 1.5 x 10-2 m/s (b) 1.5 x 10-3 m/s (c) 1.5 x 10-4 m/s (d) 1.5 x 10-6 m/s 6. A galvanometer has 30 divisions and a sensitivity 16 µ A/div. It can be converted into a voltmeter to read 3 V by connecting (a) Resistance nearly 6 k Ω in series (b) 6 k Ω in parallel (c) 500 Ω in series (d) 6.6 k Ω in series SPACE FOR ROUGH WORK 10 Ω 20 Ω 20 V 10 V 4th AIEEE 2011 Test Series 4 www.vidyadrishti.com An education portal for future IITians 7. Two particles X and Y having equal charges, after being accelerated through the same potential difference, enter a region of uniform magnetic field and describes circular path of radius R1 and R2 respectively. The ratio of mass of X to that of Y is (a) 1/212RR (b) 21RR (c) 212RR (d) 12RR 8. A circular coil of radius 0.1 m has 80 turns of wire. If the magnetic field through the coil increases from 0 to 2 tesla in 0.4 sec and the coil is connected to a 11 ohm resistor, what is the current (in A) flow through the resistor during the 0.4 sec? (a) 87 A (b) 78 A (c) 8 A (d) 7 A 9. A rectangle coil of 20 turns and area of cross-section 25 sq. cm has a resistor of 100 Ω. If a magnetic field which is perpendicular to the plane of coil changes at a rate of 1000 tesla per second, the current in the coil is (a) 1 A (b) 50 A (c) 0.5 A (d) 5 A SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 5 www.vidyadrishti.com An education portal for future IITians 10. For the LR circuit shown in figure the plane angle if frequency is 100π is (a) 30o (b) 60o (c) 45o (d) 90o 11. The figure shows the cross-section of a long conducting cylinder of inner radius a and outer radius b. The cylinder carries a current whose current density J = Cr2 where C is a constant. What is the magnitude of the magnetic field B at a point r, where a < r < b? (a) ()22024CBrarµπ=− (b) ()44024CBrarµ=− (c) ()2204CBrarµπ=− (d) ()4404CBrarµ=− SPACE FOR ROUGH WORK 5 Ω L R 0.025 H r b a 4th AIEEE 2011 Test Series 6 www.vidyadrishti.com An education portal for future IITians 12. A pure resistive circuit element X when connected to an ac supply of peak voltage 200 V gives a peak current of 5 A which its in phase with the voltage. A second circuit element Y, when connected to the same ac supply also gives the same value of peak current but the current lags behind by 90o. If the series combination of X and Y is connected to the same supply, what will be the rms value of current? (a) 102A (b) 52A (c) 52A (d) 5 A 13. For what value of R will the current in galvano-meter be zero? (a) 1 Ω (b) 2 Ω (c) 5 Ω (d) 7 Ω 14. An ion of charge +q and mass m is accelerated in a region of potential difference V volts and then it enters a region of uniform magnetic field B perpendicular to its velocity. If the ion describes a semicircle of diameter x, then its mass m is (a) 228BqxmV= (b) 228BqxmV= (c) 28BqxmV= (d) 2228BqxmV= SPACE FOR ROUGH WORK -+ 5 Ω R 12 V G -+ 7 V 4th AIEEE 2011 Test Series 7 www.vidyadrishti.com An education portal for future IITians 15. An LC circuit contains a 40 mH inductor and a 25 µF capacitor. The resistance of the circuit is negligible. The time is measured from the instant the circuit is closed. The energy stored in the circuit is completely magnetic at times (in milliseconds) (a) 0, 3.14, 6.28 (b) 0, 1.57, 4.71 (c) 1.57, 4.71, 7.85 (d) 1.57, 3.14, 4.71 16. A potentiometer has uniform potential gradient. The specific resistance of the material of the potentiometer wire is 10-7 Ω m and the current passing through it is 0.1 A, cross section of the wire is 10-6 m2. The potential gradient along the potentiometer wire is (a) 10-4 V/m (b) 10-6 V/m (c) 10-2 V/m (d) 10-8 V/m 17. Two current loops (similar) are placed with their planes one along x-axis and the other along y-axis. Then the ratio of resultant magnetic field at a common point on the axis to the individual magnetic field at the same point is (a) 2:1 (b) 1:2 (c) 3:2 (d) 3:2 18. The electromagnetic waves having speed 2.5 x 108 m/s are travelling in a medium of relative permeability 0.8. The relative permittivity of the medium is (a) 1.0 (b) 1.8 (c) 2.5 (d) 1.2 SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 8 www.vidyadrishti.com An education portal for future IITians 19. In a series resonance LCR circuit, the voltage across R is 100 V and R = 1 kΩ, C = 2 µF. The resonant frequency is 200 rad/s. At resonance, the voltage across L is (a) 40 V (b) 250 V (c) 4 x 10-3 V (d) 2.5 x 10-2 V 20. A conducting ring of radius 1 m is placed in a uniform magnetic field B of 0.01 T oscillating with frequency 100 Hz with its plane at right angles to B. What will be induced electric field? (a) πV/m (b) 0.5 V/m (c) 10 V/m (d) 62 V/m 21. Three similar cells, each of emf 2 V and internal resistance r Ω send the same current through an external resistance of 2 Ω, when connected in series or in parallel. The strength of the current flowing through the external resistance is (a) 1 A (b) 1.5 A (c) 2 A (d) 0.75 A SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 9 www.vidyadrishti.com An education portal for future IITians Passage for questions 22 to 24 Wheatstone’s bridge is a very simple but powerful concept which has made many calculations easy. If the assumption is made that current does nor flow through the galvanometer from B to C or C to B. VA – VB = VA – VC, VB – VD = VC – VD The current in R1, has to flow through BD and that in R3 through CD. This ideal has been extended to many symmetrical circuits. 22. For Wheatstone’s bridge, (a) it is necessary that R1 = R2 and R3 = R4 (b) it is necessary that the current in R1 should be equal to current in R3 (c) it is not necessary that the current in R1 should be equal to the current in R3 (d) the current in R2 should be the same that in R4 23. If R1 = 2 ohm, R3 = 4 ohm, R2 = 5 ohm, R4 = 10 ohm and 11Ri= A, then the current in R3 is (a) 0.5 A (b) 1.0 A (c) 1.5 A (d) 2.0 A 24. If the position of the galvanometer is exchanged with that of the battery in the Wheatstone’s bridge shown in the passage. (a) there will be no balancing of the galvanometer (b) there will be no current in the galvanometer (c) there will be balancing if the resistances are also (d) none of these SPACE FOR ROUGH WORK C B G R1 R2 D A R3 R4 4th AIEEE 2011 Test Series 10 www.vidyadrishti.com An education portal for future IITians 25. R1 < R2 length is the same for AB and CD as in question 1. Choose the correct statements. (a) The current density in AB is the same as in CD (b) The field in AB is greater than field in CD (c) The current in the thicker wire AB > the current in the thinner wire CD (d) The current is the same in AB and CD 26. In the open circuit, the potentiometer gave a balance point at 76.3 cm but when 9.5 Ω is used as an external resistance, this gave 64.8 cm as the balance point. The internal resistance of the battery is (a) 1.7 Ω (b) 0.5 Ω (c) negligible (d) none of these. SPACE FOR ROUGH WORK C B R1 R2 D A i 𝜺𝜺 i 𝜺𝜺 G G O’ B O 2 V 1.5 V A 9.5 Ω 4th AIEEE 2011 Test Series 11 www.vidyadrishti.com An education portal for future IITians 27. The total current which in the grounded circuit is (a) 4 A (b) 5 A (c) 4.5 A (d) 5.5 A 28. A and B are connected to a battery of emf ε. The effective resistance of the circuit is (a) r (b) 34r (c) 2r (d) 4r 29. Match similar things in Column I and Column II. Column I Column II A Electric potential difference P Current B Quantity of heat Q Reciprocal of conductivity of heat C Heat flowing per second R Temperature gradient D Specific resistance S Charge SPACE FOR ROUGH WORK 20 V I1 5 V I2 2 Ω 4 Ω I 2 Ω r A r B C r r r 4th AIEEE 2011 Test Series 12 www.vidyadrishti.com An education portal for future IITians 30. If the resistances (100 4th AIEEE 2011 Test Series 13 www.vidyadrishti.com An education portal for future IITians CHEMISTRY 1. For the conversion. 332CHCHCHCHOCHCHCHCHOHX=→= the required reagent X is (a) [(CH3)2CHO]3 Al (b) NaBH4 (c) [(CH3)3CO]3Al (d) H2/Ni 2. HO3HCN3CHCHO.XY+→→→ In the sequence, Y is (a) an acid (b) an aldehyde (c) β-hydroxy acid (d) α-hydroxy acid 3. The correct order of reactivity of PhMgBr with Ph – C – Ph H3C – C – H CH3 – C – CH3 is (a) I > II > III (b) III > II > I (c) II > III > I (d) II > I > III SPACE FOR ROUGH WORK (iii) (ii) (i) O O O 4th AIEEE 2011 Test Series 14 www.vidyadrishti.com An education portal for future IITians 4. In a Cannizaro’s reaction, the intermediate that will be best hydride donor is (a) (b) (c) (d) 5. To convert 2-butanone into propionic acid (a) NaOH + NaI/H+ (b) Fehling’s test (c) I2/NaOH/H+ (d) Tollen’s reagent SPACE FOR ROUGH WORK O-OH H O-O-H O-O-H MeO O-O-H O2N 4th AIEEE 2011 Test Series 15 www.vidyadrishti.com An education portal for future IITians 6. Which reagent can be used to bring the following transformation? CH2CH2CH2CH2CH2COCH3 CH2CH2CH2CH2CH2CH3 (a) N2H4/KOH/DMSO (b) NaBH4 (c) Zn(Hg)/HCl (d) LiAlH4 7. In the reaction, CN Product−→ the absolute configuration of the product will be (a) 2 R, 3 R (b) 2 S, 3 R (c) 2 R, 3 S (d) 2 S, 3 S SPACE FOR ROUGH WORK O O O O COOEt H Br H CH3 4th AIEEE 2011 Test Series 16 www.vidyadrishti.com An education portal for future IITians 8. In the reaction Cl CH2Br + CH3CHO + Mg Dry ether,A→ the product A is (a) (b) (c) (d) 9. Which one of the following reaction will not yield a carboxylic acid? (a) RMgX + CO2 → (b) PhOH + CHCl3 + NaOH → (c) PhCN + H3O+ → (d) PhOH + NaOH + CO2 → SPACE FOR ROUGH WORK CH2Br CH3 H OH C CH2 CH3 H OH C Cl CH2 CH3 H OH C CH3 H OH C CH2Br CH2 CH 4th AIEEE 2011 Test Series 17 www.vidyadrishti.com An education portal for future IITians 10. Which one of the following amines will give carbylamines reaction? (a) Tert-butyl amine (b) Trimethyl amine (c) Methyl aniline (d) Dimethyl amine 11. Consider the sequence of reactions given below: Aniline 2NaNO/Dil. HClnaphtholBCβ−→→ The product C has (a) only as azo group (b) phenolic and azo groups (c) only a phenolic group (d) amino and phenolic groups 12. In the reaction, C6H5CH2 – SH + H – C – H HCl,X→ the expected product X is (a) (b) (c) (d) CH2 – S – CH2Cl O CH2 Cl CH2SH CH2Cl CH2Cl SH 4th AIEEE 2011 Test Series 18 www.vidyadrishti.com An education portal for future IITians SPACE FOR ROUGH WORK 13. Compound ‘X’ has molecular formula C4H8O3. It evolves CO2 with aq. NaHCO3; ‘X’ reacts with LiAlH3 to give an achiral product. The structure of ‘X’ is (a) CH3.CH2 – CH – COOH (b) CH3. CH – COOH (c) CH3. CH. CH2 – COOH (d) CH3. CH. COOH 14. 1, 3-pentadiene is more stable than 1, 4-pentandiene because (a) in 1, 3-pentadiene the 4π AO’s overlap to give extended π system (b) the methyl attached to π bond in 1, 3-pentandiene stabilizes the π bond (c) 1, 3-pentadiene is a linear molecule (d) 1, 3-pentadiene undergoes easy cyclisation 15. Reaction of trans 2-butene with diazomethane in the presence of ultraviolet light gives (a) cis 1, 2-dimethyl cyclopropane (b) cis 2, 3-dimethyl aziridine (c) trans 2, 3-dimethyl aziridine (d) trans 1, 2-dimethyl cyclopropane SPACE FOR ROUGH WORK OH OMe OH CH2 – OH 4th AIEEE 2011 Test Series 19 www.vidyadrishti.com An education portal for future IITians 16. Which one of the following reactions will yield ? (a) (b) (c) (d) SPACE FOR ROUGH WORK COOCH3 O COOCH3 COOCH3 O + COOCH3 + COOCH3 O COOCH3 C C + COOCH3 COOCH3 C C COOCH3 COOCH3 4th AIEEE 2011 Test Series 20 www.vidyadrishti.com An education portal for future IITians 17. Consider the following statements associated with the reaction 22Br.KOH652652HOCHCONHCHNH→ (a) The reaction proceeds through a nitrene intermediate. (b) CO2 is also evolved during the reaction. (c) The reaction is called Gattermann amine synthesis. (d) If water is not used, the main product will be isocyanate. Which of the above statements are correct? (A) 1 and 2 (B) 2 and 3 (C) 1, 2 and 4 (D) 2, 3 and 4 18. 2-phenyl ethanol can be prepared by reacting C6H5 – MgBr with (a) methanol (b) ethanol (c) ethyl acetate (d) ethylene oxide 19. The alkene (C6H10) which on ozonolysis gives formaldehyde, acetaldehyde and propane-1, 3-dial is (a) 1, 2-hexadiene (b) 1, 3-hexadiene (c) 1, 4-hexadiene (d) 2-methyl-1, 3-pentadiene SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 21 www.vidyadrishti.com An education portal for future IITians 20. In the reaction, 3CHCOOH3326CHCCCHBHAB−≡−+→→ Product B is (a) C – C (b) C = C (c) C – C (d) C – C 21. In the reaction, C – N 24HSO→ the product forms (a) C6H5NHCOCH2CH3 (b) C6H5CONHCH2CH3 (c) C6H5CN (d) CH3CH2CN SPACE FOR ROUGH WORK H3C CH3 H OCOCH3 H3C COCH3 H CH3 H3C CH3 H H H3C H H CH3 C6H5 OH CH3 CH2 4th AIEEE 2011 Test Series 22 www.vidyadrishti.com An education portal for future IITians 22. An organic compound A having the molecular formula C4H8Cl2 on hydrolysis yields a compound B having molecular formula C4H8O. Compound B forms phenylhydrazone with phenylhydrazine but gives no positive test with Tollen’s reagent. The structures of compound A and B are respectively, (a) ClCH2 – CH2 – CH2 – CH2Cl and C2H5COCH3 (b) CH3 – CH – CH – CH3 and CH3CH2CH2CHO (c) CH3 – CH2 – C – CH3 and C2H5COCH3 (d) CH – CH and CH – CHO SPACE FOR ROUGH WORK Cl Cl Cl Cl Cl CH3 Cl CH3 H3C H3C 4th AIEEE 2011 Test Series 23 www.vidyadrishti.com An education portal for future IITians 23. Consider the following reaction ()NaOEt223222EtOH2 CHCOO CH CH+Br CHCHCHBrI → 2H/HOII+∆→ The structure of II will be (a) (b) (c) (d) 24. Nitration of which of the following compounds occurs at a slower rate than that of bezene? (a) Methylbenzene (b) Nitrobenzene (c) Aniline (d) Phenol SPACE FOR ROUGH WORK COOH COOH COOH COOC2H5 COOH 4th AIEEE 2011 Test Series 24 www.vidyadrishti.com An education portal for future IITians 25. 50% aq KOHXY→+ (m-chlorobenzaldehyde) X and Y respectively are (a) , (b) , (c) , (d) Cl CHO COO-OH CH2OH OH CHOH Cl CHOH Cl COO-Cl CH2OH Cl CH OH CH OH OH OH 4th AIEEE 2011 Test Series 25 www.vidyadrishti.com An education portal for future IITians SPACE FOR ROUGH WORK 26. The conversion, PhCN → PhCOCH3, can be achieved most conveniently by reaction with (a) CH3MgBr followed by hydrolysis (b) I2 + NaOH; CH3I (c) dilute H2SO4 followed by reaction with CH2N2 (d) LiAlH4 followed by reaction with CH3I 27. Reaction of RCONH2 with a mixture of Br2 and NaOH gives RNH2 as the major product. The intermediate(s) involved in the reaction is/are (a) RCONH2Br (b) R – NHBr (c) R – N = C = O (d) RCONBr2 28. The reaction is influenced by the hyperconjugative effect of – R group. Now, if R, sequentially, is (a) –CH3 (b) C2H5 – (c) (CH3)2 CH – (d) (CH3)3C – SPACE FOR ROUGH WORK R CH2Br + N + Br-R CH2 + N 4th AIEEE 2011 Test Series 26 www.vidyadrishti.com An education portal for future IITians 29. Which of the following reaction(s) will yield propan-2-ol? (a) H232CHCHCHHO+=−+→ (b) 32(i) CHMgI3(ii)HOCHCHO−→ (c) 252(i) CHMgI2(ii)HOCHO→ (d) 4neutral KMnO23CHCHCH=−→ 30. Oxidation of toluene with CrO3 in the presence of (CH3CO)2O gives a product which on treatment with aqueous solution of NaOH produces (a) C6H5CH2COOH (b) (C6H5CO)2O (c) C6H5CH2COOH (d) C6H5CH2OH SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 27 www.vidyadrishti.com An education portal for future IITians MATHEMATICS 1. If f(x) = xn, then the value ()()()()121111...,1!2!!nffffn++++ where f’(x) denotes the rth order derivative of f(x) with respect to x is (a) n (b) 2n (c) 2n – 1 (d) 1 2. Let ()()1/1/1/1/.,xxxxeefxgxee−−−=+ where g is a continuous function then ()0limxfx→ does not exist if (a) g(x) is any constant function (b) g(x) = x (c) g(x) = x2 (d) g(x) = xh(x), where h(x) is a polynomial. 3. If ()11sin,011sin,0, 0, 0xxxxfxxxxxx+>=−−+<= then f(x) is (a) continuous as well as differentiable at x = 0 (b) continuous but not differentiable at x = 0 (c) differentiable but not continuous at x = 0 (d) neither continuous nor differentiable at x = 0 SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 28 www.vidyadrishti.com An education portal for future IITians 4. The set of all values of a for which the function ()()()22232cossin1sin144xxfxaaax=−+−+−+ does not possess critical points is (a) [1, ∞) (b) (0, 1) ∪ (1, 4) (c) (-2, 4) (d) (1, 3) ∪ (3, 5) 5. The value of ()22222axbdxxcxaxb−−+∫ is (a) ()1/sinaxbxKc−++ (b) ()221/sinaxbxKc−++ (c) ()1/cosaxbxKc−++ (d) ()221/cosaxbxKc−++ 6. If ()()1cosec2 ,2xdxfgxC=+∫ then (a) range g(x) = (-∞, ∞) (b) dom f(x) = (-∞, ∞) – {0} (c) g’(x) = sec2x (d) all of these SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 29 www.vidyadrishti.com An education portal for future IITians 7. Given that h(a) = h(b). Then the value of ()(){}()(){}()()()1'''bafghxfghxghxhxdx−∫ is (a) 0 (b) ()()2hahb+ (c) g(h(b)) – g(h(a)) (d) log[f{g(h(b))}] – log[f{g(h(a))}] 8. if the area enclosed by y2 = 4ax and line y = ax is 13 sq. units, then the area enclosed by y = 4x with same parabola is (in sq. units) (a) 8 (b) 4 (c) 43 (d) 83 9. 2210.nxxedx∞+−∫ is equal to (n ∈ N) (a) n! (b) 2 (n!) (c) !2n (d) ()1!2n+ SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 30 www.vidyadrishti.com An education portal for future IITians 10. If /230,1sinxdxIx=+∫ then (a) 0 < I < 1 (b) 22Iπ> (c) 2Iπ< (d) 1 > 2π 11. The number of values of x, where the function ()()coscos2fxxx=+ attains its maximum, is (a) 0 (b) 1 (c) 2 (d) infinite 12. Let ()()2234,xfxfttdt=−+∫ then (a) f(2) = 0 (b) f(-2) = 0 (c) f’(2) = 0 (d) f’(2) = 2 SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 31 www.vidyadrishti.com An education portal for future IITians 13. The value of parameter m for which ()()20sinxmxxdxππ−−∫ is minimum is 5,kπ then k must be (a) 30 (b) 60 (c) 90 (d) 120 14. Statement – 1: The area of the curve y = sin2x from 0 to π will be more than that of the curve y = sin x from 0 to π. Statement – 2: x2 > x if x > 1. a) Statement-I is true, Statement-II is true; Statement-II is a correct explanation for Statement I. b) Statement-I is true, Statement-II is true; Statement-II is not a correct explanation for Statement-I. c) Statement-I is true, Statement-II is false d) Statement-I is false, Statement-II is true 15. Statement – 1: Line 12131111xyz−−+== lies in the plane 11x – 3z – 14 = 0 Statement – 2: A straight line lies in a plane if the line is parallel to the plane and a point of the line lies in the plane a) Statement-I is true, Statement-II is true; Statement-II is a correct explanation for Statement I. b) Statement-I is true, Statement-II is true; Statement-II is not a correct explanation for Statement-I. c) Statement-I is true, Statement-II is false d) Statement-I is false, Statement-II is true SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 32 www.vidyadrishti.com An education portal for future IITians Using the following paragraph, solve Q.16 to 18 Let a function f(x) satisfies the condition f(x). f(y) = 2f(x + y + xy), f’(0) = 1 and f(x) ≥ 0 16. The curve y = f(x) is (a) ()21yx=+ (b) ()21yx=+ (c) y = ln(x + 1) (d) y = ln(x – 1) 17. Area bounded between y = f(|x|) and y = 7 -|x| is (a) 236 sq. units (b) 116 sq. units (c) 866 sq. units (d) 7 sq. units 18. The number of points where g(x) = max. {f(x), 6, 7 -|x|} is non-differentiable ∀ x ∈ [-10, 10] are (a) 5 (b) 6 (c) 7 (d) 8 SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 33 www.vidyadrishti.com An education portal for future IITians 19. A function f is defined for all positive integers and satisfies f(1) = 2005 and f(1) + f(2) + … + f(n) = n2f(n) for all n > 1. The value of f(2004) is (a) 11002 (b) 12004 (c) 20042005 (d) 2004 20. If f(x) = x2 + bx + c and f(2 + t) = f(2 – t) for all real numbers t, then which of the following is true? (a) f(1) < f(2) < f(4) (b) f(2) < f(1) < f(4) (c) f(2) < f(4) < f(1) (d) f(4) < f(2) < f(1) 21. If f: R → R satisfies f(x + y) = f(x) + f(y) ∀ x, y ∈ R and f(1) = 4, then ()1nrfr=∑ is equal to (a) n(n + 1) (b) 4n(n + 1) (c) 2n(n + 1) (d) none of these SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 34 www.vidyadrishti.com An education portal for future IITians Paragraph for Question No. 22 to 24 By the Roll’s theorem If (1) A function f(x) is continuous in [a, b] (2) A function f(x) is Differentiable in (a, b) (3) f(a) = f(b), then there will be at least one root of f’(x) in (a, b) means f’(x) = 0 for atleast one value of x ∈ (a, b) 22. Number of distinct real roots of f”(x) if f(x) = (x – 1)25(x – 2)43(x – 3)11 is (a) 9 (b) 7 (c) 5 (d) 78 23. Number of common roots in f’(x) and f”(x) if f(x) = (x – 3)21(x + 1)19(x – 2)12 is (a) 50 (b) 49 (c) 49 (d) 46 24. Let f(x) be a polynomial such that α, β with α < β are two consecutive roots of f’(x), then the number of roots of f(x) lying in the interval [α, β] is (a) at most 1 (b) at most 2 (c) at least 1 (d) at least 2 SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 35 www.vidyadrishti.com An education portal for future IITians 25. Let f(x) be a polynomial of degree 2 such that f(1) = f(-1) and a, b, c ∈ A.P., then f’(a), f’(b), f’(c) are in (a) A.G.P (b) G.P. (c) A.P. (d) H.P. 26. If ()()()1/221/23/2sin1sin for 0 for 0 for 0axxxxfxcxxbxxxbx++<==+−> is continuous at x = 0, then a = (a) 32 (b) 32− (c) 14 (d) 14− 27. If f(x) = xn, then the value of (){}()()()2222'1"111...1!2!!nffffn+++ is (a) 2nCn (b) 2nCn – 1 (c) 2nCn + 1 (d) 2nC1 SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 36 www.vidyadrishti.com An education portal for future IITians 28. If a, b, c are real, then ()222xaabacfxabxbbcacbcxc+=++ is decreasing in (a) ()2222,03abc−++ (b) ()22220,3abc++ (c) 222,03abc++ (d) None of these 29. Let f(x) be a function satisfying the condition ()(){}2'1fxfx=− with f(0) = 0, then the value of the integral ()121xxeefxdxdxx−+−∫∫ is (a) ex cosx + c (b) ex sinx + ex sin-1x + c (c) ex sin-1 x + c (d) ex cos-1x + x + ex sin-1x + c 30. If 110,xyyx+++= then dydx is (a) ()211x+ (b) ()211x−+ (c) 11x+ (d) None of these SPACE FOR ROUGH WORK 4th AIEEE 2011 Test Series 37 www.vidyadrishti.com An education portal for future IITians SPACE FOR ROUGH WORK Please send your valuable feedback to www.vidyadrishti.com/feedback.php