JEE Main

JEE Main Online Computer Based Test (CBT) Examination Morning Held on 16-04-2018

Timing : 9 : 30 AM – 12 : 30 PM

PHYSICS 

1. The percentage errors in quantities P, Q, R and S are 0.5%, 1%, 3% and 1.5% respectively in the measurement of a physical quantity The maximum percentage error in the value of A will be :

(1)   6.0%

(2)   7.5%

(3)   8.5%

(4)   6.5%

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2. Let  The magnitude of a coplanar vector  such that  is given by :

(1) 

(2) 

(3) 

(4) 

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3. A body of mass m starts moving from rest along x-axis so that its velocity varies as  where a is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t seconds after the start of the motion is :

(1) 

(2)   8 m a⁴t²

(3)   4 m a⁴t²

(4) 

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4. Two particles of the same mass m are moving in circular orbits because of force, given by

The first particle is at a distance r=1, and the second, at r=4. The best estimate for the ratio of kinetic energies of the first and the second particle is closest to :

(1)   6 × 10⁻²

(2)   3 × 10⁻³

(3)   10⁻¹

(4)   6 × 10²

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5. An oscillator of mass M is at rest in its equilibrium position in a potential  A particle of mass m comes from right with speed u and collides completely inelastically with M and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is : (M = 10, m = 5, u = 1, k = 1)

(1) 

(2)   1/2

(3)   2/3

(4) 

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6. Suppose that the angular velocity of rotation of earth is increased. Then, as a consequence :

(1)   Weight of the object, everywhere on the earth, will increase.

(2)   Weight of the object, everywhere on the earth, will decrease.

(3)   There will be no change in weight anywhere on the earth.

(4)   Except at poles, weight of the object on the earth will decrease.

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7. A thin circular disk is in the xy plane as shown in the figure. The ratio of its moment of inertia about z and zʹ axes will be :

(1)   1 : 3

(2)   1 : 4

(3)   1 : 5

(4)   1 : 2

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8. The relative uncertainty in the period of a satellite orbiting around the earth is 10⁻². If the relative uncertainty in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is :

(1)   10⁻²

(2)   2 × 10⁻²

(3)   3 × 10⁻²

(4)   6 × 10⁻²

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9. A small soap bubble of radius 4 cm is trapped inside another bubble of radius 6 cm without any contact. Let P₂ be the pressure inside the inner bubble and P₀, the pressure outside the outer bubble. Radius of another bubble with pressure difference P₂ − P₀ between its inside and outside would be :

(1)   12 cm

(2)   2.4 cm

(3)   6 cm

(4)   4.8 cm

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10. One mole of an ideal monoatomic gas is taken along the path ABCA as shown in the PV diagram. The maximum temperature attained by the gas along the path BC is given by :

(1)  

(2) 

(3) 

(4)    

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11. Two moles of helium are mixed with n moles of hydrogen. If  for the mixture, then the value of n is :

(1)   1

(2)   3

(3)   2

(4)   3/2

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12. A particle executes simple harmonic motion and is located at x=a, b and c at times t_o, 2t_o and 3t_o The frequency of the oscillation is :

(1)    

(2) 

(3)    

(4)     

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13. Two sitar strings, A and B, playing the note ‘Dha’ are slightly out of tune and produce beats of frequency 5 Hz. The tension of the string B is slightly increased and the beat frequency is found to decrease by 3 Hz. If the frequency of A is 425 Hz, the original frequency of B is :

(1)   430 Hz

(2)   420 Hz

(3)   428 Hz

(4)   422 Hz

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14. Two identical conducting spheres A and B, carry equal charge. They are separated by a distance much larger than their diameters, and the force between them is F. A third identical conducting sphere, C, is uncharged. Sphere C is first touched to A, then to B, and then removed. As a result, the force between A and B would be equal to :

(1)   F

(2)   3F/4

(3)   3F/8

(4)   F/2

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15. A heating element has a resistance of 100 Ω at room temperature. When it is connected to a supply of 220 V, a steady current of 2 A passes in it and temperature is 500°C more than room temperature. What is the temperature coefficient of resistance of the heating element ?

(1)   0.5 × 10⁻⁴ °C⁻¹

(2)   5 × 10⁻⁴ °C⁻¹

(3)   1 × 10⁻⁴ °C⁻¹

(4)   2 × 10⁻⁴ °C⁻¹

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16. A galvanometer with its coil resistance 25 Ω requires a current of 1 mA for its full deflection. In order to construct an ammeter to read up to a current of 2 A, the approximate value of the shunt resistance should be :

(1)   2.5 × 10⁻³ Ω

(2)   1.25 × 10⁻² Ω

(3)   1.25 × 10⁻³ Ω

(4)   2.5 × 10⁻² Ω

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17. In the following circuit, the switch S is closed at t=0. The charge on the capacitor C₁ as a function of time will be given by 

(1)   C₁E [1 − exp(−tR/C₁)]

(2)   C₂E [1 − exp(−t/RC₂)]

(3)   C_eqE [1 − exp(−t/RC_eq)]

(4)   C_eqE exp (−t/RC_eq)

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18. A coil of cross-sectional area A having n turns is placed in a uniform magnetic field B. When it is rotated with an angular velocity ω, the maximum e.m.f. induced in the coil will be :

(1)   3 nBAω

(2)  

(3)   nBAω

(4)    

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19. A charge q is spread uniformly over an insulated loop of radius r. If it is rotated with an angular velocity ω with respect to normal axis then the magnetic moment of the loop is :

(1)   q ωr²

(2)    

(3)    

(4)    

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20. A power transmission line feeds input power at 2300 V to a step down transformer with its primary windings having 4000 turns, giving the output power at 230 V. If the current in the primary of the transformer is 5 A, and its efficiency is 90%, the output current would be :

(1)   50 A

(2)   45 A

(3)   25 A

(4)   20 A

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21. A plane electromagnetic wave of wavelength λ has an intensity I. It is propagating along the positive Y-direction. The allowed expressions for the electric and magnetic fields are given by :

(1)  

(2)  

(3)    

(4)  

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22. A ray of light is incident at an angle of 60° on one face of a prism of angle 30°. The emergent ray of light makes an angle of 30° with incident ray. The angle made by the emergent ray with second face of prism will be :

(1)   0°

(2)   90°

(3)   45°

(4)   30°

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23. Unpolarized light of intensity I is incident on a system of two polarizers, A followed by B. The intensity of emergent light is I/2. If a third polarizer C is placed between A and B, the intensity of emergent light is reduced to I/3. The angle between the polarizers A and C is θ. Then :

(1)    

(2)  

(3)    

(4)    

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24. The de-Broglie wavelength (λ_B) associated with the electron orbiting in the second excited state of hydrogen atom is related to that in the ground state (λ_G) by :

(1)   λ_B = 2λ_G

(2)   λ_B = 3λ_G

(3)   λ_B = λ_G/2

(4)   λ_B = λ_G/3

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25. Both the nucleus and the atom of some element are in their respective first excited states. They get de-excited by emitting photons of wavelengths λ_N, λ_A The ratio  is closest to :

(1)   10⁻⁶

(2)   10

(3)   10⁻¹⁰

(4)   10⁻¹

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26. At some instant, a radioactive sample S₁ having an activity 5 μCi has twice the number of nuclei as another sample S₂ which has an activity of 10 μCi. The half lives of S₁ and S₂ are :

(1)   20 years and 5 years, respectively

(2)   20 years and 10 years, respectively

(3)   5 years and 20 years, respectively

(4)   10 years and 20 years, respectively

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27. In the given circuit, the current through zener diode is :

(1)   5.5 mA

(2)   6.7 mA

(3)   2.5 mA

(4)   3.3 mA

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28. A carrier wave of peak voltage 14 V is used for transmitting a message signal. The peak voltage of modulating signal given to achieve a modulation index of 80% will be :

(1)   7 V

(2)   28 V

(3)   11.2 V

(4)   22.4 V

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29. In a circuit for finding the resistance of a galvanometer by half deflection method, a 6 V battery and a high resistance of 11 kΩ are used. The figure of merit of the galvanometer is 60 μA/division. In the absence of shunt resistance, the galvanometer produces a deflection of θ = 9 divisions when current flows in the circuit. The value of the shunt resistance that can cause the deflection of θ/2, is closest to :

(1)   550 Ω

(2)   220Ω

(3)   55Ω

(4)   110Ω

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30. The end correction of a resonance column is 1 cm. If the shortest length resonating with the tuning fork is 10 cm, the next resonating length should be :

(1)   28 cm

(2)   32 cm

(3)   36 cm

(4)   40 cm

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CHEMISTRY

31. An unknown chlorohydrocarbon has 3.55% of chlorine. If each molecule of the hydrocarbon has one chlorine atom only; chlorine atoms present in 1 g of chlorohydrocarbon are :

(Atomic wt. of Cl=35.5 u; Avogadro constant=6.023 × 10²³ mol⁻¹)

(1)   6.023 × 10²⁰

(2)   6.023 × 10⁹

(3)   6.023 × 10²¹

(4)   6.023 × 10²³

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32. The gas phase reaction 2NO₂(g) → N₂O₄(g) is an exothermic reaction. The decomposition of N₂O₄, in equilibrium mixture of NO₂(g) and N₂O₄(g), can be increased by :

(1)   lowering the temperature.

(2)   increasing the pressure.

(3)   addition of an inert gas at constant volume.

(4)   addition of an inert gas at constant pressure.

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33. Assuming ideal gas behaviour, the ratio of density of ammonia to that of hydrogen chloride at same temperature and pressure is : (Atomic wt. of Cl=35.5 u)

(1)   1.46

(2)   0.46

(3)   1.64

(4)   0.64

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34. When 9.65 ampere current was passed for 1.0 hour into nitrobenzene in acidic medium, the amount of p-aminophenol produced is :

(1)   9.81 g

(2)   10.9 g

(3)   98.1 g

(4)   109.0 g

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35. For which of the following processes, ΔS is negative ?

(1)   H₂(g) → 2H(g)

(2)   N₂(g, 1 atm) → N₂(g, 5 atm)

(3)   C(diamond) → C(graphite)

(4)   N₂(g, 273 K) → N₂(g, 300 K)

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36. Which one of the following is not a property of physical adsorption ?

(1)   Higher the pressure, more the adsorption

(2)   Lower the temperature, more the adsorption

(3)   Greater the surface area, more the adsorption

(4)   Unilayer adsorption occurs

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37. If 50% of a reaction occurs in 100 second and 75% of the reaction occurs in 200 second, the order of this reaction is :

(1)   Zero

(2)   1

(3)   2

(4)   3

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38. Which of the following statements is false ?

(1)   Photon has momentum as well as wavelength.

(2)   Splitting of spectral lines in electrical field is called Stark effect.

(3)   Rydberg constant has unit of energy.

(4)   Frequency of emitted radiation from a black body goes from a lower wavelength to higher wavelength as the temperature increases.

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39. At 320 K, a gas A2 is 20% dissociated to A(g). The standard free energy change at 320 K and 1 atm in J mol⁻¹ is approximately : (R=8.314 JK⁻¹ mol⁻¹; ln 2=0.693; ln 3=1.098)

(1)   4763

(2)   2068

(3)   1844

(4)   4281

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40. The mass of a non-volatile, non-electrolyte solute (molar mass=50 g mol⁻¹) needed to be dissolved in 114 g octane to reduce its vapour pressure to 75%, is :

(1)   37.5 g

(2)   75 g

(3)   150 g

(4)   50 g

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41. The incorrect statement is :

(1)   Cu²⁺ salts give red coloured borax bead test in reducing flame.

(2)   Cu²⁺ and Ni²⁺ ions give black precipitate with H2S in presence of HCl solution.

(3)   Ferricion gives blood red colour with potassium thiocyanate.

(4)   Cu²⁺ ion gives chocolage coloured precipitate with potassium ferrocyanide solution.

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42. The incorrect geometry is represented by :

(1)   BF₃ – trigonal planar

(2)   H₂O – bent

(3)   NF₃ – trigonal planar

(4)   AsF₅ – trigonal bipyramidal

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43. In Wilkinson’s catalyst, the hybridization of central metal ion and its shape are respectively :

(1)   sp³d, trigonal bipyramidal

(2)   sp³, tetrahedral

(3)   dsp², square planar

(4)   d²sp³, octahedral

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44. Among the oxides of nitrogen : N₂O₃, N₂O₄ and N₂O₅ ; the molecule(s) having nitrogen-nitrogen bond is/are :

(1)   Only N₂O₅

(2)   N­₂O₃ and N₂O₅

(3)   N₂O₄ and N₂O₅

(4)   N₂O₃­ and N₂O₄

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45. Which of the following complexes will show geometrical isomerism ?

(1)   aquachlorobis(ethylenediamine) cobalt(II) chloride

(2)   pentaaquachlorochromium(III) chloride

(3)   potassium amminetrichloroplatinate (II)

(4)   potassium tris(oxalato)chromate(III)

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46. In a complexometric titration of metal ion with ligand M(Metal ion)+L(Ligand) → C(Complex) end point is estimated spectrophotometrically (through light absorption). If ‘M’ and ‘C’ do not absorb light and only ‘L’ absorbs, then the titration plot between absorbed light (A) versus volume of ligand ‘L’ (V) would look like :

(1)  

(2) 

(3)  

(4)  

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47. In the extraction of copper from its sulphide ore, metal is finally obtained by the oxidation of cuprous sulphide with :

(1)   Fe₂O₃

(2)   Cu₂O

(3)   SO₂

(4)   CO

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48. Which of the following conversions involves change in both shape and hybridisation ?

(1)   NH₃ → NH₄⁺

(2)   CH₄ → C₂H₆

(3)   H₂O → H₃O⁺

(4)   BF­₃ → BF₄⁻

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49. A group 13 element ‘X’ reacts with chlorine gas to produce a compound XCl₃ . XCl₃ is electron deficient and easily reacts with NH3 to form Cl₃X ← NH₃ adduct; however, XCl₃ does not dimerize. X is :

(1)   B

(2)   Al

(3)   Ga

(4)   In

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50. When XO₂ is fused with an alkali metal hydroxide in presence of an oxidizing agent such as KNO₃ ; a dark green product is formed which disproportionates in acidic solution to afford a dark purple solution. X is :

(1)   Ti

(2)   V

(3)   Cr

(4)   Mn

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51. The major product of the following reaction is :

(1)  

(2)  

(3)  

(4)  

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52. For standardizing NaOH solution, which of the following is used as a primary standard ?

(1)   Ferrous Ammonium Sulfate

(2)   dil. HCl

(3)   Oxalic acid

(4)   Sodium tetraborate

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53. The most polar compound among the following is :

(1)  

(2)  

(3)  

(4)  

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54. The correct match between items of List – I and List – II is :

(1)   (A)-(R), (B)-(S), (C)-(P), (D)-(Q)

(2)   (A)-(S), (B)-(R), (C)-(P), (D)-(Q)

(3)   (A)-(S), (B)-(R), (C)-(Q), (D)-(P)

(4)   (A)-(R), (B)-(S), (C)-(Q), (D)-(P)

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55. Among the following, the incorrect statement is :

(1)   Maltose and lactose has 1, 4-glycosidic linkage.

(2)   Sucrose and amylose has 1, 2-glycosidic linkage.

(3)   Cellulose and amylose has 1, 4-glycosidic linkage.

(4)   Lactose contains β-D-galactose and β-D-glucose.

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56. Which of the following compounds will most readily be dehydrated to give alkene under acidic condition ?

(1)   1-Pentanol

(2)   4-Hydroxypentan-2-one

(3)   3-Hydroxypentan-2-one

(4)   2-Hydroxycyclopentanone

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57. Products A and B formed in the following reactions are respectively :

(1)  

(2)   

(3)  

(4)  

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58. The major product B formed in the following reaction sequence is :

(1)  

(2)  

(3)  

(4)  

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59. The major product of the following reaction is :

(1)   

(2) 

(3)   

(4) 

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60. The major product of the following reaction is :

(1)   

(2)   

(3)   

(4)  

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MATHEMATICS

61. Let N denote the set of all natural numbers. Define two binary relations on N as R₁={(x, y) ϵ N × N : 2x + y = 10} and R₂={(x, y) ϵ N × N : x + 2y = 10}. Then :

(1)   Range of R₁ is {2, 4, 8}.

(2)   Range of R₂ is {1, 2, 3, 4}.

(3)   Both R₁ and R₂ are symmetric relations.

(4)   Both R₁ and R₂ are transitive relations.

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62. Let p, q and r be real numbers (p ≠ q, r ≠ 0), such that the roots of the equation  are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to :

(1)     

(2)   p² + q²

(3)   2(p² + q²)

(4)   p² + q² + r²

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63. The least positive integer n for which  is

(1)   2

(2)   3

(3)   5

(4)   6

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64. Let  and B = A²⁰. Then the sum of the elements of the first column of B is :

(1)   210

(2)   211

(3)   231

(4)   251

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65. The number of values of k for which the system of linear equations,

(k + 2)x + 10y = k

kx + (k + 3)y = k – 1

has no solution, is :

(1)   1

(2)   2

(3)   3

(4)   infinitely many

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66. The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple of 3 is :

(1)   24

(2)   30

(3)   36

(4)   48

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67. The coefficient of x² in the expansion of the product (2− x²) ⋅ ((1 + 2x + 3x²)6+ (1 − 4x²)6) is :

(1)   107

(2)   106

(3)   108

(4)   155

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68. Let  (x_i ≠ 2 for i = 1, 2, . . . , n) be in A.P. such that x₁ = 4 and x₂₁ = 20. If n is the least positive integer for which x_n > 50, then  is equal to :

(1)   1/8

(2)   3

(3)   13/8

(4)   13/4

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69. The sum of the first 20 terms of the series  is :

(1)    

(2)    

(3)    

(4)    

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70. 

(1)   1/3

(2)   −1/3

(3)   −1/6

(4)   1/6

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71. If the function f defined as  is continuous at x = 0, then the ordered pair (k, f(0)) is equal to :

(1)   (3, 2)

(2)   (3, 1)

(3)   (2, 1)

(4)   (1/3, 2)

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72. If  then  is equal to :

(1)   y/x

(2)   x/y

(3)   −y/x

(4)   −x/y

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73. Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f (x)=2x³ − 9x² + 12x + 5 in the interval [0, 3]. Then M−m is equal to :

(1)   5

(2)   9

(3)   4

(4)   1

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74. If  (C is a constant of integration), then the ordered pair (K, A) is equal to :

(1)   (2, 1)

(2)   (−2, 3)

(3)   (2, 3)

(4)   (−2, 1)

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75. If  then :

(1)   fʹʹʹ(x) + fʹʹ(x) = sin x

(2)   fʹʹʹ(x) + fʹʹ(x) – fʹ(x) = cos x

(3)   fʹʹʹ(x) + fʹ(x) = cos x – 2x sin x

(4)   fʹʹʹ(x) – fʹʹ(x) = cos x – 2x sin x

Show Answer

76. If the area of the region bounded by the curves, y = x², y = 1/x and the lines y = 0 and x = t (t > 1) is 1 sq. unit, then it is equal to :

(1)   e^3/2

(2)   4/3

(3)   3/2

(4)   e^2/3

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77. The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, centre at the origin and passing through the point (0, 3) is :

(1)   xy yʹʹ + x(yʹ)² – y yʹ = 0

(2)   x + y yʹʹ = 0

(3)   xy yʹ + y² – 9 = 0

(4)   xy yʹ – y² + 9 = 0

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78. The locus of the point of intersection of the lines,  and   (k is any non-zero real parameter), is :

(1)   an ellipse whose eccentricity is 1/√3.

(2)   an ellipse with length of its major axis 8√2.

(3)   a hyperbola whose eccentricity is √3.

(4)   a hyperbola with length of its transverse axis 8√2.

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79. If a circle C, whose radius is 3, touches externally the circle, x² + y² + 2x − 4y – 4 = 0 at the point (2, 2), then the length of the intercept cut by this circle C, on the x-axis is equal to :

(1)   2√5

(2)   3√2

(3)   √5

(4)   2√3

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80. Let P be a point on the parabola, x² = 4y. If the distance of P from the centre of the circle, x² + y² + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is :

(1)   x + 4y – 2 = 0

(2)   x – y + 3 = 0

(3)   x + y + 1 = 0

(4)   x + 2y = 0

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81. If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is 3/2 units, then its eccentricity is :

(1)   1/2

(2)   1/3

(3)   2/3

(4)   1/9

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82. The sum of the intercepts on the coordinate axes of the plane passing through the point (−2, −2, 2) and containing the line joining the points (1, −1, 2) and (1, 1, 1), is :

(1)   4

(2)   −4

(3)   −8

(4)   12

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83. If the angle between the lines,  is  then p is  equal to :

(1)   7/2

(2)   2/7

(3)   −7/4

(4)   −4/7

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84. Let  and a vector  be such that  Then  equals :

(1)   11/3

(2)   11/√3

(3)    

(4)    

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85. The mean and the standard deviation(s.d.) of five observations are 9 and 0, respectively. If one of the observations is changed such that the mean of the new set of five observations becomes 10, then their s.d. is :

(1)   0

(2)   1

(3)   2

(4)   4

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86. Let A, B and C be three events, which are pair-wise independent and  denotes the complement of an event E. If P(A ∩ B ∩ C) = 0 and P(C) > 0, then  is equal to :

(1)     

(2)    

(3)    

(4)    

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87. Two different families A and B are blessed with equal number of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family B is 1/12, then the number of children in each family is :

(1)   3

(2)   4

(3)   5

(4)   6

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88. If an angle A of a ΔABC satisfies 5 cosA+3=0, then the roots of the quadratic equation, 9x² + 27x + 20=0 are :

(1)   sec A, cot A

(2)   sin A, sec A

(3)   sec A, tan A

(4)   tan A, cos A

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89. A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min. for the angle of depression of the car to change from 30° to 45° ; then after this, the time taken (in min.) by the car to reach the foot of the tower, is :

(1)    

(2)    

(3)    

(4)    

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90. If p→(∼p∨∼q) is false, then the truth values of p and q are respectively :

(1)   F, F

(2)   T, F

(3)   F, T

(4)   T, T

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JEE Main Online Computer Based Test (CBT) Examination Afternoon Held on 15-04-2018,

Timing : 2:30 PM – 5.30 PM

PHYSICS 

1. The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants G, h and c. Which of the following correctly gives the Planck length ?

(1)   G h² c³

(2)   G² h c

(3)   G^1/2 h² c

(4) 

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2. A man in a car at location Q on a straight highway is moving with speed υ. He decides to reach a point P in a field at a distance d from the highway (point M) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach P is minimum ?

(1)   d

(2)   d/√2

(3)   d/2

(4)   d/√3

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3. A body of mass 2 kg slides down with an acceleration of 3 m/s² on a rough inclined plane having a slope of 30°. The external force required to take the same body up the plane with the same acceleration will be : (g=10 m/s²)

(1)   14 N

(2)   20 N

(3)   6 N

(4)   4 N

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4. A proton of mass m collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of 90° with respect to each other. The mass of unknown particle is :

(1)   m/2

(2)   m

(3)   m/√3

(4)   2 m

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5. A disc rotates about its axis of symmetry in a horizontal plane at a steady rate of 3.5 revolutions per second. A coin placed at a distance of 1.25 cm from the axis of rotation remains at rest on the disc. The coefficient of friction between the coin and the disc is : (g=10 m/s²)

(1)   0.5

(2)   0.3

(3)   0.7

(4)   0.6

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6. A thin uniform bar of length L and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m are moving in the same horizontal plane from opposite sides of the bar with speeds 2υ and υ respectively. The masses stick to the bar after collision at a distance  respectively from the centre of the abr. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be :

(1) 

(2) 

(3) 

(4) 

Show Answer

7. A thin rod MN, free to rotate in the vertical plane about the fixed end N, is held horizontal. When the end M is released the speed of this end, when the rod makes an angle α with the horizontal, will be proportional to : (see figure)

(1) 

(2)   sin α

(3) 

(4)   cos α

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8. As shown in the figure, forces of 10⁵ N each are applied in opposite directions, on the upper and lower faces of a cube of side 10 cm, shifting the upper face parallel to itself by 0.5 cm. If the side of another cube of the same material is 20 cm, then under similar conditions as above, the displacement will be :

(1)   0.25 cm

(2)   0.37 cm

(3)   0.75 cm

(4)   1.00 cm

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9. When an air bubble of radius r rises from the bottom to the surface of a lake, its radius becomes 5r/4. Taking the atmospheric pressure to be equal to 10 m height of water column, the depth of the lake would approximately be (ignore the surface tension and the effect of temperature) :

(1)   11.2 m

(2)   8.7 m

(3)   9.5 m

(4)   10.5 m

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10. Two Carnot engines A and B are operated in series. Engine A receives heat from a reservoir at 600 K and rejects heat to a reservoir at temperature T. Engine B receives heat rejected by engine A and in turn rejects it to a reservoir at 100 K. If the efficiencies of the two engines A and B are represented by η_A and η_B, respectively, then what is the value of 

(1)   12/7

(2)   7/12

(3)   12/5

(4)   5/12

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11. The value closest to the thermal velocity of a Helium atom at room temperature (300 K) in ms⁻¹ : [k_B = 1.4 × 10⁻²³ J/K; m_He = 7 × 10⁻²⁷ kg]

(1)   1.3 × 10⁴

(2)   1.3 × 10³

(3)   1.3× 10⁵

(4)   1.3 × 10²

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12. Two simple harmonic motions, as shown below, are at right angles. They are combined to form Lissajous figures.

         x(t) = A sin (at + δ)

         y(t) = B sin (bt)

Identify the correct match below.

          Parameters                      Curve

(1)   A ≠ B, a = b ; δ = 0            Parabola

(2)   A = B, a = b ; δ = π/2         Line

(3)   A ≠ B, a = b ; δ = π/2         Ellipse

(4)   A = B, a = 2b ; δ = π/2       Circle

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13. 5 beats/second are heard when a tuning fork is sounded with a sonometer wire under tension, when the length of the sonometer wire is either 0.95 m or 1 m. The frequency of the fork will be :

(1)   195 Hz

(2)   150 Hz

(3)   300 Hz

(4)   251 Hz

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14. A solid ball of radius R has a charge density ρ given by ρ = ρ₀(1 – r/R) for 0 ≤ r ≤ The electric field outside the ball is :

(1) 

(2) 

(3) 

(4) 

Show Answer

15. A parallel plate capacitor with area 200 cm² and separation between the plates 1.5 cm, is connected across a battery of emf V. If the force of attraction between the plates is 25×10⁻⁶ N, the value of V is approximately :

(1)   250 V

(2)   100 V

(3)   300 V

(4)   150 V

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16. A copper rod of cross-sectional area A carries a uniform current I through it. At temperature T, if the volume charge density of the rod is ρ, how long will the charges take to travel a distance d ?

(1) 

(2) 

(3) 

(4) 

Show Answer

17. A capacitor C₁=1.0 μF is charged up to a voltage V=60 V by connecting it to battery B through switch (1). Now C₁ is disconnected from battery and connected to a circuit consisting of two uncharged capacitors C₂=3.0 μF and C₃=6.0 μF through switch (2), as shown in the figure. The sum of final charges on C₂ and C₃ is :

(1)   40 μC

(2)   36 μC

(3)   20 μC

(4)   54 μC

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18. A current of 1 A is flowing on the sides of an equilateral triangle of side 4.5×10⁻² The magnetic field at the centre of the triangle will be :

(1)   2 ×10⁻⁵ Wb/m²

(2)   Zero

(3)   8 ×10⁻⁵ Wb/m²

(4)   4 ×10⁻⁵ Wb/m²

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19. At the centre of a fixed large circular coil of radius R, a much smaller circular coil of radius r is placed. The two coils are concentric and are in the same The larger coil carries a current I. The smaller coil is set to rotate with a constant angular velocity ω about an axis along their common diameter. Calculate the emf induced in the smaller coil after a time t of its start of rotation.

(1) 

(2) 

(3) 

(4) 

Show Answer

20. 

A copper rod of mass m slides under gravity on two smooth parallel rails, with

separation l and set at an angle of θ with the horizontal. At the bottom, rails are joined by a resistance R. There is a uniform magnetic field B normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is :

(1) 

(2) 

(3) 

(4) 

Show Answer

21. A plane polarized monochromatic EM wave is traveling in vacuum along z direction such that at t = t₁ it is found that the electric field is zero at a spatial point z₁. The next zero that occurs in its neighbourhood is at z₂. The frequency of the electromagnetic wave is :

(1) 

(2) 

(3) 

(4) 

Show Answer

22. A convergent doublet of separated lenses, corrected for spherical aberration, has resultant focal length of 10 cm. The separation between the two lenses is 2 cm. The focal lengths of the component lenses are :

(1)   10 cm, 12 cm

(2)   12 cm, 14 cm

(3)   16 cm, 18 cm

(4)   18 cm, 20 cm

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23. A plane polarized light is incident on a polariser with its pass axis making angle θ with x-axis, as shown in the figure. At four different values of θ, θ=8°, 38°, 188° and 218°, the observed intensities are same. What is the angle between the direction of polarization and x-axis ?

(1)   98°

(2)   128°

(3)   203°

(4)   45°

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24. If the de Broglie wavelengths associated with a proton and an α-particle are equal, then the ratio of velocities of the proton and the α-particle will be :

(1)   4 : 1

(2)   2 : 1

(3)   1 : 2

(4)   1 : 4

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25. Muon (μ⁻) is a negatively charged (|q| = |e|) particle with a mass m_μ=200 m_e, where m_e is the mass of the electron and e is the electronic charge. If μ⁻ is bound to a proton to form a hydrogen like atom, identify the correct statements.

(A)  Radius of the muonic orbit is 200 times smaller than that of the electron.

(B)  The speed of the μ⁻ in the nth orbit is 1/200 times that of the electron in the nth orbit.

(C)  The ionization energy of muonic atom is 200 times more than that of an hydrogen atom.

(D)  The momentum of the muon in the nth orbit is 200 times more than that of the electron.

(1)   (A), (B), (D)

(2)   (A), (C), (D)

(3)   (B), (D)

(4)   (C), (D)

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26. An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of 8 : 27. The ratio of the radii of the nuclei (assumed to be spherical) is :

(1)   8 : 27

(2)   4 : 9

(3)   3 : 2

(4)   2 : 3

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27. Truth table for the following digital circuit will be :

(1) 

(2) 

(3) 

(4) 

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28. The carrier frequency of a transmitter is provided by a tank circuit of a coil of inductance 49 μH and a capacitance of 2.5 nF. It is modulated by an audio signal of 12 kHz. The frequency range occupied by the side bands is :

(1)   13482 kHz − 13494 kHz

(2)   442 kHz − 466 kHz

(3)   63 kHz − 75 kHz

(4)   18 kHz − 30 kHz

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29. A constant voltage is applied between two ends of a metallic wire. If the length is halved and the radius of the wire is doubled, the rate of heat developed in the wire will be :

(1)   Doubled

(2)   Halved

(3)   Unchanged

(4)   Increased 8 times

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30. A body takes 10 minutes to cool from 60°C to 50° The temperature of surroundings is constant at 25°C. Then, the temperature of the body after next 10 minutes will be approximately :

(1)   47°C

(2)   41°C

(3)   45°C

(4)   43°C

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CHEMISTRY

31. For per gram of reactant, the maximum quantity of N₂ gas is produced in which of the following thermal decomposition reactions ?

(Given : Atomic wt. – Cr = 52 u, Ba = 137 u)

(1)   (NH₄)₂Cr₂O₇(s) → N₂(g)+4H₂O(g) +Cr₂O₃(s)

(2)   2NH₄NO₃(s) → 2 N₂(g)+4H₂O(g) +O₂(g)

(3)   Ba(N₃)₂(s) → Ba(s)+3N₂(g)

(4)   2NH₃(g) → N₂(g)+3H₂(g)

Show Answer

32. All of the following share the same crystal structure except :

(1)   LiCl

(2)   NaCl

(3)   RbCl

(4)   CsCl

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33. The de-Broglie’s wavelength of electron present in first Bohr orbit of ‘H’ atom is :

(1)   0529 Å

(2)   2π × 0.529 Å

(3) 

(4)   4 × 0.529 Å

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34. ∆_fG° at 500 K for substance ‘S’ in liquid state and gaseous state are +100.7 kcal mol⁻¹ and +103 kcal mol−1, respectively. Vapour pressure of liquid ‘S’ at 500 K is approximately equal to :

(R – 2 cal K⁻¹ mol⁻¹)

(1)   0.1 atm

(2)   1 atm

(3)   10 atm

(4)   100 atm

Show Answer

35. Given

(i) 2Fe₂O₃(s) → 4Fe(s)+3O₂(g) ∆_rG° = +1487.0 kJ mol⁻¹

(ii) 2CO(g) + O₂(g) → 2CO₂(g); ∆_rG° = −514.4 kJ mol⁻¹

Free energy change, ∆_rG° for the reaction 2Fe₂O₃(s) + 6CO(g) → 4Fe(s) + 6CO₂(g) will  be :

(1)   −112.4 kJ mol⁻¹

(2)   −56.2 kJ mol⁻¹

(3)   −168.2 kJ mol⁻¹

(4)   −208.0 kJ mol⁻¹

Show Answer

36. Two 5 molal solutions are prepared by dissolving a non-electrolyte non-volatile solute separately in the solvents X and Y. The molecular weights of the solvents are M_X and M_Y, respectively where  The relative lowering of vapour pressure of the solution in X is “m” times that of the solution in Y. Given that the number of moles of solute is very small in comparison to that of solvent, the value of “m” is :

(1)   4/3

(2)   3/4

(3)   1/2

(4)   1/4

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37. Following four solutions are prepared by mixing different volumes of NaOH and HCl of different concentrations, pH of which one of them will be equal to 1 ?

(1) 

(2) 

(3) 

(4) 

Show Answer

38. At a certain temperature in a 5 L vessel, 2 moles of carbon monoxide and 3 moles of chlorine were allowed to reach equilibrium according to the reaction,

CO + Cl₂ ⇌ COCl₂

At equilibrium, if one mole of CO is present then equilibrium constant (K_c) for the reaction is :

(1)   2

(2)   2.5

(3)   3

(4)   4

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39. If x gram of gas is adsorbed by m gram of adsorbent at pressure P, the plot of  versus log P is linear. The slope of the plot is :

(n and k are constants and n > 1)

(1)   2 k

(2)   log k

(3)   n

(4)   1/n

Show Answer

40. For a first order reaction, A → P, t_­1/2 (half-life) is 10 days. The time required for 1/4th conversion of A (in days) is :

(ln 2 = 0.693, ln 3 = 1.1)

(1)   5

(2)   3.2

(3)   4.1

(4)   2.5

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41. Which of the following best describes the diagram below of a molecular orbital ?

(1)   A non-bonding orbital

(2)   An antibonding σ orbital

(3)   A bonding π orbital

(4)   An antibonding π orbital

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42. Biochemical Oxygen Demand (BOD) value can be a measure of water pollution caused by the organic matter. Which of the following statements is correct ?

(1)   Aerobic bacteria decrease the BOD value.

(2)   Anaerobic bacteria increase the BOD value.

(3)   Clean water has BOD value higher than 10 ppm.

(4)   Polluted water has BOD value higher than 10 ppm.

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43. In KO₂, the nature of oxygen species and the oxidation state of oxygen atom are, respectively :

(1)   Oxide and −2

(2)   Superoxide and −1/2

(3)   Peroxide and −1/2

(4)   Superoxide and −1

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44. The number of P−O bonds in P₄O₆ is :

(1)   6

(2)   9

(3)   12

(4)   18

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45. Lithium aluminium hydride reacts with silicon tetrachloride to form :

(1)   LiCl, AlH₃ and SiH₄

(2)   LiCl, AlCl₃ and SiH₄

(3)   LiH, AlCl₃ and SiCl₂

(4)   LiH, AlH₃ and SiH₄

Show Answer

46. The correct order of spin-only magnetic moments among the following is :

(Atomic number : Mn=25, Co=27, Ni=28, Zn=30)

(1)   [ZnCl₄]²⁻ > [NiCl₄]²⁻ > [CoCl₄]²⁻ > [MnCl₄]²⁻

(2)   [CoCl₄]²⁻ > [MnCl₄]²⁻ > [NiCl₄]²⁻ > [ZnCl₄]²⁻

(3)   [NiCl₄]²⁻ > [CoCl₄]²⁻ > [MnCl₄]²⁻ > [ZnCl₄]²⁻

(4)   [MnCl₄]²⁻ > [CoCl₄]²⁻ > [NiCl₄]²⁻ > [ZnCl₄]²⁻

Show Answer

47. The correct order of electron affinity is :

(1)   F > Cl > O

(2)   F > O > Cl

(3)   Cl > F > O

(4)   O > F > Cl

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48. In XeO₃F₂, the number of bond pair(s), π-bond(s) and lone pair(s) on Xe atom respectively are :

(1)   5, 2, 0

(2)   4, 2, 2

(3)   5, 3, 0

(4)   4, 4, 0

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49. In the leaching method, bauxite ore is digested with a concentrated solution of NaOH that produces ‘X’. When CO₂ gas is passed through the aqueous solution of ‘X’, a hydrated compound ‘Y’ is precipitated. ‘X’ and ‘Y’ respectively are :

(1)   NaAlO₂ and Al₂(CO₃)₃⋅ x H₂O

(2)   Al(OH)₃ and Al₂O₃⋅ x H₂O

(3)   Na[Al(OH)₄] and Al₂O₃⋅ x H₂O

(4)   Na[Al(OH)₄] and Al₂(CO₃)₃⋅ x H₂O

Show Answer

50. The total number of possible isomers for square-planar [Pt(Cl)(NO₂)(NO₃)(SCN)]²⁻ is :

(1)   8

(2)   12

(3)   16

(4)   24

Show Answer

51. Two compounds I and II are eluted by column chromatography (adsorption of I > II). Which one of following is a correct statement ?

(1)   I moves faster and has higher R_f value than II

(2)   II moves faster and has higher R_f value than I

(3)   I moves slower and has higher R_f value than II

(4)   II moves slower and has higher R_f value than I

Show Answer

52. Which of the following statements is not true ?

(1)   Step growth polymerisation requires a bifunctional monomer.

(2)   Nylon 6 is an example of step growth polymerisation.

(3)   Chain growth polymerization includes both homopolymerisation and copolymerisation.

(4)   Chain growth polymerization involves homopolymerisation only.

Show Answer

53. When 2-butyne is treated with H₂/ Lindlar’s catalyst, compound X is produced as the major product and when treated with Na/liq. NH₃ it produces Y as the major product. Which of the following statements is correct ?

(1)   X will have higher dipole moment and higher boiling point than Y.

(2)   Y will have higher dipole moment and higher boiling point than X.

(3)   X will have lower dipole moment and lower boiling point than Y.

(4)   Y will have higher dipole moment and lower boiling point than X.

Show Answer

54. The increasing order of the acidity of the following carboxylic acids is :

(1)   I < III < II < IV

(2)   IV < II < III < I

(3)   II < IV < III < I

(4)   III < II < IV < I

Show Answer

55. The major product formed in the following reaction is :

(1) 

(2) 

(3) 

(4) 

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56. On treatment of the following compound with a strong acid, the most susceptible site for bond cleavage is :

(1)   C1 – O2

(2)   O2 – C3

(3)   C4 – O5

(4)   O5 – C6

Show Answer

57. The increasing order of diazotisation of the following compounds is :

(1)   (a) < (b) < (c) < (d)

(2)   (a) < (d) < (b) < (c)

(3)   (a) < (d) < (c) < (b)

(4)   (d) < (c) < (b) < (a)

Show Answer

58. The dipeptide, Gln-Gly, on treatment with CH₃COCl followed by aqueous work up gives :

(1) 

(2) 

(3) 

(4) 

Show Answer

59. The total number of optically active compounds formed in the following reaction is :

(1)   Two

(2)   Four

(3)   Six

(4)   Zero

Show Answer

60. The major product formed in the following reaction is :

(1) 

(2) 

(3) 

(4) 

Show Answer

MATHEMATICS

61. Let f : A → B be a function defined as  where A = R – {2} and B R – {1}. Then f is :

(1)   invertible and 

(2)   invertible and 

(3)   invertible and 

(4)   not invertible

Show Answer

62. If f (x) is a quadratic expression such that f (1)+f (2)=0, and −1 is a root of f (x)=0, then the other root of f (x)=0 is :

(1)   −5/8

(2)   −8/5

(3)   5/8

(4)   8/5

Show Answer

63. If |z – 3 + 2i| ≤ 4 then the difference between the greatest value and the least value of |z| is :

(1)   2√13

(2)   8

(3)   4 + √13

(4)   √13

Show Answer

64. Suppose A is any 3×3 non-singular matrix and (A − 3I) (A − 5I) = O, where I = I₃ and O = O₃. If αA + βA⁻¹ = 4I, then α + β is equal to :

(1)   8

(2)   7

(3)   13

(4)   12

Show Answer

65. If the system of linear equations

x + ay + z = 3

x + 2y + 2z = 6

x + 5y + 3z = b

has no solution, then :

(1)   a=−1, b=9

(2)   a=−1, b ≠ 9

(3)   a ≠−1, b=9

(4)   a=1, b ≠ 9

Show Answer

66. The number of four letter words that can be formed using the letters of the word BARRACK is :

(1)   120

(2)   144

(3)   264

(4)   270

Show Answer

67. The coefficient of x¹⁰ in the expansion of (1+x)²(1+x²)³(1+x³)⁴ is equal to :

(1)   52

(2)   56

(3)   50

(4)   44

Show Answer

68. If a, b, c are in A.P. and a², b², c² are in G.P. such that a < b < c and  then the value of a is :

(1) 

(2) 

(3) 

(4) 

Show Answer

69. Let  and B_n = 1 – A_n. Then, the least odd natural number p, so that B_n > A_n, for all n ≥ p, is :

(1)   9

(2)   7

(3)   11

(4)   5

Show Answer

70. 

(1)   1/4

(2)   1

(3)   1/2

(4)   −1/2

Show Answer

71. Let 

The value of k for which f is continuous at x = 2 is :

(1)   1

(2)   e

(3)   e⁻¹

(4)   e⁻²

Show Answer

72. If  equals :

(1)   −√3 log_e √3

(2)   √3 log_e √3

(3)   −√3 log_e 3

(4)   √3 log_e 3

Show Answer

73. Let f (x) be a polynomial of degree 4 having extreme values at x = 1 and x = 2.

If  then f(−1) is equal to :

(1)   9/2

(2)   5/2

(3)   3/2

(4)   1/2

Show Answer

74. If 

(where C is a constant of integration), then the ordered pair (A, B) is equal to :

(1)   (2, 1)

(2)   (−2, −1)

(3)   (−2, 1)

(4)   (2, −1)

Show Answer

75. The value of integral  is :

(1) 

(2) 

(3) 

(4) 

Show Answer

76. If 

(1)   I₂ > I₃ > I₁

(2)   I₂ > I₁ > I₃

(3)   I₃ > I₂ > I₁

(4)   I₃ > I₁ > I₂

Show Answer

77. The curve satisfying the differential equation, (x² − y²)dx+2xydy=0 and passing through the point (1, 1) is :

(1)   a circle of radius one.

(2)   a hyperbola.

(3)   an ellipse.

(4)   a circle of radius two.

Show Answer

78. The sides of a rhombus ABCD are parallel to the lines, x−y+2=0 and 7x−y+3=0. If the diagonals of the rhombus intersect at P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the ordinate of A is :

(1)   5/2

(2)   7/4

(3)   2

(4)   7/2

Show Answer

79. The foot of the perpendicular drawn from the origin, on the line, 3x+y=λ(λ ≠ 0) is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is :

(1)   1 : 3

(2)   3 : 1

(3)   1 : 9

(4)   9 : 1

Show Answer

80. The tangent to the circle C₁ : x²+y² − 2x−1=0 at the point (2, 1) cuts off a chord of length 4 from a circle C₂ whose centre is (3, −2). The radius of C₂ is :

(1)   2

(2)   √2

(3)   3

(4)   √6

Show Answer

81. Tangents drawn from the point (−8, 0) to the parabola y² = 8x touch the parabola at P and Q. If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to :

(1)   4x² + 9y² = 121

(2)   9x² + 4y² = 169

(3)   4x² – 9y² = 121

(4)   9x² – 4y² = 169

Show Answer

82. A normal to the hyperbola, 4x² − 9y² = 36 meets the co-ordinate axes x and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is :

(1)   4x² + 9y² = 121

(2)   9x² + 4y² = 169

(3)   4x² – 9y² = 121

(4)   9x² – 4y² = 169

Show Answer

83. An angle between the lines whose direction cosines are given by the equations, l + 3m + 5n = 0 and 5lm −2mn + 6nl = 0, is :

(1)   cos⁻¹ (1/3)

(2)   cos⁻¹ (1/4)

(3)   cos⁻¹ (1/6)

(4)   cos⁻¹ (1/8)

Show Answer

84. A plane bisects the line segment joining the points (1, 2, 3) and (−3, 4, 5) at right angles. Then this plane also passes through the point :

(1)   (−3, 2, 1)

(2)   (3, 2, 1)

(3)   (−1, 2, 3)

(4)   (1, 2, −3)

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85. If the position vectors of the vertices A, B and C of a ∆ABC are respectively  and  then the position vector of the point, where the bisector of ∠A meets BC is :

(1) 

(2) 

(3) 

(4) 

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86. A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of ‘p’ is :

(1)   1/5

(2)   1/3

(3)   2/5

(4)   1/4

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87. If the mean of the data : 7, 8, 9, 7, 8, 7, λ, 8 is 8, then the variance of this data is :

(1)   7/8

(2)   1

(3)   9/8

(4)   2

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88. The number of solutions of sin 3x=cos 2x, in the interval (π/2, π) is :

(1)   1

(2)   2

(3)   3

(4)   4

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89. A tower T₁ of height 60 m is located exactly opposite to a tower T₂ of height 80 m on a straight road. From the top of T₁, if the angle of depression of the foot of T₂ is twice the angle of elevation of the top of T₂, then the width (in m) of the road between the feet of the towers T₁ and T₂ is :

(1)   10√2

(2)   10√3

(3)   20√3

(4)   20√2

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90. Consider the following two statements :

Statement p :

The value of sin 120° can be derived by taking θ = 240° in the equation 

Statement q :

The angles A, B, C and D of any quadrilateral ABCD satisfy the equation 

Then the truth values of p and q are respectively :

(1)   F, T

(2)   T, F

(3)   T, T

(4)   F, F

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