Description

Attempt WBJEE Previous Year - 2013 | 200 questions in 300 minutes | Mock test for JEE preparation | Free important questions MCQ to study WBJEE Sample Papers, Section Wise & Full Mock Tests for JEE Exam | Download free PDF with solutions

QUESTION: 1

The equation of state of a gas is given by where P, V, T are pressure, volume and temperature respectively, and a, b, c are constants. The dimensions of a and b are respectively

Solution:

QUESTION: 2

A capacitor of capacitance C_{0 }is charged to a potential V_{0} and is connected with another capacitor of capacitance C as shown. After closing the switch S, the common potential across the two capacitors becomes V. The capacitance C is given by

Solution:

Charge on isolated plates remains same

QUESTION: 3

The r.m.s. speed of the molecules of a gas at 100°C is ν. The temperature at which the r.m.s. speed will be √3ν is

Solution:

T (°C) = 846°C

QUESTION: 4

As shown in the figure below, a charge +2C is situated at the origin O and another charge +5C is on the x-axis at the point A. The later charge from the point A is then brought to a point B on the y-axis. The work done is

Solution:

QUESTION: 5

A frictionless piston-cylinder based enclosure contains some amount of gas at a pressure of 400 kPa. Then heat is transferred to the gas at constant pressure in a quasi-static process. The piston moves up slowly through a height of 10cm. If the piston has a cross-section area of 0.3 m^{2}, the work done by the gas in this process is

Solution:

Wconst – Press = P × ΔV = 400 × 10^{3} × 0.3 × 10 × 10^{–2} = 400 × 10 × 3 = 12000 = 12 kJ

QUESTION: 6

An electric cell of e.m.f. E is connected across a copper wire of diameter d and length l. The drift velocity of electrons in the wire is v_{d}. If the length of the wire is changed to 2l, the new drift velocity of electrons in the copper wire will be

Solution:

QUESTION: 7

A NOR gate and a NAND gate are connected as shown in the figure. Two different sets of inputs are given to this set up. In the first case, the input to the gates are A=0, B=0, C=0. In the second case, the inputs are A=1, B=0, C=1. The output D in the first case and second case respectively are

Solution:

QUESTION: 8

A bar magnet has a magnetic moment of 200 A.m^{2.} The magnet is suspended in a magnetic field of 0.30 NA^{–1} m^{–1}. The torque required to rotate the magnet from its equilibrium position through an angle of 30°, will be

Solution:

QUESTION: 9

Two soap bubbles of radii r and 2r are connected by a capillary tube-valve arrangement as shown in the diagram. The valve is now opened. Then which one of the following will result:

Solution:

As pressure inside smaller bubble is greater than pressure inside bigger bubble . so air flows from smaller to bigger.

QUESTION: 10

An ideal mono-atomic gas of given mass is heated at constant pressure. In this process, the fraction of supplied heat energy used for the increase of the internal energy of the gas is

Solution:

QUESTION: 11

The velocity of a car travelling on a straight road is 36 kmh^{–1 }at an instant of time. Now travelling with uniform acceleration for 10 s, the velocity becomes exactly double. If the wheel radius of the car is 25 cm, then which of the following numbers is the closest to the number of revolutions that the wheel makes during this 10 s?

Solution:

QUESTION: 12

Two glass prisms P_{1} and P_{2 }are to be combined together to produce dispersion without deviation. The angles of the prisms P_{1} and P_{2} are selected as 4° and 3° respectively. If the refractive index of prism P_{1} is 1.54, then that of P_{2} will be

Solution:

δ_{1} + δ_{2} = 0 ⇒ (μ – 1)A_{1} = (μ_{2} – 1)A_{2 }⇒ μ_{2} = 1.72

QUESTION: 13

The ionization energy of the hydrogen atom is 13.6 eV. The potential energy of the electron in n = 2 state of hydrogen atom is

Solution:

QUESTION: 14

Water is flowing in streamline motion through a horizontal tube. The pressure at a point in the tube is p where the velocity of flow is v. At another point, where the pressure is p/2, the velocity of flow is [density of water = ρ]

Solution:

QUESTION: 15

In the electrical circuit shown in figure, the current through the 4Ω resistor is

Solution:

QUESTION: 16

A wire of initial length L and radius r is stretched by a length l. Another wire of same material but with initial length 2L and radius 2r is stretched by a length 2l. The ratio of the stored elastic energy per unit volume in the first and second wire is,

Solution:

QUESTION: 17

A current of 1 A is flowing along positive x-axis through a straight wire of length 0.5 m placed in a region of a magnetic field given by B =(2ˆi+ 4ˆj) T. The magnitude and the direction of the force experienced by the wire respectively are

Solution:

QUESTION: 18

Two spheres of the same material, but of radii R and 3R are allowed to fall vertically downwards through a liquid of density σ. The ratio of their terminal velocities is

Solution:

QUESTION: 19

S_{1} and S_{2} are the two coherent point sources of light located in the xy-plane at points (0,0) and (0,3λ) respectively.Here λ is the wavelength of light. At which one of the following points (given as coordinates), the intensity of interference will be maximum?

Solution:

QUESTION: 20

An alpha particle (^{4}He)has a mass of 4.00300 amu. A proton has mass of 1.00783 amu and a neutron has mass of 1.00867 amu respectively. The binding energy of alpha particle estimated from these data is the closest to

Solution:

ΔM = 2(m_{p} + m_{n}) – m_{He} = 0.0300 amu E = ΔM C^{2} = 0.03 × 931 MeV ≈ 27.9 Mev

QUESTION: 21

The equivalent resistance between the points a and b of the electrical network shown in the figure is

Solution:

QUESTION: 22

Four small objects each of mass m are fixed at the corners of a rectangular wire-frame of negligible mass and of sides a and b (a > b). If the wire frame is now rotated about an axis passing along the side of length b, then the moment of inertia of the system for this axis of rotation is

Solution:

QUESTION: 23

The de Broglie wavelength of an electron (mass = 1 × 10^{–30} kg, charge = 1.6 × 10^{-19} C) with a kinetic energy of 200 eV is (Planck’s constant = 6.6 × 10^{–34} J s)

Solution:

QUESTION: 24

An object placed at a distance of 16 cm from a convex lens produces an image of magnification m (m > 1). If the object is moved towards the lens by 8 cm then again an image of magnification m is obtained. The numerical value of the focal length of the lens is

Solution:

m = f/f+u

As magnification can be same for two diffeent values of u only if they are of opposite sign.

QUESTION: 25

The number of atoms of a radioactive substance of half-life T is N_{0} at t = 0. The time necessary to decay from N_{0}/2 atoms to N_{0}/10 atoms will be

Solution:

N(t) = N_{o} x e^{-lt}

QUESTION: 26

A travelling acoustic wave of frequency 500 Hz is moving along the positive x-direction with a velocity of 300 ms^{–1}. The phase difference between two points x_{1 }and x_{2} is 60º. Then the minimum separation between the two pints is

Solution:

QUESTION: 27

A mass M at rest is broken into two pieces having masses m and (M-m). The two masses are then separated by a distance r. The gravitational force between them will be the maximum when the ratio of the masses [m:(M-m)] of the two parts is

Solution:

QUESTION: 28

A shell of mass 5M, acted upon by no external force and initially at rest, bursts into three fragments of masses M, 2M and 2M respectively. The first two fragments move in opposite directions with velocities of magnitudes 2V and V respectively. The third fragment will

Solution:

By conservation of momentum

QUESTION: 29

A bullet of mass m travelling with a speed v hits a block of mass M initially at rest and gets embedded in it. The combined system is free to move and there is no other force acting on the system. The heat generated in the process will be

Solution:

QUESTION: 30

A particle moves along X-axis and its displacement at any time is given by x(t) = 2t^{3} – 3t^{2} + 4t in SI units. The velocity of the particle when its acceleration is zero, is

Solution:

x(t) = (2t^{3} – 3t^{2} + 4t)

QUESTION: 31

A planet moves around the sun in an elliptical orbit with the sun at one of its foci. The physical quantity associated with the motion of the planet that remains constant with time is

Solution:

Torque about the sun, S = 0 ⇒ Angular momentum is conserved

QUESTION: 32

The fundamental frequency of a closed pipe is equal to the frequency of the second harmonic of an open pipe. The ratio of their lengths is

Solution:

QUESTION: 33

A particle of mass M and charge q is released from rest in a region of uniform electric field of magnitude E. After a time t, the distance travelled by the charge is S and the kinetic energy attained by the particle is T. Then, the ratio T/S

Solution:

QUESTION: 34

An alternating current in a circuit is given by I = 20 sin (100πt + 0.05π) A. The r.m.s. value and the frequency of current respectively are

Solution:

I = 20sin (100 πt + 0.05π)

QUESTION: 35

The specific heat c of a solid at low temperature shows temperature dependence according to the relation c = DT^{3} where D is a constant and T is the temperature in kelvin. A piece of this solid of mass m kg is taken and its temperature is raised from 20 K to 30 K. The amount of the heat required in the process in energy units is

Solution:

QUESTION: 36

Four identical plates each of area a are separated by a distance d. The connection is shown below. What is the capacitance between P and Q ?

Solution:

QUESTION: 37

The least distance of vision of a longsighted person is 60 cm. By using a spectacle lens, this distance is reduced to 12 cm. The power of the lens is

Solution:

Here, v = – 60 cm, u = – 12 cm

QUESTION: 38

A particle is acted upon by a constant power. Then, which of the following physical quantity remains constant ?

Solution:

QUESTION: 39

A particle of mass M and charge q, initially at rest, is accelerated by a uniform electric field E through a distance D and is then allowed to approach a fixed static charge Q of the same sign. The distance of the closest approach of the charge q will then be

Solution:

QUESTION: 40

In an n-p-n transistor

Solution:

QUESTION: 41

At two different places the angles of dip are respectively 30º and 45º. At these two places the ratio of horizontal component of earth’s magnetic field is

Solution:

Note : Information is not sufficient in the given question. It can be solved only when magnetic field at these two places are equal.

QUESTION: 42

Two vectors are given by A = iˆ + 2 ˆj + 2kˆ and B = 3iˆ + 6 ˆj + 2kˆ. Another vector C has the same magnitude as B but has the same direction as A . Then which of the following vectors represents C?

Solution:

QUESTION: 43

An equilateral triangle is made by uniform wires AB, BC, CA. A current I enters at A and leaves from the mid point of BC. If the lengths of each side of the triangle is L, the magnetic field B at the centroid O of the triangle is

Solution:

QUESTION: 44

A car moving at a velocity of 17 ms^{–1} towards an approaching bus that blows a horn at a frequency of 640 Hz on a straight track. The frequency of this horn appears to be 680 Hz to the car driver. If the velociy of sound in air is 340 ms^{–1}, then velocity of the approaching bus is

Solution:

QUESTION: 45

A particle is moving with a uniform speed v in a circular path of radius r with the centre at O. When the particle moves from a point P to Q on the circle such that ∠POQ = θ, then the magnitude of the change in velocity is

Solution:

QUESTION: 46

Two simple harmonic motions are given by

x_{1} = a sin ωt + a cos ωt and

x_{2} = a sin ωt + a/√3 cos ωt

Solution:

QUESTION: 47

A small mass m attached to one end of a spring with a negligible mass and an unstretched length L, executes vertical oscillations with angular frequency ω_{0}. When the mass is rotated with an angular speed ω by holding the other end of the spring at a fixed point, the mass moves uniformly in a circular path in a horizontal plane. Then the increase in length of the spring during this rotation is

Solution:

QUESTION: 48

A cylindrical block floats vertically in a liquid of density ρ_{1} kept in a container such that the fraction of volume of the cylinder inside the liquid is x_{1}. then some amount of another immiscible liquid of density ρ_{2} (ρ_{2} < ρ_{1}) is added to the liquid in the container so that the cylinder now floats just fully immersed in the liquids with x_{2} fraction of volume of the cylinder inside the liquid of density ρ_{1}. The ratio ρ_{1}/ρ_{2} will be

Solution:

p_{1}x_{1}g = p_{1}x_{2}g + p_{2}(1 – x_{2})g, as Bouyant force in both the cases are same

QUESTION: 49

A sphere of radius R has a volume density of charge ρ = kr, where r is the distance from the centre of the sphere and k is constant. The magnitude of the electric field which exists at the surface of the sphere is given by (ε_{0} = permittivity of the free space)

Solution:

By Gauss’s theorem

QUESTION: 50

A particle of mass M and charge q is at rest at the midpoint between two other fixed similar charges each of magnitude Q placed a distance 2d apart. The system is collinear as shown in the figure. The particle is now displaced by a small amount x (x<< d) along the line joining the two charges and is left to itself. It will now oscillate about the mean position with a time period (ε_{0} = permittivity of free space)

Solution:

Restoring force on displacement of x,

QUESTION: 51

A body is projected from the ground with a velocity v = (3iˆ + 10 ˆj) ms^{ –1} . The maximum height attained and the range of the body respectively are (given g = 10 ms^{–2})

Solution:

V = 3i + 10j

Vx = 3

Vy = 10

QUESTION: 52

The stopping potential for photoelectrons from a metal surface is V_{1 }when monochromatic light of frequency v_{1} is incident on it. The stopping potential becomes V_{2} when monochromatic light of another frequency is incident on the same metal surface. If h be the Planck’s constant and e be the charge of an electron, then the frequency of light in the second case is

Solution:

hν_{1} = φ_{0} + ev_{1} ——(1)

hν_{2} = φ_{0} + ev_{2} ——(2)

h (ν_{2} – ν_{1}) = e (v_{2} – v_{1})

QUESTION: 53

A cell of e.m.f. E is connected to a resistance R_{1} for time t and the amount of heat generated in it is H. If the resistance R_{1} is replaced by another resistance R_{2} and is connected to the cell for the same time t, the amount of heat generated in R_{2} is 4H. Then the internal resistance of the cell is

Solution:

QUESTION: 54

3 moles of a mono-atomic gas (γ = 5/3) is mixed with 1 mole of a diatomic gas (γ = 7/3). The value of γ for the mixture will be

Solution:

Degree of freedom

i.e. f = 7/2

QUESTION: 55

The magnetic field B = 2t^{2} + 4t^{2} (where t = time) is applied perpendicular to the plane of a circular wire of radius r and resistance R. If all the units are in SI the electric charge that flows through the circular wire during t = 0 s to t = 2 s is

Solution:

*Multiple options can be correct

QUESTION: 56

**Q. 56 – Q. 60 carry two marks each, for which one or more than one options may be correct. Marking of correct options will lead to a maximum mark of twoon pro rata basis. There willbe no negative marking for these questions. However, any marking of wrong option will lead to award of zero mark against the respective question – irrespective of the number of corredt options marked.**

Q. If E and B are the magnitudes of electric and magnetic fields respectively in some region of space, then the possibilities for which a charged particle may move in that space with a uniform velocity of magnitude v are

Solution:

*Multiple options can be correct

QUESTION: 57

An electron of charge e and mass m is moving in circular path of radius r with a uniform angular speed ω. Then which of the following statements are correct ?

Solution:

Magnetic moment μ = IA

Angular momentum = 2 m dA/dt

*Multiple options can be correct

QUESTION: 58

A biconvex lens of focal length f and radii of curvature of both the surfaces R is made of a material of refractive index n_{1}. This lens is placed in a liquid of refractive index n_{2}. Now this lens will behave like

Solution:

*Multiple options can be correct

QUESTION: 59

A block of mass m (= 0.1 kg) is hanging over a frictionless light fixed pulley by an inextensible string of negligible mass. The other end of the string is pulled by a constant force F in the verticaly downward direction. The linear momentum of the block increase by 2 kg ms–1 in 1 s after the block starts from rest. Then, (given g = 10 ms^{–2})

Solution:

F – mg = 2

F = 2 + mg = 3 N

.

∴ W by tension = F × 10 = 3 × 10 = 30 J

W against gravity = mg × s = = 1 × 10 = 10 J

*Multiple options can be correct

QUESTION: 60

A bar of length l carrying a small mass m at one of its ends rotates with a uniform angular speed ω in a vertical plane about the mid-point of the bar. During the rotation, at some instant of time when the bar is horizontal, the mass is detached from the bar but the bar continues to rotate with same ω. The mass moves vertically up, comes back and reches the bar at the same point. At that place, the acceleration due to gravity is g.

Solution:

QUESTION: 61

In diborane, the number of electrons that account for bonding in the bridges is

Solution:

Each bridging bond is formed by two electrons. Hence four electrons account for bonding in the bridges.

QUESTION: 62

The optically active molecule is

Solution:

Others are meso compound due to presence of plane of symmetry.

QUESTION: 63

A van der Waals gas may behave ideally when

Solution:

A van der waals gas may behave ideally when pressure is very low as compressibility factor (Z) approaches 1. At high temperature Z > 1.

QUESTION: 64

The half-life for decay of ^{14}C by β-emission is 5730 years. The fraction of ^{14}C decays, in a sample that is 22,920 years old, would be

Solution:

QUESTION: 65

2-Methylpropane on monochlorination under photochemical condition give

Solution:

Ratio of A: B is 5:9

QUESTION: 66

For a chemical reaction at 27°C, the activation energy is 600 R. The ratio of the rate constants at 327°C to that of at 27°C will be

Solution:

QUESTION: 67

Chlorine gas reacts with red hot calcium oxide to give

Solution:

2CaO + 2Cl_{2 }→ CaCl_{2}+O_{2} ↑ Red hot

QUESTION: 68

Correct pair of compounds which gives blue colouration/precipitate and white precipitate, respectively, when their Lassaigne’s test is separately done is

Solution:

Organic compound

QUESTION: 69

The change of entropy (dS) is defined as

Solution:

QUESTION: 70

In O_{2} and H_{2}O_{2}, the O–O bond lengths are 1.21 and 1.48 Å respectively. In ozone, the average O–O bond length is

Solution:

Bond length is nearly average of bond length of O – O in

QUESTION: 71

The IUPAC name of the compound X is

Solution:

2, 2-Dimethyl-4-oxopentanenitrile

QUESTION: 72

At 25°C, the solubility product of a salt of MX_{2 }type is 3.2 × 10^{–8} in water. The solubility (in moles/lit) of MX_{2} in water at the same temperature will be

Solution:

2× 10^{–3}

QUESTION: 73

In SOCl_{2}, the Cl–S–Cl and Cl–S–O bond angles are

Solution:

QUESTION: 74

(+)-2-chloro-2-phenylethane in toluene racemises slowly in the presence of small amount of SbCl_{5}, due to the formation of

Solution:

SbCl_{5} removes Cl– from the substrate to generate a planar carbocation, which is then subsequently attacked by Cl– from both top and bottom to result in a racemic mixture.

QUESTION: 75

Acid catalysed hydrolysis of ethyl acetate follows a pseudo-first order kinetics with respect to ester. If the reaction is carried out with large excess of ester, the order with respect to ester will be

Solution:

With large excess of ester the rate of reaction is independent of ester concentration.

QUESTION: 76

The different colours of litmus in acidic, neutral and basic solutions are, respectively

Solution:

QUESTION: 77

Baeyer’s reagent is

Solution:

QUESTION: 78

The correct order of equivalent conductances at infinite dilution in water at room temperature for H^{+}, K^{+}, CH_{3}COO^{–} and HO^{–} ions is

Solution:

QUESTION: 79

Nitric acid can be obtained from ammonia via the formations of the intermediate compounds

Solution:

QUESTION: 80

In the following species, the one which is likely to be the intermediate during benzoin condensation of benzaldehyde, is

Solution:

QUESTION: 81

The correct order of acid strength of the following substituted phenols in water at 28°C is

Solution:

As order of electron withdrawing nature from benzene ring : –NO_{2}>–Cl>–F

QUESTION: 82

For isothermal expansion of an ideal gas, the correct combination of the thermodynamic parameters will be

Solution:

For isothermal process, ΔT=0

∴ ΔU= nC_{v}ΔT=0

ΔH = nC_{p}ΔT= 0

From first law of thermodynamics

ΔU= Q +W

As ΔU = 0

∴ Q= W ≠ 0

QUESTION: 83

Addition of excess potassium iodide solution to a solution of mercuric chloride gives the halide complex

Solution:

HgCl_{2} + 4KI → K_{2}[HgI_{4}] + 2KCl

QUESTION: 84

Amongst the following, the one which can exist in free state as a stable compound is

Solution:

n = no. of atoms of a particular type

v = valency of the atom

Molecules with fractional degree of unsaturation cannot exist with stability

QUESTION: 85

A conducitivity cell has been calibrated with a 0.01 M 1:1 electrolyte solution (specific conductance, k=1.25 x 10^{–3} S cm^{-1}) in the cell and the measured resistance was 800 ohms at 25°C. The constant will be

Solution:

QUESTION: 86

The orange solid on heating gives a colourless gas and a greensolid which can be reduced to metal by aluminium powder. The orange and the green solids are, respectively

Solution:

QUESTION: 87

The best method for the preparationof 2,2 -dimethylbutane is via the reaction of

Solution:

Corey-House alkane synthesis gives the alkane in best yield

QUESTION: 88

The condition of spontaneity of process is

Solution:

dG_{P,T} = –ve is the criterion for spontaneity

QUESTION: 89

The increasing order of O-N-O bond angle in the species NO_{2}^{+ }, NO_{2}and NO_{2}^{–} is

Solution:

No option is correct correct ans : NO2+ > NO2 > NO2–

QUESTION: 90

The correct structure of the dipeptide gly-ala is

Solution:

QUESTION: 91

Equivalent conductivity at infinite dilution for sodium-potassium oxalate ((COO^{–})_{2}Na^{+}K^{+}) will be [given, molar conductivities of oxalate, K^{+ }and Na^{+ }ions at infinite dilution are 148.2, 50.1, 73.5 S cm2mol^{–1}, respectively]

Solution:

QUESTION: 92

For BCl_{3}, AlCl_{3 }and GaCl_{3} the increasing order of ionic character is

Solution:

Ionic character is inversely proportional to polarising power of cation.

BCl_{3}<GaCl_{3}<AlCl_{3}

QUESTION: 93

At 25°C, pH of a 10^{–8 }M aqueous KOH solution will be

Solution:

QUESTION: 94

The reaction of nitroprusside anion with sulphide ion gives purple colouration due to the formation of

Solution:

⇒ Tetra anionic complex of iron (II) co-ordinating to one NOS– ion

QUESTION: 95

An optically active compound having molecular formula C_{8}H_{16} on ozonolysis gives acetone as one of the products. The structure of the compound is

Solution:

QUESTION: 96

Mixing of two different ideal gases under istohermal reversible condition will lead to

Solution:

QUESTION: 97

The ground state electronic configuration of CO molecule is

Solution:

QUESTION: 98

When aniline is nitrated with nitrating mixturte in ice cold condition, the major product obtained is

Solution:

QUESTION: 99

The measured freezing point depression for a 0.1 m aqueous CH_{3}COOH solution is 0.19°C. The acid dissociation constant Ka at this concentration will be (Given Kf, the molal cryoscopic constant = 1.86 K kg mol^{–1})

Solution:

ΔT_{f} = i × k_{f} × m

QUESTION: 100

The ore chromite is

Solution:

Chromite ore is FeCr_{2}O_{4}

QUESTION: 101

‘Sulphan’ is

Solution:

Sulphan is pure liquid SO_{3}

QUESTION: 102

Pressure-volume (PV) work done by an ideal gaseous system at constant volume is (where E is internal energy of the system)

Solution:

From 1st law of thermodynamic

ΔE = q+w.

Now w = PΔV. for Δv = 0 w = 0

QUESTION: 103

Amongst [NiCl_{4}]^{2–}, [Ni(H_{2}O_{)6}]^{2+},[Ni(PPh_{3})_{2}Cl_{2}], [Ni(CO)_{4}] and [Ni(CN)_{4}]^{2–}, the paramagnetic species are

Solution:

QUESTION: 104

Number of hydrogen ions present in 10 millionth part of 1.33 cm^{3} of pure water at 25°C is

Solution:

QUESTION: 105

Ribose and 2-deoxyribose can be differentiated by

Solution:

In deoxyribose, one –OH group is missing, which will prevent the formation of osazone.

QUESTION: 106

The standard Gibbs free energy change (ΔG^{0}) at 25°C for the dissociation of N_{2}O_{4}(g) to NO_{2}(g) is (given, equilibrium constant = 0.15, R=8.314 JK/mol)

Solution:

ΔG^{0} = – RTlnk

QUESTION: 107

Bromination of PhCOMe in acetic acid medium produces mainly

Solution:

QUESTION: 108

Silicone oil is obtained from the hydrolysis and polymerisation of

Solution:

QUESTION: 109

Solution:

Reaction proceeds via benzyne mechanism with intermediate as

QUESTION: 110

Identify the CORRECT statement

Solution:

QUESTION: 111

In borax the number of B–O–B links and B–OH bonds present are, respectively,

Solution:

QUESTION: 112

Reaction of benzene with Me_{3}COCl in the presence of anhydrous AlCl_{3} gives

Solution:

It is because of rearrangement during which initially formed acyl cation loses CO to form stable tertiary butyl cation

QUESTION: 113

1 × 10^{–3} mole of HCl is added to a buffer solution made up of 0.01 M acetic and 0.01 M sodium acetate. The final pH of the buffer will be (given, pK_{a} of acetic acid is 4.75 at 25°C)

Solution:

QUESTION: 114

The best method for preparation of Me_{3}CCN is

Solution:

It’s a SN_{2 }reaction where Me_{3}C–MgBr + Cl – CN → Me_{3}C – CN + Mg (Cl)Br

QUESTION: 115

On heating, chloric acid decompose to

Solution:

*Multiple options can be correct

QUESTION: 116

**Q. 116 – Q. 120 carry two marks each, for which one or more than one options may be correct. Marking of correct options will lead to a maximum mark of two on pro rata basis. There will be no negative marking for these questions. However, any marking of wrong option will lead to award of zero mark against the respective question-irrespective of the number of correct options marked.**

**Q. **Consider the following reaction for 2NO2(g) + F2(g) → 2NO2F(g). The expression for the rate of reaction interms of the rate of change of partial pressures of reactant and product is/are

Solution:

*Multiple options can be correct

QUESTION: 117

Tautomerism is exhibited by

Solution:

Availability of acidic H-atoms at these positions(shown by arrow marks) enable the compounds to show keto-enol tautomerism

*Multiple options can be correct

QUESTION: 118

The important advantage(s) of Lintz and Donawitz (L.D.) process for the manufacture of steel is (are)

Solution:

*Multiple options can be correct

QUESTION: 119

In basic medium the amount of Ni^{2+} in a solution can be estimated with the dimethylglyoxime reagent. The correct statement(s) about the reaction and the product is(are)

Solution:

*Multiple options can be correct

QUESTION: 120

Correct statement(s) in cases of n-butanol and t-butanol is (are)

Solution:

More branching means less boiling point and high solubility

QUESTION: 121

A point P lies on the circle x^{2 }+ y^{2} = 169. If Q = (5,12) and R = (–12,5), then the angle ∠QPR is

Solution:

Q (5,12) R (–12,5) 0 (0,0)

m_{OQ} . m_{OR }= –1, so ∠QOR = π/2 Hence ∠QPR = π/4

QUESTION: 122

A circle passing through (0,0), (2,6), (6,2) cuts the x-axis at the point P ≠ (0,0). Then the length of OP, where O is origin, is

Solution:

Circle passes through (0,0)

so, x^{2}+y^{2}+2gx+2fy = 0

(2,6) & (6,2) lies on it so,

2^{2}+6^{2}+4g+12f = 0 — (1)

2^{2}+6^{2}+12g+4f = 0 — (2)

⇒ From (1) & (2), g = f = –5/2

Eqn. of circle is x^{2}+y^{2}–5x–5y=0

For y = 0, x(x–5)=0 ⇒ x=0, x=5

OP = 5

QUESTION: 123

The locus of the midpoints of the chords of an ellpse x^{2}+4y^{2} = 4 that are drawn form the positive end of the minor axis, is

Solution:

Positive end of minor axis (0,1) but mid-pt be (h,k) x=2h, y=2k–1 lies on ellipse

Here on ellipse of centre (0,1/2), major axis 2, minor axis 1

QUESTION: 124

A point moves so that the sum of squares of its distances from the points (1,2) and (–2,1) is always 6. Then its locus is

Solution:

Let (h,k) be co-ordinates of the point

(h–1)^{2 }+ (k–2)^{2} + (h+2)^{2} + (k–1)^{2}=

6 ⇒ h^{2}+k^{2}+h – 3k + 2 = 0

a circle with centre (-1/2,3/2) and radius 1/√2

QUESTION: 125

For the variable t, the locus of the points of intersection of lines x–2y = t and x+2y = 1/t is

Solution:

(x–2y) (x+2y) = 1 ⇒ x^{2} - 4y^{2} = 1

QUESTION: 126

Solution:

QUESTION: 127

The number of solutions of the equation x+y+z = 10 in positive integers x,y,z is equal to

Solution:

^{10–1 }C_{3–1} = ^{9}C_{2} = 36

QUESTION: 128

For 0 ≤ P,Q ≤ π/2 , if sin P + cos Q=2, then the value of tan(P+Q/2) is

Solution:

P = π/2 , Q = 0

QUESTION: 129

If α and β are the roots of x^{2 }– x+1 = 0, then the value of α^{2013} + β^{2013} is equal to

Solution:

α = – ω, – ω^{2}

– ω^{2013 }– ω^{2x2013 }= – (ω^{3})^{671} – (ω^{3})^{2x671} = – 2

QUESTION: 130

Solution:

QUESTION: 131

Let

f(x) = 2^{100 }x+1,

g(x) = 3^{100 }x+1.

Then the set of real numbers x such that f(g(x)) = x is

Solution:

f(x) = 2^{100}x+1 ; g(x) = 3^{100 }x+1

QUESTION: 132

The limit of x sin(e^{1/x}) as x → 0

Solution:

–1≤ sine^{1/x}≤1, -x≤ xsin(e^{1/x}) ≤x

QUESTION: 133

Then the matrix P^{3} + 2P^{2} is equal to

Solution:

|P −λI = 0 , characteristics equation of P is P^{3}+2P^{2}–P–2I = 0

P^{3}+2P^{2}=P+2I

QUESTION: 134

If α, β are the roots of the quadratic equation x^{2}+ax+b=0, (b≠0); then the quadratic equation whose roots are α - 1/β, β - 1/α

Solution:

α+β = –a, αβ = b

QUESTION: 135

The value of is equal to

Solution:

= 999

QUESTION: 136

Solution:

(1+a^{2}+b^{2})^{3}

= C_{1} → C_{1} – bC_{3}, C_{2} → aC_{3} + C_{2}

QUESTION: 137

If the distance between the foci of an ellipse is equal to the length of the latus rectum, then its eccentricity is

Solution:

2ae = 2b^{2}/a

e = b^{2}/a^{2} = 1-e^{2}

e^{2 }+ e–1 = 0

= e = 1/2 (√5-1)

QUESTION: 138

For the curve x^{2}+4xy+8y^{2}=64 the tangents are parallel to the x-axis only at the points

Solution:

x^{2}+4xy+8y^{2} = 64

⇒ 2x+4xy′+4y+16yy′=0

⇒ (4x+16y)y′ = – (2x+4y)

2x + 4y = 0

QUESTION: 139

The value of

Solution:

QUESTION: 140

Let f(θ) = (1+sin^{2}θ)(2–sin^{2}θ). Then for all values of θ

Solution:

f(θ) = (1+sin^{2}θ) (2–sin^{2}θ)

f(θ)=(1+sin^{2}θ)(1+cos^{2}θ)

=2+sin^{2}θcos^{2}θ

= 2 + 1/2sin^{2}θ

2≤ f(θ)≤ 9/4

QUESTION: 141

Then

Solution:

so L.H.D at x = 2 is 9, R.H.D at x = 2 is –3

so f(x) is continuous but not differentiable at x = 2

QUESTION: 142

Solution:

QUESTION: 143

If f(x) = e^{x} (x – 2)^{2 }then

Solution:

f′(x) = e^{x} [(x –2)^{2} + 2(x – 2)]

= e^{x} [x^{2 }– 2x] = e^{x}.x(x – 2)

sign scheme of f′(x) will be

so f is increasing in (–∞, 0) and (2, ∞) and decreasing in (0, 2)

QUESTION: 144

Let f : R → R be such that f is injective and f(x)f(y) = f(x +y) for all x, y ∈ R. If f(x), f(y), f(z) are in G.P., then x, y, z are in

Solution:

f (x + y) = f(x).f(y), so f(x) = a^{kx }

a^{kx}, a^{ky}, a^{kz} are in G.P

a^{2ky }= a^{k(x + z)} ⇒ 2y

= x + z, so x, y, z are in A.P

QUESTION: 145

The number of solutions of the equation

Solution:

(x + 1) = 3, x = 2 so only one solution

QUESTION: 146

The area of the region bounded by the parabola y = x^{2 }–4x + 5 and the straight line y = x + 1 is

Solution:

QUESTION: 147

The value of the integral

Solution:

QUESTION: 148

Then

Solution:

QUESTION: 149

Solution:

f(x) = sinx + 2cos^{2}x

= – 2sin^{2}x + sinx + 2

= – 2 (sin^{2}x – ½ sinx) + 2

f(x) will be minimum when (sinx-1/4)^{2} is maximum at x = π/2

QUESTION: 150

Each of a and b can take values 1 or 2 with equal probability. The probability that the equation ax^{2} + bx + 1 = 0 has real roots, is equal to

Solution:

ax^{2} +bx + 1 = 0 has real roots for b^{2} –4a ≥ 0

So a has to be 1 and b has to be 2

so probability is = 1/2 x 1/2 = 1/4

QUESTION: 151

There are two coins, one unbiased with probability 1/2 of getting heads and the other one is biased with probability 3/4 of getting heads. A coin is selected at random and tossed. It shows heads up. Then the probability that the unbiased coin was selected is

Solution:

H → Event of head showing up

B → Event of biased coin chosen

UB → Event of unbiased coin chosen

QUESTION: 152

For the variable t, the locus of the point of intersection of the lines 3tx – 2y + 6t =0 and 3x + 2ty – 6 = 0 is

Solution:

The point of intersection of 3tx – 2y + 6t = 0 and 3x + 2ty – 6 = 0 is

Considering t = tan θ, x = 2cos 2θ, y = 3.sin 2θ

so locus of point of intersection is

QUESTION: 153

Cards are drawn one-by-one without replacement from a well shuffled pack of 52 cards. Then the probability that a face card (Jack, Queen or King) will appear for the first time on the third turn is equal to

Solution:

P (face card on third turn) = P (no face card in first turn) × P (no face card in 2nd turn) × P (face card in 3rd turn)

QUESTION: 154

Lines x + y = 1 and 3y = x + 3 intersect the ellipse x^{2} + 9y^{2 }= 9 at the points P,Q,R. The are of the triangle PQR is

Solution:

QUESTION: 155

The number of onto functions from the set {1, 2, ..........., 11} to set {1, 2, .... 10} is

Solution:

QUESTION: 156

Solution:

QUESTION: 157

Let z_{1} = 2 + 3i and z_{2} = 3 + 4i be two points on the complex plane. Then the set of complex numbers z satisfying |z – z_{1}|^{2 }+ |z – z_{2}|^{2 }= |z_{1} – z_{2}|^{2} represents

Solution:

Clearly the locus of Z is a circle with Z_{1 }& Z_{2} as end point of diameter.

QUESTION: 158

Let p(x) be a quadratic polynomial with constant term 1. Suppose p(x) when divided by x – 1 leaves remainder 2 and when divided by x + 1 leaves remainder 4. Then the sum of the roots of p(x) = 0 is

Solution:

P(x) = ax^{2} + bx + 1

P(1) = a + b + 1 =

2P(–1) = a – b + 1 =

4 so b = –1, a = 2

sum of roots of P(x) is -b/a = 1/2

QUESTION: 159

Eleven apples are distributed among a girl and a boy. Then which one of the following statements is true ?

Solution:

QUESTION: 160

Five numbers are in H.P. The middle term is 1 and the ratio of the second and the fourth terms is 2:1. Then the sum of the first three terms is

Solution:

Let a, b, 1, c, d are H.P

so, sum of the first three terms 3 + 3/2 + 1 = 11/2

QUESTION: 161

Solution:

R.H.L. ≠ L.H.L

QUESTION: 162

The maximum and minimum values of cos^{6}θ + sin^{6}θ are respectively

Solution:

sin^{6}θ + cos^{6}θ = 1 – 3sin^{2}θ.cos^{2}θ

1-3/4sin^{2}θ

QUESTION: 163

If a, b, c are in A.P., then the straight line ax + 2by + c = 0 will always pass through a fixed point whose co-ordinates are

Solution:

a(x + y) + c( y + 1) = 0

x = 1, y = – 1

QUESTION: 164

If one end of a diameter of the circle 3x^{2 }+ 3y^{2} – 9x + 6y + 5 = 0 is (1, 2) then the other end is

Solution:

Center (3/2,-1)

Let the other point be (h, k)

QUESTION: 165

The value of cos^{2}75^{0} + cos^{2}45^{0} + cos^{2}15^{0} – cos^{2}30^{0} – cos^{2}60^{0} is

Solution:

cos15º= sin75º

cos275º + cos245º + cos215º – cos230º – cos260º

= cos275º + sin275º + cos2 45º – cos230º – cos260º

= 1 +1/2-3/4-1/4

= 1/2

QUESTION: 166

Suppose z= x + iy where x and y are real numbers and i = √-1, The points (x,y) for which z-1/z+1 is real,lie on

Solution:

QUESTION: 167

The equation 2x^{2} + 5xy – 12y^{2 }= 0 represents a

Solution:

2x^{2} + 5xy – 12y^{2} = 0

(x + 4y) (2x – 3y) = 0

QUESTION: 168

The line y = x intersects the hyperbola x^{2}/9 - y^{2}/25 = 1,at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length 5/√2 is

Solution:

QUESTION: 169

The equation of the circle passing through the point (1, 1) and the points of intersection of x^{2} + y^{2 }– 6x – 8 = 0 and x^{2} + y^{2} – 6 = 0 is

Solution:

QUESTION: 170

Six positive numbers are in G.P., such that their product is 1000. If the fourth term is 1, then the last term is

Solution:

QUESTION: 171

In the set of all 3×3 real matrices a relation is defined as follows. A matrix A is related to a matrix B if and only if there is a non-singular 3×3 matrix P such that B = P^{–1}AP. This relation is

Solution:

R = {(A, B) | B = P^{–1} AP}

A = I^{–1}AI ⇒ (A, A) ∈ R ⇒ R is reflexive

Let (A, B) ∈ R , B = P^{–1}AP PB = AP

⇒ PBP–1 = A ⇒ A = (P^{–1})^{–1} B(P^{–1})

⇒ (B, A) ∈R, ⇒ R is symmetric

Let (A, B) ∈ R, (B, C) ∈ R

A = P^{–1}BP and B = Q–1CQ

A = P^{–1}Q^{–1}C QP = (QP)^{–1} C(QP)

⇒ (A, C) ∈R

QUESTION: 172

The number of lines which pass through the point (2, –3) and are at the distance 8 from the point (–1, 2) is

Solution:

The maximum distance of the line passing through (2, –3) from (–1, 2) is √34 . So there is no possible line

QUESTION: 173

If α, β are the roots of the quadratic equation ax^{2} + bx + c = 0 and 3b^{2} = 16ac then

Solution:

3b^{2} = 16ac

QUESTION: 174

For any two real numbers a and b, we define a R b if and only if sin^{2} a + cos^{2}b = 1. The relation R is

Solution:

sin^{2}a + cos^{2}b = 1

Reflexive : sin^{2}a + cos^{2}a = 1

⇒ aRa

sin^{2}a + cos^{2}b = 1,

1 – cos^{2}a + 1 – sin^{2}b

= 1

sin^{2}b + cos^{2}a = 1

⇒ bRa

Hence symmetric Let aRb, bRc

sin^{2}a + cos^{2}b = 1 ............. (1)

sin^{2}b + cos^{2}c = 1 ............... (2)

(1) + (2)

sin^{2}a + cos^{2}c = 1

Hence transitive therefore equivalence relation.

QUESTION: 175

Let n be a positive even integer. The ratio of the largest coefficient and the 2nd largest coefficient in the expansion of (1 + x)^{n} is 11:10. The the number of terms in the expansion of (1 + x)^{n} is

Solution:

Let n = 2m

QUESTION: 176

Let exp(x) denote exponential function ex. If f(x) = exp(x^{1/x}), x > 0 then the minimum value of f in the interval [2, 5] is

Solution:

f( x ) = e^{x1/x}

g(x) = log f(x) = x^{1/x}

g(x) increases is (0, e) & decreases in (e, ∞) it will be minimum at either 2 or 5

2^{1/2}_{ }> 5^{1/5}_{ } ⇒ minimum value of f(x) = exp(5^{1/5})

QUESTION: 177

The sum of the series

Solution:

On integrate (1 + x)^{25} twice 1st under the limit 0 to x & then 0 to 1 we get sum =

QUESTION: 178

Five numbers are in A.P. with common difference ≠ 0. If the 1st, 3rd and 4th terms are in G.P., then

Solution:

Let a, a + d, a + 2d, a + 3d, a + 4d are five number in A.P.

QUESTION: 179

The minimum value of the function f(x) = 2|x – 1| + |x – 2| is

Solution:

f(x) will be minimum at x = 1

QUESTION: 180

If P, Q, R are angles of an isosceles triangle and P, ∠= π/2 then the value of (cosP/3 - isinP/3)^{3} + (cosQ + isinQ) (cosR – isinR) + (cosP – isinP) (cosQ – isinQ) (cosR – isinR) is equal to

Solution:

(cosP/3 - isinP/3)^{3} + (cosQ + isinQ) (cosR – isinR) + (cosP – isinP) (cosQ – isinQ) (cosR – isinR) =

QUESTION: 181

A line passing through the point of intersection of x + y = 4 and x – y = 2 makes an angle tan^{–1}(3/4) with the x-axis. It intersects the parabola y2 = 4(x–3) at points (x_{1}, y_{1}) and (x_{2}, y_{2}) respectively. Then |x_{1}–x_{2}| is equal to

Solution:

QUESTION: 182

Let [a] denote the greatest integer which is less than or equal to a. Then the value of the integral

Solution:

QUESTION: 183

Solution:

QUESTION: 184

If sin^{2} θ+ 3 cos θ = 2 , then cos^{3} θ+ sec^{3} θ is

Solution:

Cos^{2}θ− 3Cosθ + 1 = 0

cosθ + 1/cosθ = 3

QUESTION: 185

Then the value of log_{e}y is

Solution:

QUESTION: 186

The value of the infinite series

Solution:

QUESTION: 187

The value of the integral

Solution:

QUESTION: 188

Solution:

QUESTION: 189