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QUESTION: 1

A paramagnetic substance in the form of a cube with sides 1 cm has a magnetic dipole moment of 20 × 10^{–6} J/T when a magnetic intensity of 60 × 10^{3} A/m is applied. Its magnetic susceptibility is :-

Solution:

= 0.33 × 10^{–3} = 3.3 × 10^{–4}

QUESTION: 2

A particle of mass m is moving in a straight line with momentum p. Starting at time t = 0, a force F = kt acts in the same direction on the moving particle during time interval T so that its momentum changes from p to 3p. Here k is a constant. The value of T is :-

Solution:

QUESTION: 3

Seven capacitors, each of capacitance 2μF, are to be connected in a configuration to obtain an effective capacitance of (6/13) μF. Which of the combinations, shown in figures below, will achieve the desired value ?

Solution:

Therefore three capacitors most be in parallel to get 6 in

QUESTION: 4

An electric field of 1000 V/m is applied to an electric dipole at angle of 45°. The value of electric dipole moment is 10^{–29} C.m. What is the potential energy of the electric dipole?

Solution:

= –PE cos θ

= –(10^{–29}) (10^{3}) cos 45º

= – 0.707 × 10^{–26} J

= –7 × 10^{–27} J.

QUESTION: 5

A simple pendulum of length 1 m is oscillating with an angular frequency 10 rad/s. The support of the pendulum starts oscillating up and down with a small angular frequency of 1 rad/s and an amplitude of 10^{–2} m. The relative change in the angular frequency of the pendulum is best given by :-

Solution:

Angular frequency of pendulum

[ω_{s} = angular frequency of support]

Δω = 10^{-3} rad/sec.

QUESTION: 6

Two rods A and B of identical dimensions are at temperature 30°C. If A is heated upto 180°C and B upto T°C, then the new lengths are the same. If the ratio of the coefficients of linear expansion of A and B is 4 : 3, then the value of T is :-

Solution:

T = 230º C

QUESTION: 7

In a double-slit experiment, green light (5303 Å) falls on a double slit having a separation of 19.44 μm and a width of 4.05 μm. The number of bright fringes between the first and the second diffraction minima is :-

Solution:

For diffraction

location of 1^{st} minime

Now for interference Path difference at P.

path difference at Q

So orders of maxima in between P & Q is

5, 6, 7, 8, 9

So 5 bright fringes all present between P & Q.

QUESTION: 8

An amplitude modulated signal is plotted below :-

Which one of the following best describes the above signal?

Solution:

Analysis of graph says

(1) Amplitude varies as 8 – 10 V or 9 ± 1

(2) Two time period as 100 μs (signal wave) & 8 μs (carrier wave)

= 9 ± 1sin (2π × 10^{4}t) sin 2.5π × 10^{5} t

QUESTION: 9

In the circuit, the potential difference between A and B is :-

Solution:

Potential difference across AB will be equal to battery equivalent across CD

QUESTION: 10

A 27 mW laser beam has a cross-sectional area of 10 mm^{2}. The magnitude of the maximum electric field in this electromagnetic wave is given by [Given permittivity of space ∈_{0} = 9 × 10^{-12} SI units, Speed of light c = 3 × 10^{8} m/s]:-

Solution:

Intensity of EM wave is given by

QUESTION: 11

A pendulum is executing simple harmonic motion and its maximum kinetic energy is K_{1}. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is K_{2} Then :-

Solution:

Maximum kinetic energy at lowest point B is given by

K = mgl (1 – cos θ)

where θ= angular amp.

K_{1} = mgℓ (1 - cos θ)

K_{2} = mg(2ℓ) (1 - cos θ)

K_{2} = 2K_{1}.

QUESTION: 12

In a hydrogen like atom, when an electron jumps from the M - shell to the L - shell, the wavelength of emitted radiation is λ. If an electron jumps from N-shell to the L-shell, the wavelength of emitted radiation will be :-

Solution:

For M → L steel

for N → L

QUESTION: 13

If speed (V), acceleration (A) and force (F) are considered as fundamental units, the dimension of Young's modulus will be :-

Solution:

Now from dimension

QUESTION: 14

A particle moves from the point m, at t = 0, with an initial velocity ms^{–1}.It is acted upon by a constant force which produces a constant acceleration What is the distance of the particle from the origin at time 2s?

Solution:

QUESTION: 15

A monochromatic light is incident at a certain angle on an equilateral triangular prism and suffers minimum deviation. If the refractive index of the material of the prism is √3, then the angle of incidence is :-

Solution:

i = e

by Snell's law

i = 60

QUESTION: 16

A galvanometer having a resistance of 20 Ω and 30 divisions on both sides has figure of merit 0.005 ampere/division. The resistance that should be connected in series such that it can be used as a voltmeter upto 15 volt, is :-

Solution:

R_{g} = 20Ω

N_{L} = N_{R} = N = 30

Ig_{max} = 0.005 × 30

= 15 × 10^{–2} = 0.15

15 = 0.15 [20 + R]

100 = 20 + R

R = 80

QUESTION: 17

The circuit shown below contains two ideal diodes, each with a forward resistance of 50Ω. If the battery voltage is 6 V, the current through the 100 Ω resistance (in Amperes) is :-

Solution:

I = 6/300 = 0.002 (D_{2} is in reverse bias)

QUESTION: 18

When 100 g of a liquid A at 100°C is added to 50 g of a liquid B at temperature 75°C, the temperature of the mixture becomes 90°C. The temperature of the mixture, if 100 g of liquid A at 100°C is added to 50 g of liquid B at 50°C, will be :-

Solution:

100 × S_{A} × [100 – 90] = 50 × S_{B} × (90 – 75)

2S_{A} = 1.5 S_{B}

Now, 100 × S_{A} × [100 – T] = 50 × S_{B} (T – 50)

300 – 3T = 2T – 100

400 = 5T

T = 80

QUESTION: 19

The mass and the diameter of a planet are three times the respective values for the Earth. The period of oscillation of a simple pendulum on the Earth is 2s. The period of oscillation of the same pendulum on the planet would be :-

Solution:

QUESTION: 20

The region between y = 0 and y = d contains a magnetic field A particle of mass m and charge q enters the region with a velocity the acceleration of the charged particle at the point of its emergence at the other side is :-

Solution:

In equation, entry point of particle is no given Assuming particle center from (0, d)

QUESTION: 21

A thermometer graduated according to a linear scale reads a value x_{0} when in contact with boiling water, and x_{0}/3 when in contact with ice. What is the temperature of an object in 0 °C, if this thermometer in the contact with the object reads x_{0}/2 ?

Solution:

QUESTION: 22

A string is wound around a hollow cylinder of mass 5 kg and radius 0.5 m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string) :-

Solution:

40 + f = m(Rα) .....(i)

40 × R – f × R = mR^{2}α

40 – f = mRα ...... (ii)

From (i) and (ii)

QUESTION: 23

In a process, temperature and volume of one mole of an ideal monoatomic gas are varied according to the relation VT = K, where K is a constant. In this process the temperature of the gas is incresed by ΔT. The amount of heat absorbed by gas is (R is gas constant) :

Solution:

VT = K

QUESTION: 24

In a photoelectric experiment, the wavelength of the light incident on a metal is changed from 300 nm to 400 nm. The decrease in the stopping potential is close to :

Solution:

......(i)

......(ii)

(i) – (ii)

= 1V

QUESTION: 25

A metal ball of mass 0.1 kg is heated upto 500°C and dropped into a vessel of heat capacity 800 JK^{-1} and containing 0.5 kg water. The initial temperature of water and vessel is 30°C. What is the approximate percentage increment in the temperature of the water ?

[Specific Heat Capacities of water and metal are, respectively, 4200 Jkg^{–1}K^{–1} and 400 JKg^{–1}K^{–1}]

Solution:

0.1 × 400 × (500 – T) = 0.5 × 4200 × (T – 30) + 800 (T – 30)

⇒ 40(500 - T) = (T - 30) (2100 + 800)

⇒ 20000 - 40T = 2900 T - 30 × 2900

⇒ 20000 + 30 × 2900 = T(2940)

T = 30.4°C

QUESTION: 26

The magnitude of torque on a particle of mass 1kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5m, the angle between the force and the position vector is (in radians) :-

Solution:

2.5 = 1 × 5 sin θ

sin θ = 0.5 = 1/2

θ = π/6

QUESTION: 27

In the experimental set up of metre bridge shown in the figure, the null point is obtained at a distance of 40 cm from A. If a 10Ω resistor is connected in series with R_{1}, the null point shifts by 10 cm. The resistance that should be connected in parallel with (R_{1} + 10) Ω such that the null point shifts back to its initial position is

Solution:

.......(1)

QUESTION: 28

A circular disc D_{1} of mass M and radius R has two identical discs D_{2 }and D_{3} of the same mass M and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis OO', passing through the centre of D_{1}, as shown in the figure, will be:-

Solution:

= 3 MR^{2}

QUESTION: 29

A copper wire is wound on a wooden frame, whose shape is that of an equilateral triangle. If the linear dimension of each side of the frame is increased by a factor of 3, keeping the number of turns of the coil per unit length of the frame the same, then the self inductance of the coil :

Solution:

Total length L will remain constant

L = (3a) N (N = total turns) and length of winding = (d) N (d = diameter of wire)

QUESTION: 30

A particle of mass m and charge q is in an electric and magnetic field given by

The charged particle is shifted from the origin to the point P(x = 1 ; y = 1) along a straight path. The magnitude of the total work done is :-

Solution:

QUESTION: 31

The correct option with respect to the Pauling electronegativity values of the elements is :-

Solution:

QUESTION: 32

Along the period electronegativity increases The homopolymer formed from 4-hydroxy-butanoic acid is :-

Solution:

QUESTION: 33

The correct match between Item I and Item II is :-

Solution:

(A) Ester test (Q) Aspartic acid (Acidic amino acid)

(B) Carbylamine (S) Lysine [NH2 group present]

(C) Phthalein dye (P) Tyrosine {Phenolic group present)

QUESTION: 34

Taj Mahal is being slowly disfigured and discoloured. This is primarily due to

Solution:

Taj mahal is slowely disfigured and discoloured due to acid rain.

QUESTION: 35

The major product obtained in the following conversion is :-

Solution:

QUESTION: 36

The number of bridging CO ligand (s) and Co-Co bond (s) in CO_{2}(CO)_{8}, respectively are :-

Solution:

Bridging CO are 2 and Co – Co bond is 1.

QUESTION: 37

In the following compound,

the favourable site/s for protonation is/are :-

Solution:

Localised lone pair e^{–}.

QUESTION: 38

The higher concentration of which gas in air can cause stiffness of flower buds?

Solution:

Due to acid rain in plants high concentration of SO_{2} makes the flower buds stiff and makes them fall.

QUESTION: 39

The correct match between item I and item II is :-

Solution:

QUESTION: 40

The radius of the largest sphere which fits properly at the centre of the edge of body centred cubic unit cell is: (Edge length is represented bv 'a')

Solution:

r = 0.067 a

QUESTION: 41

Among the colloids cheese (C), milk (M) and smoke (S), the correct combination of the dispersed phase and dispersion medium, respectively is :-

Solution:

QUESTION: 42

The reaction that does NOT define calcination is:-

Solution:

Calcination in carried out for carbonates and oxide ores in absence of oxygen. Roasting is carried out mainly for sulphide ores in presence of excess of oxygen.

QUESTION: 43

MgO(s) + C(s) → Mg(S) + CO(g), for which ΔrHº = + 491.1 kJ mol^{–1} and Δ_{r}Sº = 198.0 JK^{–1} mol^{–1 }, is not feasible at 298 K. Temperature above which reaction will be feasible is :-

Solution:

= 2480.3 K

QUESTION: 44

Given the equilibrium constant :

K_{C} of the reaction :

Cu (s) + 2Ag^{+}(aq) → Cu^{2+}(aq) + 2Ag(s ) is 10 × 10^{15}, calculate the E^{0}_{cell }of this reaction at 298 K

Solution:

At equilibrium

= 0.059×8

= 0.472 V

QUESTION: 45

The hydride that is NOT electron deficient is:-

Solution:

(1) B_{2}H_{6} : Electron deficient

(2) A_{1}H_{3} : Electron deficient

(3) SiH_{4} : Electron precise

(4) GaH_{3} : Electron deficient

QUESTION: 46

The standard reaction Gibbs energy for a chemical reaction at an absolute temperature T is given by

ΔrGº = A – BT

Where A and B are non-zero constants. Which of the following is TRUE about this reaction ?

Solution:

QUESTION: 47

K_{2}HgI_{4} is 40% ionised in aqueous solution. The value of its van't Hoff factor (i) is :-

Solution:

For K_{2}[HgI_{4}]

i = 1+ 0.4 (3–1)

= 1.8

QUESTION: 48

The de Broglie wavelength (λ) associated with a photoelectron varies with the frequency (v) of the incident radiation as, [v_{0} is thre shold frequency] :

Solution:

For electron

(de broglie wavelength)

By photoelectric effect

hν = hν_{0} + KE

KE = hν –hν_{0}

QUESTION: 49

The reaction 2X → B is a zeroth order reaction. If the initial concentration of X is 0.2 M, the half-life is 6h. When the initial concentration of X is 0.5 M, the time required to reach its final concentration of 0.2 M will be:

Solution:

For zero order

[A_{0}]–[A_{t}] = kt

0.2 – 0.1 = k×6

t = 18 hrs.

QUESTION: 50

A compound 'X' on treatment with Br_{2}/NaOH, provided C_{3}H_{9}N, which gives positive carbylamine test. Compound 'X' is :-

Solution:

Thus [X] must be amide with one carbon more than in amine.

Thus [X] is CH_{3}CH_{2}CH_{2}CONH_{2}

QUESTION: 51

Which of the following compounds will produce a precipitate with AgN0_{3}?

Solution:

as it can produce aromatic cation so will produce precipitate with AgNO_{3}.

QUESTION: 52

The relative stability of +1 oxidation state of group 13 elements follows the order :-

Solution:

Due to inert pair effect as we move down the group in 13^{th} group lower oxidation state becomes more stable.

Al < Ga < In < Tℓ

QUESTION: 53

Which of the following compounds reacts with ethylmagnesium bromide and also decolourizes bromine water solution

Solution:

QUESTION: 54

Match the following items in column I with the corresponding items in column II.

Solution:

Na_{2}CO_{3}.10H_{2}O → Solvay process

Mg(HCO_{3})_{2} → Temporary hardness

NaOH → Castner-kellner cell

Ca_{3}Al_{2}O_{6} → Portland cement

QUESTION: 55

25 ml of the given HCl solution requires 30 mL of 0.1 M sodium carbonate solution. What is the volume of this HCl solution required to titrate 30 mL of 0.2 M aqueous NaOH solution?

Solution:

HCl with Na_{2}CO_{3}

Eq. of HCl = Eq. of Na_{2}CO_{3}

Eq of HCl = Eq. of NaOH

V = 25 ml

QUESTION: 56

In the above sequence of reactions, respectively, are :-

Solution:

A → MnO_{2}

D → KIO_{3}

QUESTION: 57

The coordination number of Th in K_{4}[Th(C_{2}O_{4})_{4}(OH_{2})_{2}] is :-

Solution:

QUESTION: 58

The major product obtained in the following reaction is :-

Solution:

LiAlH_{4} will not affect C=C in this compound.

QUESTION: 59

The major product of the following reaction is

Solution:

QUESTION: 60

For the equilibrium, 2H_{2}O ⇔H_{3}O^{+} + OH^{–}, the value of ΔGº at 298 K is approximately :-

Solution:

QUESTION: 61

If the point (2, α, β) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane 2x – 5y = 15, then 2α – 3β is equal to :-

Solution:

Normal vector of plane

equation of plane is 5(x–7)+ 2y–3(z– 6) = 0 5x + 2y – 3z = 17

QUESTION: 62

Let α and β be the roots of the quadratic equation x^{2 }sin θ – x (sin θ cos θ + 1) + cos θ = 0 (0 < θ < 45º), and α < β. Then is equal to :-

Solution:

QUESTION: 63

Let K be the set of all real values of x where the function f(x) = sin |x| – |x| + 2(x – π) cos |x| is not differentiable. Then the set K is equal to :-

Solution:

ƒ(x) = sin|x|–|x| + 2(x – π) cosx

∵ sin|x| – |x| is differentiable function at x=0

∴ k = f

QUESTION: 64

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it ?

Solution:

QUESTION: 65

If the area of the triangle whose one vertex is at the vertex of the parabola, y^{2} + 4(x – a^{2})= 0 and the other two vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is :-

Solution:

Vertex is (a^{2},0)

y^{2 }= –(x – a^{2}) and x = 0 ⇒ (0,±2a)

Area of triangle is = 1/2 4a. ( a^{2}) 250

⇒ a^{3} = 125 or a= 5

QUESTION: 66

The integral equals :-

Solution:

QUESTION: 67

Let (x + 10)^{50} + (x – 10)^{50 }= a_{0} + a_{1}x + a_{2}x^{2} + ..... + a_{50 }x^{50}, for all x∈R, then a_{2}/a_{0 }is equal to:-

Solution:

QUESTION: 68

Let a function f : (0, ∞) → (0, ∞) be defined by Then f is :-

Solution:

⇒ ƒ(x) is not injective

but range of function is [0,∞)

Remark : If co-domain is [0,∞), then ƒ(x) will be surjective

QUESTION: 69

Let S = {1, 2, ...... , 20}. A subset B of S is said to be "nice", if the sum of the elements of B is 203. Then the probability that a randomly chosen subset of S is "nice" is :-

Solution:

QUESTION: 70

Two lines intersect at the point R. The reflection of R in the xy-plane has coordinates :-

Solution:

Point on L_{1} (λ+ 3, 3λ – 1, –λ+ 6)

Point on L_{2} (7μ – 5, –6μ + 2, 4μ + 3)

⇒ λ + 3 = 7μ – 5 ...(i)

3λ – 1 = –6μ + 2 ...(ii) ⇒ λ = –1, μ=1

point R(2,–4,7)

Reflection is (2,–4,–7)

QUESTION: 71

The number of functions f from {1, 2, 3, ..., 20} onto {1, 2, 3, ....., 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :-

Solution:

ƒ(k) = 3m (3,6,9,12,15,18)

for k = 4,8,12,16,20 6.5.4.3.2 ways

For rest numbers 15! ways

Total ways = 6!(15!)

QUESTION: 72

Contrapositive of the statement "If two numbers are not equal, then their squares are not equal." is :-

Solution:

Contrapositive of p → q is ~q → ~p

QUESTION: 73

The solution of the differential equation, dy/dx = (x-y)^{2} , when y(1) = 1, is :-

Solution:

QUESTION: 74

Let A and B be two invertible matrices of order 3 × 3. If det(ABA^{T}) = 8 and det(AB^{–1}) = 8, then det (BA^{–1} B^{T}) is equal to :-

Solution:

QUESTION: 75

If where C is a constant of integration, then f(x) is equal to :-

Solution:

QUESTION: 76

A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, the is equal to :-

Solution:

p (probability of getting white ball) = 30/40

and standard diviation

QUESTION: 77

If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3, 4) and (2, 5), then the equation of the diagonal AD is:-

Solution:

co-ordinates of point D are (4,7)

⇒ line AD is 5x – 3y + 1 = 0

QUESTION: 78

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is :-

Solution:

2b = 5 and 2ae = 13

QUESTION: 79

The area (in sq. units) in the first quadrant bounded by the parabola, y = x^{2} + 1, the tangent to it at the point (2, 5) and the coordinate axes is :-

Solution:

QUESTION: 80

Let and respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is 3√2 , then the sum of all possible values of β is :-

Solution:

Angle bisector is x – y = 0

QUESTION: 81

If = (a + b + c) (x + a + b + c)^{2}, x ≠ 0 and a + b + c ≠ 0, then x is equal to :-

Solution:

= (a + b + c)(a + b + c)^{2}

⇒ x = –2(a + b + c)

QUESTION: 82

Let S_{n} = 1 + q + q^{2} + ....... + q^{n} and where q is a real number and q ≠ 1. If ^{101}C_{1} + ^{101}C_{2}.S_{1} + ...... + ^{101}C_{101}.S_{100} = αT_{100}, then α is equal to :-

Solution:

QUESTION: 83

A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is :-

Solution:

Let equation of circle is

x^{2} + y^{2} + 2ƒx + 2ƒy + e = 0, it passes through (0, 2b)

⇒ 0 + 4b^{2} + 2g × 0 + 4ƒ + c = 0

⇒ 4b^{2} + 4ƒ + c = 0 ...(i)

g^{2} – c = 4a^{2} ⇒ c = ( g^{2}- 4a^{2} )

Putting in equation (1)

⇒ 4b^{2} + 4ƒ + g^{2} – 4a^{2} = 0

⇒ x^{2} + 4y + 4(b^{2} – a^{2}) = 0, it represent a parabola.

QUESTION: 84

If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is :-

Solution:

a + 18d = 0 ...(1)

QUESTION: 85

Let x∈R, where a, b and d are non-zero real constants. Then :-

Solution:

ƒ(x) is an increasing function.

QUESTION: 86

Let z be a complex number such that |z| + z = 3 + i (where i = √-1). Then |z| is equal to :-

Solution:

|z| + z = 3 + i

z = 3 – |z| + i

Let 3 – |z| = a ⇒ |z| = (3 – a)

QUESTION: 87

All x satisfying the inequality (cot^{–1} x)^{2 }– 7 (cot^{–1} x) + 10 > 0, lie in the interval:-

Solution:

cot^{–1}x > 5, cot^{–1}x < 2

⇒ x < cot5, x > cot2

QUESTION: 88

Given for a ΔABC with usual notation. If then the ordered triad (α, β, γ) has a value :-

Solution:

b + c = 11λ, c + a = 12λ, a + b = 13λ

⇒ a = 7λ, b = 6λ, c = 5λ

(using cosine formula)

α : β : γ ⇒ 7 : 19 : 25

QUESTION: 89

Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression

Solution:

using AM ≥ GM

QUESTION: 90

Solution:

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