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QUESTION: 1

A current loop, having two circular arcs joined by two radial lines is shown in the figure. It carries a current of 10 A. The magnetic field at point O will be close to :

Solution:

QUESTION: 2

A gas can be taken from A to B via two different processes ACB and ADB.

When path ACB is used 60 J of heat flows into the system and 30 J of work is done by the system. If path ADB is used work done by the system is 10 J. The heat Flow into the system in path ADB is :

Solution:

QUESTION: 3

A plane electromagnetic wave of frequency 50 MHz travels in free space along the positive xdirection. At a particular point in space and time, The corresponding magnetic field at that point will be:

Solution:

QUESTION: 4

Two coherent sources produce waves of different intensities which interfere. After interference, the ratio of the maximum intensity to the minimum intensity is 16. The intensity of the waves are in the ratio:

Solution:

Using componendo & dividendo.

QUESTION: 5

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If AB = BC, and the angle made by AB with downward vertical is θ, then :

Solution:

Let mass of one rod is m.

Balancing torque about hinge point.

mg (C_{1}P) = mg (C_{2}N)

QUESTION: 6

A mixture of 2 moles of helium gas (atomic mass = 4 u), and 1 mole of argon gas (atomic mass = 40 u) is kept at 300 K in a container. The ratio of their rms speeds is close to

Solution:

QUESTION: 7

When the switch S, in the circuit shown, is closed, then the value of current i will be :

Solution:

Let voltage at C = xv

QUESTION: 8

A resistance is shown in the figure. Its value and tolerance are given respectively by:

Solution:

Color code:

Red violet orange silver

R = 27 x 10^{3} Ω, 10%

= 27 kΩ, 10%

QUESTION: 9

A bar magnet is demagnetized by inserting it inside a solenoid of length 0.2 m, 100 turns, and carrying a current of 5.2 A. The coercivity of the bar magnet is :

Solution:

QUESTION: 10

A rod, of length L at room temperature and uniform area of cross section A, is made of a metal having coefficient of linear expansion α/ °C. It is observed that an external compressive force F, is applied on each of its ends, prevents any change in the length of the rod, when its temperature rises by ΔT K. Young's modulus, Y, for this metal is :

Solution:

QUESTION: 11

A block of mass m, lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant k. The other end of the spring is fixed, as shown in the figure. The block is initally at rest in its equilibrium position. If now the block is pulled with a constant force F, the maximum speed of the block is :

Solution:

Maximum speed is at mean position (equilibrium). F = kx

QUESTION: 12

Three charges +Q, q, + Q are placed respectively, at distance, 0, d/2 and d from the origin, on the x-axis. If the net force experienced by + Q, placed at x= 0, is zero, then value of q is :

Solution:

QUESTION: 13

A conducting circular loop made of a thill wire, has area 3.5 x 10^{-3} m^{2} and resistance 10Ω . It is placed perpendicular to a time dependent magnetic field B(t) = (0.4T)sin(50πt). The field is uniform in space. Then the net charge flowing through the loop during t = 0 s and t = 10 ms is close to:

Solution:

QUESTION: 14

Two masses m and m/2 are connected at the two ends of a massless rigid rod of length l.The rod is suspended by a thin wire of torsional constant k at the centre of mass of the rod-mass system(see figure). Because of torsional constant k, the restoring torque is π=kθ for angular displacement 0. If the rod is rotated by θ_{0} and released, the tension in it when it passes through its mean position will be.

Solution:

QUESTION: 15

A copper wire is stretched to make it 0.5% longer. The percentage change in its electrical resistance if its volume remains unchanged is:

Solution:

QUESTION: 16

A parallel plate capacitor is made of two square plates of side 'a', separated by a distance d (d<<a). The lower triangular portion is filled with a dielectric of dielectric constant K, as shown in the figure. Capacitance of this capacitor is :

Solution:

QUESTION: 17

Mobility of electrons in a semiconductor is defined as the ratio of their drift velocity to the applied electric field. If, for an n-type semiconductor, the density of electrons is 10^{19}m^{-3} and their mobility is 1.6 m^{2}/(V.s) then the resistivity of the semiconductor (since it is an n-type semiconductor contribution of holes is ignored) is close to:

Solution:

= 0.4 Ωm

QUESTION: 18

If the angular momentum of a planet of mass m, moving around the Sun in a circular orbit is L, about the center of the Sun, its areal velocity is :

Solution:

QUESTION: 19

A block of mass 10 kg is kept on a rough inclined plane as shown in the figure. A force of 3 N is applied on the block. The coefficient of static friction between the plane and the block is 0.6. What should be the minimum value of force P, such that the block doesnot move downward ?

Solution:

QUESTION: 20

Temperature difference of 120°C is maintained between two ends of a uniform rod AB of length 2L. Another bent rod PQ, of same cross- section as AB and length 3L/2 is connected across AB (See figure). In steady state, temperature difference between P and Q will be close to :

Solution:

QUESTION: 21

A heavy ball of mass M is suspended from the ceiling of a car by a light string of mass m (m<<M). When the car is at rest, the speed of transverse waves in the string is 60 ms^{-1}. When the car has acceleration a, the wave-speed increases to 60.5 ms^{-1}. The value of a, in terms of gravitational acceleration g, is closest to :

Solution:

QUESTION: 22

A sample of radioactive material A, that has an activity of 10 mCi(1 Ci = 3.7 x 10^{10} decays/s), has twice the number of nuclei as another sample of a different radioactive material B which has an activity of 20 mCi. The correct choices for half-lives of A and B would then be respectively :

Solution:

QUESTION: 23

Consider a tank made of glass(reiractive index 1.5) with a thick bottom. It is filled with a liquid of refractive index μ. A student finds that, irrespective of what the incident angle i (see figure) is for a beam of light entering the liquid, the light reflected from the liquid glass interface is never completely polarized. For this to happen, the minimum value of μ is :

Solution:

C < ib here i_{b} is "brewester angle" and c is critical angle

sin_{c} < sini_{b}

slab µ = 1.5

QUESTION: 24

An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The radius of the loop is a and distance of its centre from the wire is d (d»a). If the loop applies a force F on the wire then :

Solution:

Eqvilent dipole of given loop

QUESTION: 25

Surface of certain metal is first illuminated with light of wavelength λ_{1} =350 nm and then, by light of wavelength λ_{2}=540 nm. It is found that the maximum speed of the photo electrons in the two cases differ by a factor of 2. The work function of the metal (in eV) is close to :

(Energy of photon = )

Solution:

QUESTION: 26

A particle is moving with a velocity where K is a constant. The general equation for its path is:

Solution:

QUESTION: 27

A convex lens is put 10 cm from a light source and it makes a sharp image on a screen, kept 10 cm from the lens. Now a glass block (refractive index 1.5) of 1.5 cm thickness is placed in contact with the light source. To get the sharp image again, the screen is shifted by a distance d. Then d is :

Solution:

Shift due to slab = in the direction of incident ray

QUESTION: 28

For a uniformly charged ring of radius R, the electric field on its axis has the largest magnitude at a distance h from its centre. Then value of h is :

Solution:

Electric field on axis of ring

for maximum electric field

QUESTION: 29

Three blocks A, B and C are lying on a smooth horizontal surface, as shown in the figure. A and B have equal masses, m while C has mass M. Block A is given an brutal speed v towards B due to which it collides with B perfectly inelastically. The combined mass collides with C, also perfectly inelastically 5/6 th of the initial kinetic energy is lost in whole process. What is value of M/m ?

Solution:

QUESTION: 30

Drift speed of electrons, when 1.5 A of current flows in a copper wire of cross section 5 mm^{2}, is v. If the electron density in copper is 9 ×10^{28} /m^{3} the value of v in mm/s is close to (Take charge of electron to be =1.6 × 10^{–19}C)

Solution:

QUESTION: 31

Which one of the following statements regarding Henry's law not correct?

Solution:

Liquid solution

P_{gas} = K_{H} × X_{gas}

More is K_{H} less is solubility, lesser solubility is at higher temperature. So more is temperature more is K_{H}.

QUESTION: 32

The correct decreasing order for acid strength is :-

Solution:

EWG increasea acidic strength

NO_{2}CH_{2}COOH > NCCH_{2}COOH >FCH_{2}COOH > CICH_{2}COOH

QUESTION: 33

Two complexes [Cr(H_{2}O_{6})Cl_{3}] (A) and [Cr(NH_{3})_{6}]Cl_{3} (B) are violet and yellow coloured, respectively. The incorrect statement regarding them is :

Solution:

Δ_{0} order will be compared by spectro chemical series not by energies of violet & yellow light so Δ_{0} order is

[Cr(H_{2}O)_{6}]Cl_{3} < [Cr(NH_{3})_{6}]Cl_{3}

QUESTION: 34

Adsorption of a gas follows Freundlich h adsorption isotherm. In the given plot, x is the mass of the gas adsorbed on mass m of the adsorbent at pressure p. x/m is proportional to

Solution:

QUESTION: 35

Correct statements among a to d regarding silicones are :

(a) They are polymers with hydrophobic character

(b) They are biocompatible.

(c) In general, they have high thermal stability and low dielectric strength.

(d) Usually, they are resistant to oxidation and used as greases.

Solution:

These are properties and uses of silicones.

QUESTION: 36

For emission line of atomic hydrogen from n_{i} = 8 to n_{f} =n, the plot of wave number against will be (The Rydbergconstant, R_{H} is in wave number unit).

Solution:

m = R_{H}

Linear with slope R_{H}

QUESTION: 37

The major product of the following reaction is:

Solution:

QUESTION: 38

The alkaline earth metal nitrate that does not crystallise with water molecules, is :

Solution:

Smaller in size of center atoms more water molecules will crystallize hence Ba(NO_{3})_{2} is answer due to its largest size of '+ve' ion.

QUESTION: 39

Major product of the following reaction is :

Solution:

NH_{2}(a) will wact as nucleophile as (b) is having delocalised lonepair.

QUESTION: 40

The highest value of the calculated spin only magnetic moment (in BM) among all the transition metal complexs is :

Solution:

n = Number of unpaired electrons

n = Maximum number of unpaired electron = 5

Ex : Mn^{2+} complex.

QUESTION: 41

20 mL of 0.1 MH_{2}SO_{4} solution is added to 30 mL of 0.2 M NH_{4}OH solution. The pH of the resulatant mixture is : [pk_{b} of NH_{4}OH = 4.7].

Solution:

QUESTION: 42

0.5 moles of gas A and x moles of gas B exert a pressure of 200 Pa in a a container of volume 10 m^{3} at 1000 K. given R is the gas constant in JK^{–1} mol^{–1}m, x is :

Solution:

QUESTION: 43

Consider the reversible isothermal expansion of an ideal gas in a closed system at two different temperatures T_{1} and T_{2} (T_{1} < T_{2}). The correct graphical depiction of the dependence of work done (w) on the final volume (V) is:

Solution:

Y = m x – C

So, slope of curve 2 is more than curve 1 and intercept of curve 2 is more negative then curve 1.

QUESTION: 44

The major product of following reaction is :

Solution:

QUESTION: 45

In general, the properties that decrease and increase down a group in the periodic table, respectively, are :

Solution:

Electronegativity decreases as we go down the group and atomic radius increases as we go down the group.

QUESTION: 46

A solution of sodium sulfate contains 92 g of Na^{+} ions per kilogram of water. The molality of Na^{+ }ions in that solution in mol kg^{–1} is:

Solution:

So molality = 4

QUESTION: 47

A water sample has ppm level concentration of the following metals: Fe= 0.2; Mn = 5.0; Cu = 3.0; Zn = 5.0. The metal that makes the water sample unsuitable drinking is :

Solution:

(i) Zn = 0.2

(ii) Fe = 0.2

(iii) Mn = 5.0

(iv) Cu = 3.0

QUESTION: 48

The increasing order of pKa of the following amino acids in aqueous solution is : Gly Asp Lys Arg

Solution:

Order of acidic strength :

So, pK_{a}

Asp < Gly < Arg < Lys

QUESTION: 49

According to molecular orbital theory, which of the following is true with respect to Li_{2}^{+} and Li_{2}^{- }?

Solution:

Both Li_{2}^{+} and Li_{2}^{-} has 0.5 bond order and hence both are stable.

QUESTION: 50

The following results were obtained during kinetic studies of the reaction :

2A + B → Products

The time (in minutes) required to consume half of A is :

Solution:

6.93×10^{-3}=K×(0.1)^{x}(0.2)^{y}

6.93×10^{-3}=K×(0.1)^{x}(0.25)^{y}

So y = 0

and 1.386×10^{-2}=K×(0.2)^{x}(0.30)^{y}

1/2=(1/2)^{x} x=1

So r=K×(0.1)×(0.2)^{0}

6.93×10−3=K×0.1×(0.2)^{0}

K=6.93×10^{-2}

t_{1/2}=0.693/2K

= 0.693/(0.693×10^{-1}×2)

= 10/2

= 5

QUESTION: 51

The major product of the following reaction is:

Solution:

During AES Br is o/p directing and major product will be formed on less hindrance p position :

QUESTION: 52

Arrange the following amines in the decreasing order of basicity:

Solution:

Order of basic strength :

QUESTION: 53

Which amongst the following is the strongest acid ?

Solution:

CN makes anino most stable so answer is CH(CN)_{3}

QUESTION: 54

The anodic half-cell of lead-acid battery is recharged unsing electricity of 0.05 Faraday. The amount of PbS0_{4} electrolyzed in g during the process is: (Molar mass of PbS0_{4} = 303 g mol^{-}^{1})

Solution:

QUESTION: 55

The one that is extensively used as a piezoelectric material is :

Solution:

Quartz (Information)

QUESTION: 56

Aluminium is usually found in +3 oxidation stagte. In contarast, thallium exists in +1 and +3 oxidation states. This is due to :

Solution:

Inert pair effect is promenent character of p block element.

QUESTION: 57

The correct match between Item -I and ItemII is :

Solution:

QUESTION: 58

The ore that contains both iron and copper is:

Solution:

Copper pyrites : CuFeS_{2}

Malachite : Cu(OH)_{2} . CuCO_{3}

Azurite Cu(OH)_{2} . 2CuCO_{3}

Dolomite CaCO_{3} . MgCO_{3}

QUESTION: 59

The compounds A and B in the following reaction are, respectively:

Solution:

QUESTION: 60

The isotopes of hydrogen are :

Solution:

Isotopes of hydrogen are: Proteium, Deuterium, Tritium

QUESTION: 61

The area (in sq. units) bounded by the parabola y = x^{2 }- 1, the tangent at the point (2, 3 ) to it and the y-axis is :

Solution:

Equation of tangent at (2,3) on

y = x^{2} – 1, is y = (4x – 5) ....(i)

∴ Requireds haded area

QUESTION: 62

The maximum volume (in cu. m) of the right circular cone having slant height 3m is :

Solution:

∴ h = 3 cos θ

r = 3 sin θ

Now,

⇒ Volume is maximum,

QUESTION: 63

For x^{2 }≠ nπ + 1, n ∈ N (the set of natural numbers), the integral is equal to :

(where c is a constant of integration)

Solution:

QUESTION: 64

Let α and β be two roots of the equation x^{2} + 2x + 2 = 0, then α^{15} + β^{15} is equal to :

Solution:

We have

(x + 1)^{2} + 1 = 0

⇒ (x + 1)^{2 }– (i)^{2 }= 0

⇒ (x + 1 + i) (x + 1 – i) = 0

So, α^{15} + β^{15} = (α^{2})^{7} α + (β^{2})^{7} β

= –128 (–i + 1 + i + 1)

= – 256

QUESTION: 65

If y = y( x) is the solution of the differential equation.satisfying y(1) = 1, then is equal to :

Solution:

QUESTION: 66

Equation of a common tangent to the circle, x^{2} + y^{2} - 6x = 0 and the parabola, y^{2} = 4x, is:

Solution:

Let equation of tangent to the parabola y^{2} = 4x is

QUESTION: 67

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be fomied from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is:

Solution:

Required number of ways

= Total number of ways – When A and B are always included.

= ^{5}C_{2} . ^{7}C_{3} - ^{5}C_{1} . ^{5}C_{2} = 300

QUESTION: 68

Three circles of radii a, b, c( a < b < c) touch each other externally. If they have x-axis as a common tangent, then :

Solution:

AB = AC + CB

QUESTION: 69

If the fractional part of the number 2^{403}/15 is k/15, then k is equal to :

Solution:

∵ 8λ is integer

⇒ fractional part of

QUESTION: 70

Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it ?

Solution:

equation of parabola is

y^{2} = 8(x – 2)

(8, 6) does not lie on parabola.

QUESTION: 71

The plane through the intersection of the planes x + y + z = 1 and 2x + 3y-z + 4 = 0 and parallel to y-axis also passes through the point :

Solution:

Equation of plane

(x + y + z – 1) + λ(2x + 3y – z + 4) = 0

⇒ (1 + 2λ)x + (1 + 3λ)y + (1 – λ)z – 1 +4λ = 0

dr's of normal of the plane are

1 + 2λ, 1+ 3λ, 1 – l

Since plane is parallel to y - axis, 1 + 3λ = 0

⇒ λ = –1/3

So the equation of plane is

x + 4z – 7 = 0

Point (3, 2, 1) satisfies this equation

Hence Answer is (3)

QUESTION: 72

If a, b and c be three distinct real numbers in G. P. and a + b + c = xb, then x cannot be :

Solution:

given a + b + c = xb

⇒ b/r + b + br = xb

⇒ b = 0 (not possible)

⇒ x – 1 > 2 or x – 1 < –2

⇒ x > 3 or x < – 1

So x can't be '2'

QUESTION: 73

Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements is true ?

Solution:

Given set of lines px + qy + r = 0

given condition 3p + 2q + 4r = 0

⇒ All lines pass through a fixed point

.

QUESTION: 74

The system of linear equations.

x + y + z = 2

2x + 3y + 2z = 5

2x + 3y + (a^{2} – 1)z = a + 1

Solution:

QUESTION: 75

Let be a vector such that is equal to :-

Solution:

QUESTION: 76

Let a_{1}, a_{2},.........,a_{30} be an A. P., S = and If a_{5} = 27 and S – 2T = 75, then a_{10 }is equal to :

Solution:

S = a_{1} + a_{2} + ...... + a_{30}

S = 15(a_{1} + a_{30}) = 15 (a_{1} + a_{1} + 29d)

T = a_{1} + a_{3} + ..... + a_{29}

= (a_{1}) + (a_{1} + 2d) ..... + (a_{1} + 28d)

= 15a_{1} + 2d(1 + 2 + ..... + 14)

T = 15a_{1} + 210 d

Now use S – 2T = 75

⇒ 15 (2a_{1} + 29d) – 2 (15a_{1} + 210 d) = 75

⇒ d = 5

Given a_{5} = 27 = a_{1} + 4d ⇒ a_{1} = 7

Now a_{10} = a_{1} + 9d = 7 + 9 × 5 = 52

QUESTION: 77

5 students of a class have an average height 150 cm and variance 18 cm^{2}. A new student, whose height is 156 cm, joined them. The variance (in cm^{2}) of the height of these six students is:

Solution:

(i)

(ii)

Given height of new student

x_{6} = 156

= 22821- 22801 = 20

QUESTION: 78

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X = 1) + P(X = 2) eauals :

Solution:

Two cards are drawn successively with replacement

4 Aces 48 Non Aces

QUESTION: 79

For x ∈ R - {0, 1}, let f_{1}(x) = 1/x, f_{2}(x) = 1 – x and f_{3}(x) = be three given functions. If a function, J(x) satisfies (f_{2}°J°f_{1})(x) = f_{3}(x) then J(x) is equal to :-

Solution:

QUESTION: 80

Let Then the sum of the elements in A is :

Solution:

is purely img

so real part becomes zero.

Now Re(z) = 0

then sum of the elements in A is

QUESTION: 81

If θ denotes the acute angle between the curves,y = 10 – x^{2} and y = 2 + x^{2} at a point of their intersection, then |tan θ| is equal to :

Solution:

Point of intersection is P (2, 6).

QUESTION: 82

If then the matrix A^{–50 }when is equal to :

Solution:

Here, AA^{T} = I

QUESTION: 83

Let 0 < θ < π/2. If the eccentricity of the hyperbola is greater than 2, then the length of its latus rectum lies in the interval :

Solution:

= 2(sec θ – cos θ)

Which is strictly increasing, so

ℓ (L.R) ∈(3, ∞).

QUESTION: 84

The equation of the line passing through (–4, 3, 1), parallel to the plane x + 2y – z – 5 = 0 and intersecting the line is:

Solution:

Normal vector of plane containing two intersecting lines is parallel to vector.

∴ Required line is parallel to vector

⇒ Required equation of line is

QUESTION: 85

For any θ ∈ , the expression 3(sinθ – cosθ)^{4} + 6(sinθ + cosθ)^{2} + 4sin^{6}θ equals :

Solution:

We have,

3(sinθ – cosθ)^{4} + 6(sinθ + cosθ)^{2} + 4 sin^{6}θ

= 3(1 – sin2θ)^{2} + 6(1 + sin2θ) + 4sin^{6}θ

= 3(1 – 2sin2θ + sin^{2}2θ) + 6 + 6 sin2θ + 4sin^{6}θ

= 9 + 12 sin^{2}θ · cos^{2}θ + 4(1 – cos^{2}θ)^{3}

= 13 – 4 cos^{6}θ

QUESTION: 86

If then x is equal to :

Solution:

QUESTION: 87

The value of

Solution:

QUESTION: 88

If the Boolean expression is equivalent to p ^ q, where then the ordered pair is:

Solution:

(given)

from truth table

QUESTION: 89

Solution:

QUESTION: 90

Let f : R → R be a function defined as :

Then, f is :

Solution:

f(1) = 5, f(1^{–}) = 5, f(1^{+}) = a + b

f(3^{–}) = a + 3b, f(3) = b + 15, f(3^{+}) = b + 15

f(5^{–}) = b + 25 ; f(5) = 30 f(5^{+}) = 30

from above we concluded that f is not continuous for any values of a and b.

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