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QUESTION: 1

Kinetic energy of the particle is E and it's De–Broglie wavelength is λ. On increasing it's KE by ΔE, it's new De–Broglie wavelength becomes λ/2 . Then ΔE is

Solution:

QUESTION: 2

The dimensional formula of is

Solution:

QUESTION: 3

Two immiscible liquids of refractive index √2 and 2√2 are filled with equal height h in a vessel. Then apparent depth of bottom surface of the container given that outside medium is air:

Solution:

QUESTION: 4

Three identical solid spheres each having mass 'm' and diameter 'd' are touching each other as shown in figure. Calculate ratio of moment of inertia about an axis (perpendicular to plane of paper) passing through point P and B as shown in figure. Given P is centroid of triangle ABC.

Solution:

M.I about P =

M.I about B =

Now ratio = 13 / 23

QUESTION: 5

A solid sphere having radius R and Uniform charge density ρ has a cavity of radius R/2 as shown in figure. Find the ratio of magnitude of electric field at point A and B i.e.

Solution:

For a solid sphere

Electric field at point B = E_{B} = E_{1A} + E_{2A }

E_{1A} = Electric Field Due to solid sphere of radius R at point B =

E_{2A} = Electric Field Due to solid sphere of radius R/2 (which having charge density –ρ)

E_{2A} = R/2

E_{B} = E_{1A} + E_{2A} =

QUESTION: 6

Consider an infinitely long current carrying cylindrical straight wire having radius 'a'. Then the ratio of magnetic field at distance a/3 and 2a from axis of wire is.

Solution:

QUESTION: 7

Find current in the wire BC.

Solution:

QUESTION: 8

Two electromagnetic waves are moving in free space whose electric field vectors are given by . A charge q is moving with velocity .Find the

net Lorentz force on this charge at t = 0 and when it is at origin.

Solution:

Magnetic field vectors associated with this electromagnetic wave are given by

by putting the value of

The net Lorentz force on the charged particle is

at t = 0 and at x = y = 0

t = 0 , x = y = 0

QUESTION: 9

Two ideal di-atomic gases A and B. A is rigid, B has an extra degree of freedom due to vibration. Mass of A is m and mass of B is m/4. The ratio of molar specific heat of A to B at constant volume is :

Solution:

Molar heat capacity of A at constant volume = 5R/2

Molar heat capacity of B at constant volume = 7R/2

Dividing both

QUESTION: 10

An ideal liquid (water) flowing through a tube of non uniform cross section area at A and B are 40 cm^{2} and 20 cm^{2} respectively. If pressure difference between A & B is 700 N/m^{2} then volume flow rate is :

Solution:

using equation of continuity

40 V_{A} = 20 V_{B}

2V_{A} = V_{B}

Using Bernoullies equation

Volume flow rate = 20 × 100 × V_{B} = 2732 cm^{3}/s

QUESTION: 11

A screw gauge adv ances by 3mm in 6 rotations. There are 50 divisions on circular scale. Find least count of screw gauge ?

Solution:

Pitch = 3/6 = 0.5 mm

L.C. = = = 0.01 mm = 0.001 cm

QUESTION: 12

A telescope of aperture diameter 5m is used to observe the moon from the earth. Distance between the moon and earth is 4 × 10^{5} km. Determine the minimum distance between two points on the moon's surface which can be resolved using this telescope. (Wave length of light is 5893 Å).

Solution:

distance = O_{1}O_{2} = dθ

=

distance = O_{1}O_{2} = ≈ 57.5 m

∴ answer from options = 60m

(minimum distance)

QUESTION: 13

A particle of mass m is revolving around a planet in a circular orbit of radius R. At the instant the particle has velocity another particle of mass m/2 moving at velocity collides perfectly in-elastically with the first particle. The new path of the combined body will take is

Solution:

Conserving momentum

v_{f }< v_{orb} (= v) thus the combined mass will go on to an elliptical path.

QUESTION: 14

Two particles of same mass 'm' moving with velocities and collide in-elastically. Find the loss in kinetic energy.

Solution:

Conserving momentum

on solving

Change in K.E

=

QUESTION: 15

Three wav es of same intensity (I_{0}) having initial phases rad respectively interfere at a point.Find the resultant Intensity

Solution:

QUESTION: 16

Particle moves from point A to point B along the line shown in figure under the action of force. . Determine the work done on the particle by in moving the particle from point A to point B

Solution:

QUESTION: 17

For the given P-V graph for an ideal gas, chose the correct V- T graph. Process BC is adiabatic.

Solution:

For process A – B Volume is constant

PV = nRT ; as P increases T increases

For process B – C

PV^{γ} = Constant

TV^{γ-1} = Constant

For process C – A pressure is constant

V = kT

QUESTION: 18

Given. Find vector parallel to electric field at position

[Note that ]

Solution:

Since

must be antiparallel to

So ,

where λ is a arbitrary positive constant

Now

so

QUESTION: 19

Which of the following statements are correct for moving charge as shown in figure.

Solution:

QUESTION: 20

Photons of wav elength 6556 Å falls on a metal surface. If ejected electrons with maximum K.E. moves in magnetic field of 3 × 10^{–4} T in circular orbit of radius 10^{–2}m, then work function of metal surface is

Solution:

= 1.1 eV

*Answer can only contain numeric values

QUESTION: 21

A rod of length 1 m is released from rest as shown in the figure below.

If ω of rod is √n at the moment it hits the ground, then find n.

Solution:

*Answer can only contain numeric values

QUESTION: 22

If reversible voltage of 100 V is applied across an inductor, current in it reduces from 0.25A to 0A in 0.025ms. Find inductance of inductor (in mH).

Solution:

∴ L = 100 × 10^{–4} H

= 10 mH

*Answer can only contain numeric values

QUESTION: 23

A wire of length l = 3m and area of cross section 10^{–2}cm^{2} and breaking stress 48×10^{7}N/m^{2} is attached with block of mass 10kg. Find the maximum possible value of angular velocity with which block can be moved in circle with string fixed at one end.

Solution:

Solving

ω = 4 rad/s

*Answer can only contain numeric values

QUESTION: 24

Position of a particle as a function of time is given as x^{2} = at^{2} + 2bt + c, where a, b, c are constants. Acceleration of particle varies with x^{–n} then value of n is.

Solution:

*Answer can only contain numeric values

QUESTION: 25

In the given circuit both diodes having zero forward resistance and built-in potential of 0.7 V. Find the potential of point E in volts.

Solution:

Let V_{B} = 0

Right diode is reversed biased and left diode is forward biased

∴ V_{E} = 12.7 – 0.7

= 12 Volt

QUESTION: 26

Determine wavelength of electron in 4th Bohr's orbit ?

Solution:

QUESTION: 27

Which of the following species have one unpaired electron each?

Solution:

QUESTION: 28

For Br_{2}(l) Enthalpy of atomisation = x kJ/mol Bond dissociation enthalpy of bromine = y kJ/mole

then

Solution:

ΔH_{atomisation} = ΔH_{vap} + Bond energy

Hence x > y

QUESTION: 29

Which of the following oxides are acidic, Basic Amphoteric Respectively.

Solution:

Non-metal oxides are acidic in nature alkali metal oxides are basic in nature Al_{2}O_{3} is amphoteric.

QUESTION: 30

Complex Cr(H_{2}O)_{6}Cl_{n} shows geometrical isomerism and also reacts with AgNO_{3} solution.

Given : Spin only magnetic moment = 3.8 B.M.

What is the IUPAC name of the complex.

Solution:

Cr(H_{2}O)_{6}Cl_{n} (μ_{complex})_{spin} = 3.8 B.M.

From data of magnetic moment oxidation number of Cr should be +3

Hence complex is Cr(H_{2}O)_{6}Cl_{3}.

Complex shows geometrical isomerism therefore formula of complex is [Cr(H_{2}O)_{4}Cl_{2}]Cl . 2H_{2}O.

It's IUPAC Name: Tetraaquadichloridochromium(III) chloride dihydrate.

QUESTION: 31

The electronic configuration of bivalent Europium and trivalent cerium respectively is: (Atomic Number : Xe = 54, Ce = 58, Eu = 63)

Solution:

Eu^{2+} : [Xe]4f^{7}

Ce^{3+} : [Xe]4f^{1}

QUESTION: 32

Ksp of PbCl_{2} = 1.6 × 10^{–5}

On mixing 300 mL, 0.134M Pb(NO_{3})_{2}(aq.) + 100 mL, 0.4 M NaCl(aq.)

Solution:

= 0.105 × 10^{–2}

=1.005 ×10^{–3}

Q > Ksp

QUESTION: 33

Which of the following can not act as both oxidising and reducing agent ?

Solution:

As in H_{3}PO_{4} Phosphorous is present it's maximum oxidation number state hence it cannot act as reducing agent.

QUESTION: 34

First Ionisation energy of Be is higher than that of Boron.

Select the correct statements regarding this

(i) It is easier to extract electron from 2p orbital than 2s orbital

(ii) Penetration power of 2s orbital is greater than 2p orbital

(iii) Shielding of 2p electron by 2s electron

(iv) Radius of Boron atom is larger than that of Be

Solution:

Theory Based

QUESTION: 35

[PdFClBrI]^{2–} Number of Geometrical Isomers = n. For [Fe(CN)_{6}]^{n–6}, Determine the spin only magnetic moment and CFSE (Ignore the pairing energy)

Solution:

Number of Geometrical Isomers in square planar [PdFClBrI]^{2–} are = 3

Hence, n = 3

Fe^{3+} = 3d^{5},

According to CFT configuration is

CFSE = - 0 .4 Δ_{0} x nt_{2g} + 0 .6 Δ_{0} x n_{eg} = -0.4 Δ_{0} × 5 = -2.0Δ_{0}

QUESTION: 36

A can reduce BO_{2} under which conditions.

Solution:

A + BO_{2 }---> B + AO_{2}

ΔG = -ve

Only above 1400°C

QUESTION: 37

Rate of reaction in absence of catalyst at 700 K is same as in presence of catalyst at 500 K. If catalyst decreases activation energy barrier by 30 kJ/mole, determine activation energy in presence of catalyst. (Assume 'A' factor to be same in both cases)

Solution:

Activation energy of the catalysed reaction = 105 – 30 = 75 kJ/mole

QUESTION: 38

A substance 'X' having low melting point, does not conduct electricity in both solid and liquid state. 'X' can be :

Solution:

CCl4 ---> Non-conductor in solid and liquid phase.

QUESTION: 39

The major product for above sequence of reaction is :

Solution:

QUESTION: 40

Which of the following can give highest yield in Friedel craft reaction?

Solution:

Aniline form anilinium complex with lewis acid so phenol is most reactive among the given compounds for electrophilic substitution reaction.

QUESTION: 41

What will be the major product ?

Solution:

QUESTION: 42

Which of the following is correct order for heat of combustion?

Solution:

In isomers of hydrocarbon heat of combustion depends upon their stabilities. As the stability increases heat of combustion decreases.

Stability order

QUESTION: 43

Write the correct order of basicity.

Solution:

Basicity is inversely proportional to electronegativity.

QUESTION: 44

A, B, C, and D are four artificial sweetners.

(i) A & D give positive test with ninhydrin.

(ii) C form precipitate with AgNO_{3} in the lassaigne extract of the sugar.

(iii) B & D give positive test with sodium nitroprusside.

Correct option is :`

Solution:

A – Aspartame

B – Saccharine

C – Sucralose

D – Alitame

(i) A & D give positive test with ninhydrin because both have free carboxylic and amine groups.

(ii) C form precipitate with AgNO_{3} in the lassaigne extract of the sugar because it has chlorine atoms.

(iii) B & D give positive test with sodium nitroprusside because both have sulphur atoms.

QUESTION: 45

Predict the compound (P) on the basis of above sequence of the reactions?

Where compound (P) gives positive Iodoform test.

Solution:

*Answer can only contain numeric values

QUESTION: 46

Given a solution of HNO_{3} of density 1.4 g/mL and 63% w/w. Determine molarity of HNO_{3} solution.

Solution:

*Answer can only contain numeric values

QUESTION: 47

Determine degree of hardness in term of ppm of CaCO_{3} of 10^{–3} molar MgSO_{4} (aq).

Solution:

10^{–3} molar MgSO_{4} = 10^{–3} moles of MgSO_{4} present in 1 L solutions.

*Answer can only contain numeric values

QUESTION: 48

Determine the amount of NaCl to be dissolved in 600g H_{2}O to decrease the freezing point by 0.2°C

Given : k_{f }of H_{2}O = 2 k-m^{–1} density of H_{2}O(l) = 1 g/ml

Solution:

*Answer can only contain numeric values

QUESTION: 49

On passing a particular amount of electricity in AgNO_{3} solution, 108 g of Ag is deposited. What will be the volume of O_{2}(g) in litre liberated at 1 bar, 273K by same quantity of electricity?

Solution:

1F charge is required to deposit 1 mole of Ag

2F charge deposit → 1/2 mole

1F charge will deposit → 1/4 mole

*Answer can only contain numeric values

QUESTION: 50

Find percentage nitrogen by mass in Histamine ?

Solution:

Structure of Histamine is

Molecular formula of Histamine is C_{5}H_{9}N_{3}

Molecular mass of Histamine is 111

Percentage nitrogen by mass in Histamine =

QUESTION: 51

Find the number of solution of log_{1/2} |sinx| = 2 – log_{1/2} |cosx| , x ∈ [0,2π]

Solution:

log_{1/2} |sinx| = 2 – log_{1/2} |cosx|

log_{1/2} |sinx cosx| = 2

|sinx cosx| = 1/4

sin2x = ± 1/2

Number of solution = 8

QUESTION: 52

If e_{1} and e_{2} are eccentricities of and , respectively and if the point (e_{1}, e_{2}) lies on ellipse 15x^{2} + 3y^{2} = k. Then find value of k

Solution:

∴ k = 16

QUESTION: 53

Find integration

Solution:

QUESTION: 54

If = 1, |z| = 5/2 then value of |z + 3i| is

Solution:

QUESTION: 55

Solution:

QUESTION: 56

Value of is

Solution:

QUESTION: 57

Find the value of

Solution:

QUESTION: 58

If f(x) = a + bx + cx^{2} where a, b, c ∈ R then is

Solution:

f(1) = a + b + c

f(0) = a

Now

QUESTION: 59

If number of 5 digit numbers which can be formed without repeating any digit while tenth place of all of the numbers must be 2 is 336 k find value of k

Solution:

Number of numbers

= 8 x 8 x 7 x 6 = 2688 = 336k => k = 8

QUESTION: 60

A (3,–1), B(1,3), C(2,4) are vertices of ΔABC if D is centroid of ΔABC and P is point of intersection of lines x + 3y - 1 = 0 and 3x - y + 1 = 0 then which of the following points lies on line joining D and P

Solution:

D (2,2)

Point of intersection P

and equation of line DP

8x – 11y + 6 = 0

QUESTION: 61

If f(x) is twice differentiable and continuous function in x ∈ [a,b] also f'(x) > 0 and f ''(x) < 0 and c ∈ (a,b)

then is greater than

Solution:

QUESTION: 62

If plane

x + 4y – 2z = 1

x + 7y – 5z = β

x + 5y + αz = 5

intersects in a line (R × R × R) then α + β is equal to

Solution:

(7α + 25) - (4α + 10) + (-20 + 14) = 0

3α + 9 = 0

α = - 3

Also D_{z} = 0 = 0

1(35 – 5β) - (15) + 1 (4β - 7) = 0

β = 13

QUESTION: 63

For observations xi given and . If mean and variance of observations is λ & μ respectively then ordered pair (λ, μ) is

Solution:

Mean

∴ λ = {Mean (xi – 5)} + 2 = 3

μ = var (xi – 5) =

QUESTION: 64

In a bag there are 20 cards 10 names A and another 10 names B. Cards are drawn randomly one by one with replacement then find probability that second A comes before third B.

Solution:

AA + ABA + BAA + ABBA + BBAA + BABA

=

QUESTION: 65

The negation of ‘ √5 is an integer or 5 is an irrational number’ is

Solution:

√5 is not an integer and 5 is not an irrational Number ~ (p v q) = ~ p ∧ ~ q

QUESTION: 66

If a circle touches y-axis at (0, 4) and passes through (2, 0) then which of the following can not be the tangent to the circle

Solution:

equation of family of circle

(x – 0)^{2} + (y – 4)^{2} + λx = 0

passes (2, 0)

4 + 16 + 2λ = 0

λ= - 10

x^{2} + y^{2} – 10x – 8y + 16 = 0

centre (5, 4). R = = 5

QUESTION: 67

If f'(x) = tan^{–1 }(secx + tanx), x ∈ and f(0) = 0 then the value of f(1) is

Solution:

f'(x) = tan^{–1} (secx + tanx) =

tan^{–1}

=

= tan^{–1}

QUESTION: 68

A sphere of 10cm radius has a uniform thickness of ice around it. Ice is melting at rate 50cm^{3}/min when thickness is 5cm then rate of change of thickness

Solution:

Let thickness = x cm

Total volume v = 4/3 π(10 + x)^{3}

Given dv/dt = 50cm^{3}/min

At x = 5cm

50 = 4π (10 + 5)^{2}dx/dt

QUESTION: 69

Find number of real roots of equation e^{4x} + e^{3x} – 4e^{2x} + e^{x} + 1 = 0 is

Solution:

Let e^{x} = t ∈ (0, ∞)

Given equation t^{4} + t^{3} – 4t^{2} + t + 1 = 0

QUESTION: 70

If A = B = adj(A) and C = 3A then is

Solution:

= 13 + 1 – 8 = 6

*Answer can only contain numeric values

QUESTION: 71

Solution:

∴ at x = 2, y = 0

∴ 0 = 3 (2+1+c) => c = -3

at x = 3 , y = 3

*Answer can only contain numeric values

QUESTION: 72

Function is continuous at x = 0, find a + 2b.

Solution:

LHL = a + 3

f(0) = b

RHL =

∴ a = -2 b = 1 ∴ a + 2b = 0

*Answer can only contain numeric values

QUESTION: 73

Find the coefficient of x^{4} in (1 + x + x^{2})^{10}

Solution:

General term

for coefficient of x^{4}

=> β + 2γ = 4

γ = 0, β = 4 , α = 6

γ = 1, β = 2 , α = 7

γ = 2, β = 0 , α = 8

Total = 615

*Answer can only contain numeric values

QUESTION: 74

and are coplanar vectors and then value of λ is

Solution:

= 0

a + 1 + a + a = 0 => a =

*Answer can only contain numeric values

QUESTION: 75

If points A (2, 4, 0), B(3, 1, 8), C(3, 1, –3), D(7, –3, 4) are four points then projection of line segment AB on line CD.

Solution:

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