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QUESTION: 1

A mass m attached to spring of natural lengthand spring constant k.One end of string is attached to centre of disc in horizontal plane which is being rotated by constant angular speed ω. Find extension per unit length in spring (given k >> mω^{2}) :

Solution:

QUESTION: 2

A loop of radius R and mass m is placed in a uniform magnetic field B with its plane perpendicular to the field. Current I is flowing in it. Now loop is slightly rotated about its diameter and released. Find time period of oscillation.

Solution:

QUESTION: 3

A string of mass per unit length μ = 6 × 10^{–3} kg/m is fixed at both ends under the tension 540 N. If the string is in resonance with consecutive frequencies 420 Hz and 490 Hz. Then find the length of the string?

Solution:

Fundamental frequency = 490 – 420

= 70 Hz

QUESTION: 4

Ratio of energy density of two steel rods is 1 : 4 when same mass is suspended from the rods. If length of both rods is same then ratio of diameter of rods will be.

Solution:

stress x

QUESTION: 5

A particle is projected from the ground with speed u at angle 60° from horizontal. It collides with a second particle of same mass moving with horizontal speed u in same direction at highest point of its trajectory. If collision is perfectly inelastic then find horizontal distance travelled by them after collision when they reached at ground

Solution:

so horizontal range after collision

QUESTION: 6

H-like atom with ionization energy of 9R. Find the wavelength of light emitted (in nm) when electron jumps from second excited state to ground state. (R is Rydberg constant)

Solution:

wavelength = nm

λ = 11.39 nm

QUESTION: 7

Two planets of masses M and M/2 have radii R and R/2 respectively. If ratio of escape velocities from their surfaces v_{1}/v_{2} is n/4, then find n :

Solution:

QUESTION: 8

Find centre of mass of given rod of linear mass density λ = , x is distance from one of its end. Length of the rod is l.

Solution:

QUESTION: 9

If a point source is placed at a depth h in a liquid of refractive index 4/3. Find percentage of energy of light that escapes from liquid. (assuming 100% transmission of emerging light)

Solution:

Solid angle dΩ = 2pR^{2} (1 - cos β)

Percentage of light =

QUESTION: 10

System is released from rest. Moment of inertia of pulley 'I'. Find angular speed of pulley when m_{1} block falls by 'h'. (Given m_{1} > m_{2} and assume no slipping between string and pulley).

Solution:

k_{1}+ U_{1} = k_{1} + k_{2}

QUESTION: 11

Find the current supplied by the battery?

Solution:

Both diodes are in reverse biased

QUESTION: 12

An AC source is connected to the LC series circuit with V = 10 sin (314t). Find the current in the circuit as function of time ? (L = 40 mH, C = 100 μF)

Solution:

QUESTION: 13

There is a long solenoid of radius ‘R’ having ‘n’ turns per unit length with current i flowing in it. A particle having charge ‘q’ and mass ‘m’ is projected with speed 'v' in the perpendicular direction of axis from a Point on its axis Find maximum value of 'v' so that it will not collide with the solenoid.

Solution:

QUESTION: 14

A Capacitor C and resister R are connected to a battery of 5V in series. Now battery is disconnected and a diode is connected as shown in figure (a) and (b) respectively. Then charge on the capacitor after time RC in (a) and (b) respectively is Q_{A} and Q_{B}. Their value are

Solution:

Maximum charge on capacitor = 5CV

(a) is forward biased and

(b) is reverse biased for case (a)

QUESTION: 15

A sphere of density ρ is half submerged in a liquid of density σ and surface tension T. The sphere remains in equilibrium. Find radius of the sphere (assume the force due to surface tension acts tangentially to surface of sphere)

Solution:

QUESTION: 16

An EM wave is travelling in direction. Axis of polarization of EM wave is found to be Then equation of magnetic field will be

Solution:

EM wave is in direction

Electric field is in direction ---> kˆ

--> direction of propagation of EM wave

QUESTION: 17

Different value of a, b and c are given and their sum is d. Arrange the value of d in increasing order

Solution:

QUESTION: 18

Two gases Ar (40) and Xe (131) at same temperature have same number density. Their diameters are 0.07 nm and 0.10 nm respectively. Find the ratio of their mean free time

Solution:

Mean free time =

QUESTION: 19

A particle starts moving from origin with velocityfrom origin and acceleration . Find x-coordinate at the instant when y-coordinate of the particle is 32 m.

Solution:

t = 4 sec

= 60 m.

QUESTION: 20

An electron is released in Electric field E from rest. Rate of change of de-Broglie wavelength with time will be.

Solution:

*Answer can only contain numeric values

QUESTION: 21

In YDSE pattern with light of wavelength λ_{1} = 500nm, 15 fringes are obtained on a certain segment of screen. If number of fringes for light of wavelength λ_{2} on same segment of screen is 10, then the value of λ_{2} (in nm) is-

Solution:

*Answer can only contain numeric values

QUESTION: 22

If in a meter bridge experiment, the balancing length ' l ' was 25 cm for the situation shown in the figure. If the length and diameter of the of wire of resistance R is made half, then find the new balancing length in centimetre is

Solution:

l = 40.00 cm

*Answer can only contain numeric values

QUESTION: 23

Find the power loss in each diode (in mW), if potential drop across the zener diode is 8V.

Solution:

Power loss in each diode = (4)(10^{–2}) W = 40 mW

*Answer can only contain numeric values

QUESTION: 24

An ideal gas at initial temperature 300 K is compressed adiabatically of its initial volume. The gas is then expanded isobarically to double its volume. Then final temperature of gas round to nearest integer is:

Solution:

PV^{y} = constant

TV^{y–1} = constant

T_{B} = 300 × 2^{8/5}

Now for BC process

T_{c} = 1819 K

*Answer can only contain numeric values

QUESTION: 25

If electric field in the space is given by and electric flux through ABCD is φ_{1} and electric flux through BCEF is φ_{2} , then find (φ_{1} - φ_{2})

Solution:

Flux via ABCD

Flux via BCEF

QUESTION: 26

5 g of Zn reacts with

(I) Excess of NaOH (II) Dilute HCl, then volume ratio of H_{2} gas evolved in (I) and (II) is

Solution:

Zn + 2NaOH ---> Na_{2}ZnO_{2} + H_{2}

Zn + 2HCl ---> ZnCl_{2} + H_{2}

According to stoichiometry in both the reactions, equal number of moles of H_{2} are evolved.

QUESTION: 27

Given K_{sp} for Cr(OH)_{3} is 6 × 10^{–31} then determine [OH^{–}].

(Neglect the contribution of OH^{–} ions from H_{2}O)

Solution:

QUESTION: 28

Select the correct statements among the followings

(A) LiCl does not dissolve in pyridine

(B) Li does not react ethyne to form ethynide.

(C) Li and Mg react slowly with water.

(D) Among alkali metals Li has highest hydration tendency.

Solution:

Theory based

QUESTION: 29

Given an element having following ionisation enthalpies IE_{1} = 496 and IE_{2} = 4562 one mole

hydroxide of this element is treated separated with HCl and H_{2}SO_{4} respectiv ely. Moles of HCl and H_{2}SO_{4 }reacted respectively is

Solution:

According to the given data of I.E, This element must belong to group 1 and thus is monovalent & form hydroxide of the type M(OH).

MOH + HCl --> MCl + H_{2}O

1 mole 1 mole

2MOH + H_{2}SO_{4} ---> M_{2}SO_{4} + H_{2}O

1 mole 1/2 mole

QUESTION: 30

Reactant A represented by square is in equilibrium with product B represented by circles. Then value of equilibrium constant is

Solution:

QUESTION: 31

Given following complexes

(I) Na_{4}[Fe(CN)_{6}]

(II) [Cr(H_{2}O)_{6}] Cl_{2}

(III) (NEt_{4})_{2} [CoCl_{4}]

(IV) Na_{3}[Fe(C_{2}O_{4})_{3}] (Δ_{0} > P)

Correct order of spin only magnetic moment for the above complexes is.

Solution:

QUESTION: 32

Select the correct option :

Solution:

Entropy is function of temperature and also entropy change is function of temperature.

ΔS = ∫dq/T

S_{T} = ∫_{0}^{T} ncdT/T

QUESTION: 33

A compound (A ; B_{3}N_{3}H_{3}Cl_{3}) reacts with LiBH_{4} to form inorganic benzene (B). (A) reacts with (C) to form B_{3}N_{3}H_{3}(CH_{3})_{3}. (B) and (C) are respectively.

Solution:

QUESTION: 34

In a box a mixture containing H_{2}, O_{2} and CO along with charcoal is present then variation of pressure with the time will be as follows :

Solution:

Theory based.

QUESTION: 35

Given complex [Co(NH_{3})_{4}Cl_{2}]. In it if Cl – Co – Cl bond angle is 90º then it is :

Solution:

QUESTION: 36

Amongst the following which has minimum conductivity.

Solution:

Theory based.

QUESTION: 37

Number of sp^{2} hybrid orbitals in Benzene is :

Solution:

In benzene total six sp^{2 }hybrid carbon atoms are present. Each carbon atom has 3 sp^{2} hybrid orbitals.

Therefore total sp^{2} hybrid orbitals are 18 in benzene.

QUESTION: 38

Which of the following reaction will not form racemic mixture as product?

Solution:

(CH_{3})_{2}CH–CH=CH_{2 }--^{HCl}----> ---^{Hybride shift}--->

QUESTION: 39

In which compound C–Cl bond length is shortest?

Solution:

Resonance form of Cl–CH=CH–NO_{2} is more stable than resonance form of any other given compounds. Hence, double bond character in carbon-chlorine bond is maximum and bond length is shortest.

QUESTION: 40

Biochemical oxygen demand (BOD) is defined as ............ in ppm of O_{2}.

Solution:

The amount of oxygen required by bacteria to break down the organic matter present in a certain volume of a sample of water, is called Biochemical Oxygen Demand (BOD).

QUESTION: 41

Monomer(s) of which of the given polymer is chiral?

Solution:

In PHBV, both monomers have chiral centre.

QUESTION: 42

Which of the following options is correct ?

Solution:

QUESTION: 43

The order of basic character is :

(I) (II) (III) (IV)

Solution:

Basic strength depends upon availability of lone pairs. Greater the resonance of lone pairs lesser the basic strength.

QUESTION: 44

Compound A Compound A will be :

Solution:

QUESTION: 45

Compound X

Compound X will be :

Solution:

*Answer can only contain numeric values

QUESTION: 46

Total number of Cr–O bonds in Chromate ion and dichromate ion is.

Solution:

Chromate ion --->

Dichromate ion --->

=> Total number of Cr and O bonds is 12.

*Answer can only contain numeric values

QUESTION: 47

Lacto bacillus has generation time 60 min. at 300 K and 40 min. at 400 K. Determine activation energy in (given wrong in paper)

Solution:

ln(3/2) × 8.3 × 1200 = E_{a}

=> E_{a} = 0.4 × 8.3 × 1200

=> Ea = 3984 J/mol.

=> Ea = 3.984 kJ/mol.

*Answer can only contain numeric values

QUESTION: 48

One litre sea water (d = 1.03g/cm^{3}) contains 10.3 mg O_{2} gas. Determine concentration of O_{2} in ppm.

Solution:

*Answer can only contain numeric values

QUESTION: 49

0.1 ml of an ideal gas has volume 1 dm^{3} in a locked box with friction less piston. The gas is in thermal equilibrium with excess of 0.5 m aqueous ethylene glycol at its freezing point. If piston is released all of a sudden at 1 atm then determine the final volume of gas in dm^{3} (R = 0.08 atm L mol^{–1} K^{–1} K_{f} = 2.0 K molal^{–1}).

Solution:

K_{f} = 2.0

m = 0.5 m

ΔT_{f} = K_{fm} = 0.5 × 2

T_{initial} = 272 K

n = 0.1 mol

V = 1 dm^{3}

= 2.176 atm

After releasing piston P_{1}V_{1} = P_{2}V_{2}

2.176 × 1 = 1 × V_{2}

V_{2} = 2.176 dm^{3}

*Answer can only contain numeric values

QUESTION: 50

Compound A

Percentage carbon in compound A is:

Solution:

Compound A is

Percentage carbon in compound A =

QUESTION: 51

If f(x) = g(x) = then find the area bounded by f(x) and g(x) from x = .

Solution:

Required area = Area of trapezium ABCD –

QUESTION: 52

z is a complex number such that |Re(z)| + |Im (z)| = 4 then |z| can't be

Solution:

z = x + iy

|x| + |y| = 4

Minimum value of |z| = 2√2

Maximum value of |z| = 4

So |z| can't be √7

QUESTION: 53

If f(x) = and a – 2b + c = 1 then

Solution:

Apply R_{1} = R_{1} + R_{3} – 2R_{2}

f(x) = 1

f(50) = 1

QUESTION: 54

Let a_{n} is a positive term of a GP and

Solution:

Let GP is a, ar, ar^{2} ……..

= a_{3} + a_{5} + ….. a_{201} = 200

............(1)

= a_{2} + a_{4} + ……. a_{200} = 100

........... (2)

Form (1) and (2) r = 2

add both

a_{2} + a_{3} + ……….. a_{200} + a_{201} = 300

r(a_{1} + ………… a_{200}) = 300

= 300/r = 150

QUESTION: 55

If y(1) = 1 and y(x) = e then x = ?

Solution:

Put y = vx

When x = 1, y = 1 then

So x = √3e

QUESTION: 56

Let probability distribution is

then value of p(x > 2) is

Solution:

6k^{2} + 5k = 1

6k^{2} + 5k – 1 = 0

6k^{2} + 6k – k – 1 = 0

(6k – 1) (k + 1) = 0

k = – 1 (rejected); k = 1/6

P(x > 2) = k + 2k + 5k^{2}

QUESTION: 57

= λtanq + 2log f(x) + c then ordered pair (λ, f(x)) is

Solution:

tanθ = t => sec^{2}θ dθ = dt

= – t + 2 log (1+t) + C

= –tanθ + 2 log (1 + tanθ) + C

= λ = -1 and f(x) = 1 + tanθ

QUESTION: 58

If p ---> (p ∧ ~ q) is false. Truth value of p & q will be

Solution:

QUESTION: 59

If = A then the value of x at which f(x) = [x^{2}] sinπx is discontinuous.

(where [.] denotes greatest integer function)

Solution:

4 – 0 = A

check when

(A) x = => x = √5 => discontinuous

(B) x = A + 21 => x = 5

=> continuous

(C) x = A => x = 2

=> continuous

QUESTION: 60

Let one end of focal chord of parabola y^{2} = 8x is , then equation of tangent at other end of this focal chord is

Solution:

Let is (2t^{2}, 4t)

=>

Parameter of other end of focal chord is 2

=> point is (8, 8)

=>equation of tangent is 8y – 4(x+8) = 0

=> 2y - x = 8

QUESTION: 61

Let x + 6y = 8 is tangent to standard ellipse where minor axis is , then eccentricity of ellipse is

Solution:

,

Equation of tangent ≡ y = mx ±

comparing with

and

QUESTION: 62

if f(x) and g(x) are continuous functions, fog is identity function, g'(b) = 5 and g(b) = a then f'(a) is

Solution:

QUESTION: 63

If 7x + 6y – 2z = 0

3x + 4y + 2z = 0

x – 2y – 6z = 0 then which option is correct

Solution:

= 7(–20) – 6(–20) – 2(–10)

= – 140 + 120 + 20 = 0

so infinite non-trivial solution exist

now equation (1) + 3 equation (3)

10x – 20z = 0

x = 2z

QUESTION: 64

Let x = 2sinθ - sin2θ and

y = 2 cosθ - cos2θ

find at θ = π

Solution:

QUESTION: 65

then correct choice is

Solution:

F'(x) = x^{2}g(x)

=> F'(1) = 1.g(1) = 0 ....(1)

(∴ g(1) = 0)

F''(1) = 0 + 1 × 3

=> F''(1) = 3 .....(2)

From (1) and (2) F(x) has local minimum at x = 1

QUESTION: 66

Let both root of equation ax^{2} – 2bx + 5 = 0 are α and root of equation x^{2} - 2bx - 10 = 0 are α and β. Find the value of α^{2} + β^{2}

Solution:

b^{2} = 5a ..... (i) (a ≠ 0)

α + β = 2b ..... (ii)

and αβ = – 10 ......(iii)

α = b/a is also root of x^{2} - 2bx - 10 = 0

b^{2} – 2ab^{2} – 10a^{2} = 0

by (i) 5a – 10a^{2} – 10a^{2 }= 0

20a^{2} = 5a

a = 1/4 and b^{2} = 5/4

α^{2 }= 20 and β^{2} = 5

Now α^{2} + β^{2}

= 5 + 20

= 25

QUESTION: 67

Let A = {x : |x| < 2} and B = {x : |x – 2| ≥ 3} then

Solution:

QUESTION: 68

Let x = and y = where θ ∈ (0,π/4), then

Solution:

y = 1 + cos^{2}θ + cos^{4}θ + .......

= > y (1 – x) = 1

QUESTION: 69

Let the distance between plane passing through lines and and plane 23x – 10y – 2z + 48 = 0 is then k is equal to

Solution:

Lines must be intersecting

(2s – 1, 3s + 3, 8s –1)

= (2t – 3, t – 2 , λt + 1)

2s - 1 = 2t - 3,

3s + 3 = t - 2 ,

8s - 1 = λt + 1

=> t = -1, s = -2 , λ = 18

distance of plane contains given lines from given plane is same as distance between point (–3, –2,1) from given plane.

Required distance equal to

k = 3

*Answer can only contain numeric values

QUESTION: 70

If ^{25}C_{0} + 5 ^{25}C_{1} + 9 ^{25}C_{2} ……… 101 ^{25}C_{25} = 2^{25} k find k = ?

Solution:

So k = 51

*Answer can only contain numeric values

QUESTION: 71

Let circles (x – 0)^{2} + (y – 4)^{2} = k and (x – 3)^{2 }+ (y – 0)^{2 }= 1^{2} touches each other than find the maximum value of 'k'

Solution:

Two circles touches each other if C_{1}C_{2} = = |r_{1} ± r2|

Distance between C_{2}(3, 0) and C_{1}(0, 4) is either √k + 1 or (C_{1}C_{2} = 5)

=> √k + 1 = 5 or

=> k = 16 or k = 36

=> maximum value of k is 36

*Answer can only contain numeric values

QUESTION: 72

Let angle between equal to π/3

If is perpendicular to then find the value of

Solution:

Also

*Answer can only contain numeric values

QUESTION: 73

Number of common terms in both sequence 3, 7, 11, ………….407 and 2, 9, 16, ……..905 is

Solution:

First common term = 23

common difference = 7 × 4 = 28

Last term ≤ 407

=> 23 + (n-1) × 28 ≤ 407

=> (n - 1) × 28 ≤ 384

=> n ≤ 13.71 + 1

n ≤ 14.71

So n = 14

*Answer can only contain numeric values

QUESTION: 74

If minimum value of term free from x for is L_{1} in θ ∈ and L_{2} in θ ∈ find

Solution:

∵ {Min value of L_{1} at θ = π/4}

= ^{16}C_{8} . 2^{8}.2^{4} {∵ min value of L_{2} at θ = π/8]

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