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QUESTION: 1

A block of mass m is suspended from a pulley in form of a circular disc of mass m & radius R. The system is released from rest, find the angular velocity of disc when block has dropped by height h. (there is no slipping between string & pulley)

Solution:

v = ωR (no slipping)

QUESTION: 2

Three point masses 1kg, 1.5 kg, 2.5 kg are placed at the vertices of a triangle with sides 3cm, 4cm and 5cm as shown in the figure. The location of centre of mass with respect to 1kg mass is :

Solution:

Take 1kg mass at origin

QUESTION: 3

In a single slit diffraction set up, second minima is observed at an angle of 60°. The expected position of first minima is

Solution:

For 2^{nd }minima

d sinθ = 2λ

sinθ = (given)

......(1)

So for 1^{st} minima is

d sinθ = λ

sinθ = (from equation (i))

θ = 25.65° (from sin table)

θ ≈ 25°

QUESTION: 4

There are two infinite plane sheets each having uniform surface charge density +σ C/m^{2}. They are inclined to each other at an angle 30° as shown in the figure. Electric field at any arbitrary point P is:

Solution:

QUESTION: 5

A parallel plate capacitor with plate area A & plate separation d is filled with a dielectric material of dielectric constant given by k = k_{0}(1 + αx). Calculate capacitance of system: (given αd << 1)

Solution:

Capacitance of element

Capacitance of element

Given αd << 1

QUESTION: 6

A long solenoid of radius R carries a time dependent current I = I_{0} t(1 – t). A ring of radius 2R is placed coaxially near its centre. During the time interval 0 ≤ t ≤ 1, the induced current I_{R} and the induced emf V_{R} in the ring vary as:

Solution:

and

QUESTION: 7

If 10% of intensity is passed from analyser, then, the angle by which analyser should be rotated such that transmitted intensity becomes zero. (Assume no absorption by analyser and polarizer).

Solution:

So θ > 45° and 90 – θ < 45º so only one option is correct i.e. 18.4º

angle rotated should be = 90° – 71.6° = 18.4°

QUESTION: 8

Three moles of ideal gas A with is mixed with two moles of another ideal gas B with . The of mixture is (Assuming temperature is constant)

Solution:

on rearranging we get

γ_{mix }= 1.42

QUESTION: 9

Given magnetic field equation is B = 3 × 10^{–8} sin(ωt + kx + φ)then appropriate equation for electric field (E) will be :

Solution:

(speed of light in vacuum)

E_{0} = B_{0}C = 3 × 10^{–8} × 3 × 10^{8}

= 9 N/C

So E = 9 sin (ωt + kx + φ)

QUESTION: 10

There is a LCR circuit, If it is compared with a damped oscillation of mass m oscillating with force constant k and damping coefficient 'b'. Compare the terms of damped oscillation with the devices in LCR circuit.

Solution:

In damped oscillation

ma + bv + kx = 0

In the circuit

Comparing equation (i) and (ii)

m = L, b = R, k = 1/c

QUESTION: 11

A lift can hold 2000kg, friction is 4000N and power provided is 60HP. (1 HP = 746W) Find the maximum speed with which lift can move up.

Solution:

QUESTION: 12

A H–atom in ground state has time period T = 1.6 × 10^{–16} sec. find the frequency of electron in first excited state

Solution:

QUESTION: 13

Magnification of compound microscope is 375. Length of tube is 150mm. Given that focal length of objective lens is 5mm, then value of focal length of eyepiece is:

Solution:

Case-I

If final image is at least distance of clear vision

Case-II

If final image is at infinity

f_{e} = 22 mm

QUESTION: 14

1 litre of a gas at STP is expanded adiabatically to 3 litre. Find work done by the gas. Given γ = 1.40 and 3^{1.4}= 4.65

Solution:

now work done

Closest ans is 90.5 J

QUESTION: 15

A string of length 60 cm, mass 6gm and area of cross section 1mm^{2} and velocity of wave 90m/s. Given young's modulus is Y = 1.6 × 10^{11} N/m^{2}. Find extension in string.

Solution:

after substituting value of μ,v,l,A and Y we get

Δl = 0.3 mm

QUESTION: 16

Which of the following gate is reversible

Solution:

A logic gate is reversible if we can recover input data from the output eg. NOT gate

QUESTION: 17

A thin uniform rod is of mass M and length L. Find the radius of gyration for rotation about an axis passing through a point at a distance of L/4 from centre and perpendicular to rod.

Solution:

QUESTION: 18

A satellite of mass 'M' is projected radially from surface of earth with speed 'u'. When it reaches a height equal to radius of earth, it ejects a rocket of mass M/10 and itself starts orbiting the earth in circular path

of radius 2R, find the kinetic energy of rocket.

Solution:

Kinetic energy

QUESTION: 19

The current 'i' in the given circuit is

Solution:

QUESTION: 20

A current carrying circular loop is placed in an infinite plane if φ_{1} is the magnetic flux through the inner region and φ_{0} is magnitude of magnetic flux through the outer region, then

Solution:

As magnetic field lines always form a closed loop, hence every magnetic field line creating magnetic flux in the inner region must be passing through the outer region. Since flux in two regions are in opposite direction,

∴ φ_{i} = - φ_{0}

*Answer can only contain numeric values

QUESTION: 21

Consider a loop ABCDEFA with coordinates A (0, 0, 0), B(5, 0, 0), C(5, 5, 0), D(0, 5, 0) E(0, 5, 5) and F(0, 0, 5). Find magnetic flux through loop due to magnetic field

Solution:

φ = (3 × 25) + (4 × 25) = 175 weber

*Answer can only contain numeric values

QUESTION: 22

A Carnot's engine operates between two reservoirs of temperature 900K and 300K. The engine performs 1200 J of work per cycle. The heat energy delivered by the engine to the low temperature reservoir in a cycle is:

Solution:

*Answer can only contain numeric values

QUESTION: 23

A non-isotropic solid metal cube has coefficients of linear expansion as 5 × 10^{–5}/°C along the x-axis and 5 × 10^{–6}/°C along y-axis and z-axis. If coefficient of volume expansion of the solid is C × 10^{–6}/°C then the value of C is

Solution:

*Answer can only contain numeric values

QUESTION: 24

A particle is released at point A. Find its kinetic energy at point P. (Given m = 1 kg and all surfaces are frictionless)

Solution:

As only consevative internal force acts upon the mass and earth system, thus we can say mechanical energy is conserved. Thus we get that net loss in PE = net gain in KE

Loss in PE = mg Δh = 1 x 10 x 1 = 10J

Thus gain in KE = 10J

*Answer can only contain numeric values

QUESTION: 25

On a photosensitive metal of area 1 cm^{2} and work function 2eV, light of intensity 6.4 × 10^{–5} W/cm^{2} and wavelength 310 nm is incident normally. If 1 out of every 10^{3} photons are successful, then number of photoelectrons emitted in one second is 10^{x}. Find x

Solution:

Energy of photon. E == 4eV > 2eV (so photoelectric effect will take place)

= 4 × 1.6 × 10^{–19} = 6.4 × 10^{–19} Joule

No. of photons falling per second

No. of photoelectron emitted per second

QUESTION: 26

Solution:

2 × 0.34 = + 1 x 0.522

E^{º}_{1} = 0.68 – 0.522

E^{º}_{1} = 0.158

QUESTION: 27

Correct order of electron gain enthalpy (kJ/mole) of F, Cl, Br, I

Solution:

QUESTION: 28

Arrange the following in order of their pK_{b} value

(B) CH_{3}–NH–CH_{3 }(C) CH_{3}–CH=NH

Solution:

Option "A" represent Guanadine, the conjugate acid of which is resonance stabilised. The option 'B' is aliphatic amine, here the 'N' is sp^{3} whereas in option 'C' the 'N' is sp^{2}, hence B is more basic than C.

QUESTION: 29

1-Methylethylene oxide Product 'X' will be –

Solution:

--^{excess HBr}--->

QUESTION: 30

Correct order of Intermolecular forces

Solution:

QUESTION: 31

Hex-3-ynal (X), formed product X will be:

Solution:

QUESTION: 32

, formed product 'X' is used as:

Solution:

Methyl orange is used as an indicator in acid base titration.

QUESTION: 33

In which of the following Saytzeff product will not be formed as major product ?

(A) (B)

(C) (D)

Solution:

QUESTION: 34

Match the column

Column-I Column-II

(A) Thiamine (P) Scurvy

(B) Riboflavin (Q) Beri Beri

(C) Pyridoxine (R) Cheilosis

(D) Ascorbic acid (S) Convulsions

Solution:

QUESTION: 35

Atomic radius of Ag is similar to

Solution:

Atomic radius of Ag is closest to Au.

QUESTION: 36

Correct IUPAC name of [Pt(NH_{3})_{2}Cl(CH_{3}NH_{2} )]Cl is:

Solution:

The IUPAC name of [Pt(NH3)2Cl(CH3NH2 )]Cl is diamminechlorido(methylamine)platinum(II)chloride.

The ligands present are ammine, chloro and methyl amine. The ligands are named according to the alphabetical order.

The prefix di indicates two.

The oxidation state of platinum is +2. The oxidation state is written in roman numerals inside parenthesis.

QUESTION: 37

Vapour pressure of pure CS_{2} and CH_{3}COCH_{3} are 512 mm of Hg and 312 mm of Hg respectively. Total vapour pressure of mixture is 600 mm of Hg then find incorrect statement:

Solution:

Above mixture of liquids show positive deviation from Raoult's Law

QUESTION: 38

Purest form of commercial iron is:

Solution:

Purest form is wrought iron.

QUESTION: 39

Mixture of above three organic compound was subjected to aq NaHCO_{3} and followed by dil NaOH. compounds which will be soluble in given solvent will be :

Solution:

QUESTION: 40

Which theory can explain bonding of Ni(CO)_{4}:

Solution:

QUESTION: 41

n = 5, m_{s} = + 1/2 How many orbitals are possible:

Solution:

QUESTION: 42

In zeolites & synthetic resin method which will be more efficient in removing permanent hardness of water :

Solution:

QUESTION: 43

Oxidation state of potassium in K_{2}O, K_{2}O_{2} & KO_{2} are respectively –

Solution:

QUESTION: 44

Decreasing order of dipole moment in CHCl_{3}, CCl_{4} & CH_{4} is –

Solution:

QUESTION: 45

Amongst the following which is not a postulate of Dalton's atomic theory

Solution:

1) Elements are composed of extremely small particles called atoms that are indivisible and indestructible

2)All atoms of a given element are identical; they have the same size, mass, and chemical properties

3) Atoms of 1 element are different from the atoms of all other elements

4)Compounds are composed of atoms of more than 1 element. The relative number of atoms of each element in a given compound is always the same.

5)Chemical reactions only involve the rearrangement of atoms. Atoms are not created or destroyed during chemical reactions.

*Answer can only contain numeric values

QUESTION: 46

Half life of _{90}Sr is 6.93 years. In a child body 1 μg of _{90}Sr dopped in place of calcium, how many years will it take to reduce its concentration by 90% (Assume no involvement of Sr in metabolism).

Solution:

*Answer can only contain numeric values

QUESTION: 47

Each of solution A and B of 100 L containing 4 g NaOH and 9.8 g H_{2}SO_{4}. Find pH of solution which is obtain by mixing 40 L solution of A and 10 L solution of B.

Solution:

*Answer can only contain numeric values

QUESTION: 48

A(l) → 2B(g)

ΔU = 2.1 kcal, ΔS = 20 cal/k, T = 300 K.

Find ΔG (in kcal)

ΔU = 2.1 kcal, ΔS = 20 cal/k, T = 300 K.

Solution:

ΔH = ΔU + ΔngRT

= 2.1 × 10^{3} + 2(2) (300)

= 2100 + 1200

= 3300 cal

ΔG = ΔH - TΔS

= 3300 – (300) (20)

= 3300 – 6000

= –2700 cals = –2.7 kcal

*Answer can only contain numeric values

QUESTION: 49

Cl_{2} on reaction with hot & conc. NaOH gives two chlorine having products X and Y. On treatment with AgNO_{3}, X gives precipitate. Determine average bond order of Cl and O bond in 'Y' ?

Solution:

*Answer can only contain numeric values

QUESTION: 50

Number of chiral centers in chloramphenicol is :

Solution:

QUESTION: 51

If f(x) is continuous and differentiable in x ∈ [ -7, 0] and f'(x) ≤ 2∈ [-7, 0], also f(-7) = -3 then range of f(-1) + f(0)

Solution:

Lets use LMVT for x ∈ [-7, -1]

Also use LMVT for x ∈ [-7, 0]

∴ f(0) + f(-1) ≤ 20

QUESTION: 52

If y = mx + 4 is common tangent to parabolas y^{2} = 4x and x^{2} = 2by. Then value of b is

Solution:

y = mx + 4 ……(i)

y^{2} = 4x tangent

y = mx + a/m

⇒ y = mx + 1/m ………(ii)

from (i) and (ii)

4 = 1/m ⇒ m =1/4

So line y = + 4 is also tangent to parabola x^{2} = 2by, so solve

⇒ 2x^{2} – bx – 16b = 0

⇒ D = 0

⇒ b^{2} - 4 × 2 × (-16b) = 0

⇒ b^{2} + 32 × 4b = 0

b = –128, b = 0 (not possible)

QUESTION: 53

If α and β are the roots of equation (k+1) tan^{2}x - √2λ tanx = 1 - k and tan^{2} (α+β) = 50. Find value of λ

Solution:

λ = 10

QUESTION: 54

Find image of point (2, 1, 6) in the plane containing points (2, 1, 0), (6, 3, 3) and (5, 2, 2)

Solution:

Plane is x + y – 2z = 3

⇒ (x, y, z) = (6, 5, -2)

QUESTION: 55

Let (x)^{k} + (y)^{k} = (a)^{k }where a, k > 0 and then find k

Solution:

k – 1 =

k == 2/3

QUESTION: 56

If g(x) = x^{2} + x – 1 and g(f(x)) = 4x^{2} – 10x + 5, then find f.

.

Solution:

QUESTION: 57

If z = x + iy and real part = 1 then locus of z is

Solution:

QUESTION: 58

Let y = f(x) is a solution of differential equation and f(0) = 0 then f(1) is equal to :

Solution:

e^{y} = t

Put x = 0, y = 0 then c = 1

e^{y–x} = x + 1

y = x + ln(x + 1)

at x = 1 , y = 1 + ln(2)

QUESTION: 59

If α is a roots of equation x^{2} + x + 1 = 0 and A = then A^{31} equal to :

Solution:

=> A^{4} = I

=> A^{30} = A^{28 }× A^{3} = A^{3}

*Answer can only contain numeric values

QUESTION: 60

The six digit numbers that can be formed using digits 1, 3, 5, 7, 9 such that each digit is used at least once.

Solution:

1, 3, 5, 7, 9

For digit to repeat we have ^{5}C_{1} choice And six digits can be arrange in ways.

Hence total such numbers =

QUESTION: 61

The area that is enclosed in the circle x^{2} + y^{2} = 2 which is not common area enclosed by y = x & y^{2} = x is

Solution:

Total area – enclosed area

QUESTION: 62

If sum of all the coefficient of even powers in (1– x + x^{2} – x^{3} ……..x^{2n}) (1 + x + x^{2 }+x^{3} ……….+ x^{2n}) is 61 then n is equal to

Solution:

Let (1– x + x^{2} …..) (1 + x + x^{2} ……) = a_{0} + a_{1} x + a_{2}x^{2} +……

put x = 1

1(2n+1) = a0 + a1 + a2 +...,.a2n ...(i)

put x = –1

(2n+1) x 1 = a0 - a1 + a2 +....a2n.....(ii)

Form (i) + (ii)

4n + 2= 2(a_{0} + a_{2} +….)

= 2 x 16

⇒ 2n+1 = 61

⇒ n = 30

*Answer can only contain numeric values

QUESTION: 63

If variance of first n natural numbers is 10 and variance of first m even natural numbers is 16 then the value of m + n is

Solution:

⇒ n^{2} - 1 = 120

⇒ n = 11

Var (2, 4, 6,.....,2m) = 16

⇒ var (1, 2,....,m) = 4

⇒ m^{2 }- 1 = 48

⇒ m = 7

⇒ m + n = 18

*Answer can only contain numeric values

QUESTION: 64

Evaluate

Solution:

Put 3^{x/2} = t

QUESTION: 65

If A(1, 1), B(6, 5), C are vertices of ΔABC. A point P is such that area of ΔPAB, ΔPAC, ΔPBC are equal, also , then length of PQ is

Solution:

P will be centroid of ΔABC

QUESTION: 66

(p → q) ∧ (q → ~p) is equivalent to

Solution:

Clearly (p → q) ∧ (q → ~p) is equivalent to ~p

QUESTION: 67

Find greatest value of k for which 49^{k} + 1 is factor of 1 + 49 + 49^{2} …..(49)^{125}

Solution:

*Answer can only contain numeric values

QUESTION: 68

If f(x) = |2 – |x – 3|| is non differentiable in x ∈ S.Then value of is

Solution:

∵ f(x) is non differentiable at x = 1,3,5

∑ f(f(x)) = f(f(1) + f(f(3)) + f (f(5))

= 1 + 1 + 1

= 3

QUESTION: 69

If system of equations

2x + 2ay + az = 0

2x + 3by + bz = 0

2x + 4cy + cz = 0

have non-trivial solution

then

Solution:

For non-trivial solution

(3bc – 4bc) – (2ac – 4ac) + (2ab – 3ab) = 0

–bc + 2ac – ab = 0

ab + bc = 2ac

a, b, c in H.P.

in A.P

QUESTION: 70

If sum of 5 consecutive terms of 'an A.P is 25 & product of these terms is 2520. If one of the terms is – 1/2 then the value of greatest term is

Solution:

Let terms be a – 2d, a –d, a, a + d , a + 2d .

sum = 25

⇒ 5a = 25

⇒ a = 5

Product = 2520

⇒ (5–2d) (5 – d) 5(5+d) (5+2d) =2520

⇒ (25 - 4d^{2}) (25 - d^{2}) = 504

⇒ 625 -100d^{2 }- 25d^{2} + 4d^{4} = 504

⇒ 4d^{4} - 125d^{2} + 625 -504 = 0

⇒ 4d^{4} - 125d^{2} + 121 = 0

⇒ 4d^{4} - 121d^{2} - 4d^{2} +121 = 0

⇒ (d^{2}-1) (4d^{2} - 121) = 0

d = ±1, d = ± 11/2

d = ±1, does not give -1/2 as a term

∴ d = 11/2

∴ Largest term = 5 + 2d = 5 + 11 = 16

QUESTION: 71

Let

lies in plane of&

of a bisectors angle between& then

Solution:

angle bisector can be or

Compare with

Not in option so now consider

Compare with

QUESTION: 72

Given f(a + b + 1 – x) = f(x) R then the value of is equal to

Solution:

x → a + b - x

[∵ put x → x + 1 in given equation]

(1) + (2)

2I =

QUESTION: 73

If distance between the foci of an ellipse is 6 and distance between its directrices is 12, then length of its latus rectum is

Solution:

2ae = 6 and

⇒ ae = 3 and a/e = 6

⇒ a^{2} = 18

⇒ b^{2} = a^{2} - a^{2}e^{2} = 18 - 9 = 9

QUESTION: 74

An unbiased coin is thrown 5 times. Let X be a random variable and k be the value assigned to X for k = 3, 4, 5 times Head occurs consecutively and otherwise the value of X is assigned –1. What is value of expectation.

Solution:

k = no. of times head occur consecutively

Now expectation

QUESTION: 75

If when then find dy/dα at α = 5π/6

Solution:

= –1 – cota

⇒ will be = 4

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