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QUESTION: 1

Two forces P and Q of magnitude 2F and 3F, respectively, are at an angle θ with each other. If the force Q is doubled, then their resultant also gets doubled. Then, the angle is :

Solution:

4F^{2} + 9F^{2} + 12F^{2} cos θ = R^{2}

4F^{2} + 36 F^{2} + 24 F^{2} cos θ = 4R^{2}

4F^{2} + 36 F^{2} + 24 F^{2} cos θ

= 4(13F^{2} + 12F^{2}cosθ) = 52 F^{2} + 48F^{2}cosθ

QUESTION: 2

The actual value of resistance R, shown in the figure is 30Ω. This is measured in an experiment as shown using the standard formula R = V/1 , where V and I are the readings of the voltmeter and ammeter, respectively. If the measured value of R is 5% less, then the internal resistance of the voltmeter is:

Solution:

0.95 × 30 = 0.05 R_{υ}

R_{υ} = 19 × 30 = 570 Ω

QUESTION: 3

An unknown metal of mass 192 g heated to a temperature of 100ºC was immersed into a brass calorimeter of mass 128 g containing 240 g of water a temperature of 8.4ºC Calculate the specific heat of the unknown metal if water temperature stabilizes at 21.5ºC (Specific heat of brass is 394 J kg^{–1} K^{–1})

Solution:

192 × S × (100 – 21.5)

= 128 × 394 × (21.5 – 8.4)

+ 240 × 4200 × (21.5 – 8.4)

⇒ S = 916

QUESTION: 4

A particle starts from the origin at time t = 0 and moves along the positive x-axis. The graph of velocity with respect to time is shown in figure. What is the position of the particle at time t = 5s?

Solution:

S = Area under graph

1/2× 2 × 2 + 2 × 2 + 3 × 1 = 9 m

QUESTION: 5

The self induced emf of a coil is 25 volts. When the current in it is changed at uniform rate from 10 A to 25 A in 1s, the change in the energy of the inductance is:

Solution:

QUESTION: 6

A current of 2 mA was passed through an unknown resistor which dissipated a power of 4.4 W. Dissipated power when an ideal power supply of 11V is connected across it is :

Solution:

P = I^{2}R

4.4 = 4 × 10^{–6} R

R = 1.1 × 10^{6} Ω

QUESTION: 7

The diameter and height of a cylinder are measured by a meter scale to be 12.6 ± 0.1 cm and 34.2 ± 0.1 cm, respectively. What will be the value of its volume in appropriate significant figures ?

Solution:

QUESTION: 8

At some location on earth the horizontal component of earth's magnetic field is 18 x 10^{-6} T. At this location, magnetic neeedle of length 0.12 m and pole strength 1.8 Am is suspended from its mid-point using a thread, it makes 45° angle with horizontal in equilibrium. To keep this needle horizontal, the vertical force that should be applied at one of its ends is :

Solution:

QUESTION: 9

The modulation frequency of an AM radio station is 250 kHz, which is 10% of the carrier wave. If another AM station approaches you for license what broadcast frequency will you allot?

Solution:

∴ Range of signal = 2250 Hz to 2750 Hz

Now check all options : for 2000 KHZ

f_{mod} = 200 Hz

∴ Range = 1800 KHZ to 2200 KHZ

QUESTION: 10

A hoop and a solid cylinder of same mass and radius are made of a permanent magnetic material with their magnetic moment parallel to their respective axes. But the magnetic moment of hoop is twice of solid cylinder. They are placed in a uniform magnetic field in such a manner that their magnetic moments make a small angle with the field. If the oscillation periods of hoop and cylinder are T_{h} and T_{c} respectively, then :

Solution:

QUESTION: 11

The electric field of a plane polarized electromagnetic wave in free space at time t= 0 is given by an expression

The magnetic field : (cis the velocity of light)

Solution:

i.e. direction of 'c'.

QUESTION: 12

Condiser the nuclear fission

Ne^{20} → 2He^{4} + C^{12}

Given that the binding energy/nucleon of Ne^{20}, He^{4} and C^{12} are, respectively, 8.03 MeV, 7.07 MeV and 7.86 MeV, identify the correct statement :

Solution:

QUESTION: 13

Two vectors have equal magnitudes. The magnitude of is 'n' times the magnitude of The angle between is :

Solution:

QUESTION: 14

A particle executes simple harmonic motion with an amplitude of 5 cm. When the particle is at 4 cm from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is :

Solution:

QUESTION: 15

Consider a Young's double slit experiment as shown in figure. What should be the slit separation d in terms of wavelength λ such that the first minima occurs directly in front of the slit (S_{1}) ?

Solution:

QUESTION: 16

The eye can be regarded as a single refracting surface. The radius of curvature of this surface is equal to that of cornea (7.8 mm). This surface separates two media of refractive indices 1 and 1.34. Calculate the distance from the refracting surface at which a parallel beam of light will come to focus.

Solution:

QUESTION: 17

Half mole of an ideal monoatomic gas is heated at constant pressure of 1atm from 20ºC to 90ºC. Work done by gas is close to :

(Gas constant R = 8.31 J /mol.K)

Solution:

QUESTION: 18

A metal plate of area 1 × 10^{–4} m^{2} is illuminated by a radiation of intensity 16 mW/m^{2}.The work function of the metal is 5eV. The energy of the incident photons is 10 eV and only 10% of it produces photo electrons. The number of emitted photo electrons per second and their maximum energy, respectively, will be :

[1 eV = 1.6 × 10^{–19}J]

Solution:

QUESTION: 19

Charges -q and +q located at A and B, respectively, constitute an electric dipole. Distance AB = 2a, O is the mid point of the dipole and OP is perpendicular to AB. A charge Q is placed at P where OP = y and y >> 2a. The charge Q experiences and electrostatic force F. If O is now moved alons the equatorial line to P' such that OP = (y/3), the force on Q will be close to (y/3 >> 2a)

Solution:

Electric field of equitorial plane of dipole

QUESTION: 20

Two stars of masses 3 × 10^{31} kg each, and at distance 2 × 10^{11}m rotate in a plane about their common centre of mass O. A meteorite passes through O moving perpendicular to the star's rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at O is : (Take Gravitational constant G = 6.67 ×10^{–11} Nm^{2} kg^{–2})

Solution:

By energy convervation between 0 & ∞.

[M is mass of star m is mass of meteroite)

QUESTION: 21

A closed organ pipe has a fundamental frequency of 1.5 kHz. The number of overtones that can be distinctly heard by a person with this organ pipe will be : (Assume that the highest frequency a person can hear is 20,000 Hz)

Solution:

For closed organ pipe, reson ate frequency is odd multiple of fundamental frequency.

∴ (2n + 1) f_{0} ≤ 20,000

(f_{o} is fundamental frequency = 1.5 KHz)

∴ n = 6

QUESTION: 22

A rigid massless rod of length 3l has two masses attached at each end as shown in the figure. The rod is pivoted at point P on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be :

Solution:

Applying torque equation about point P.

QUESTION: 23

For the circuit shown below, the current through the Zener diode is:

Solution:

Assuming zener diode doesnot undergo breakdown, current in circuit = 120/15000 = 8mA

∴ Voltage drop across diode = 80 V > 50 V.

The diode undergo breakdown.

QUESTION: 24

Four equal point charges Q each are placed in the xy plane at (0, 2), (4, 2), (4, –2) and (0, –2). The work required to put a fifth charge Q at the origin of the coordinate system will be :

Solution:

(Potential at ∞ = 0)

∴Work required to put a fifth charge Q at origin is equal to

QUESTION: 25

A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency ω. If the radius of the bottle is 2.5 cm then ω close to : (density of water = 10^{3} kg / m^{3})

Solution:

Extra Boyant force = δAxg

B_{0} + B × mg = ma

B = ma

QUESTION: 26

A parallel plate capacitor having capacitance 12 pF is charged by a battery to a potential difference of 10 V between its plates. The charging battery is now disconnected and a porcelain slab of dielectric constant 6.5 is slipped between the plates the work done by the capacitor on the slab is :

Solution:

QUESTION: 27

Two kg of a monoatomic gas is at a pressure of 4 × 10^{4} N/m^{2} . The density of the gas is 8 kg /m^{3}. What is the order of energy of the gas due to its thermal motion ?

Solution:

Thermal energy of N molecule

order will 10^{4}

QUESTION: 28

A particle which is experiencing a force, given by undergoes a displacement of If the particle had a kinetic energy of 3 J at the beginning of the displacement, what is its kinetic energy at the end of the displacement ?

Solution:

QUESTION: 29

The Wheatstone bridge shown in Fig. here, gets balanced when the carbon resistor used as R_{1} has the colour code ( Orange, Red, Brown). The resistors R_{2} and R_{4} are 80Ω and 40Ω, respectively. Assuming that the colour code for the carbon resistors gives their accurate values, the colour code for the carbon resistor, used as R_{3}, would be :

Solution:

QUESTION: 30

Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is :

Solution:

For Ball

using parallel axis theorem.

QUESTION: 31

An ide al ga s under goes isothe rma l compression from 5 m^{3} against a constant external pressure of 4 Nm^{–2}. Heat released in this process is used to increase the temperature of 1 mole of Al. If molar heat capacity of Al is 24 J mol^{–1} K^{–1}, the temperature of Al increases by :

Solution:

Work done on isothermal irreversible for ideal gas

= –P_{ext} (V_{2 }– V_{1})

= –4 N/m^{2} (1m^{3} – 5m^{3})

= 16 Nm

Isothermal process for ideal gas

ΔU = 0

q = –w

= –16 Nm

= – 16 J

Heat used to increase temperature of Aℓ

q = n C_{m} ΔT

QUESTION: 32

The 71^{st }electron of an element X with an atomic number of 71 enters into the orbital :

Solution:

Filling of electrons usually follow many rules, but what orbit the electron will enter is given the **Aufbau rule.**

This rule states that electrons will be filled in the atomic orbitals which are low in energy than the atomic orbitals which have high energy. **The filling of atomic orbitals follow the order 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s and so on....**

For the element X having atomic number 71 has 71 electrons.

**The electronic configuration of this element is: **

**From the above configuration, it is visible that the 71st electron is entering the 5d orbital.**

QUESTION: 33

The number of 2-c entre-2-electron and 3-centre-2-electron bonds in B_{2}H_{6}, respectively, are :

Solution:

QUESTION: 34

The amount of sugar (C_{12}H_{22}O_{11}) required to prepare 2 L of its 0.1 M aqueous solution is :

Solution:

wt (C_{12}H_{22}O_{11}) = 68.4 gram

QUESTION: 35

Among the following reactions of hydrogen with halogens, the one that requires a catalyst is :

Solution:

QUESTION: 36

Sodium metal on dissolution in liquid ammonia gives a deep blue solution due to the formation of:

Solution:

QUESTION: 37

What will be the major product in the following mononitration reaction?

Solution:

amine is o-p directing

QUESTION: 38

In the cell Pt(s)|H_{2}(g, 1bar|HCl (aq)|Ag(s)|Pt(s) the cell potential is 0.92 when a 10^{–6} molal HCl solution is used. THe standard electrode potential of (AgCl/Ag,Cl^{–}) electrode is :

Solution:

QUESTION: 39

The major product of the following recation is:

Solution:

QUESTION: 40

The pair that contains two P–H bonds in each of the oxoacids is

Solution:

QUESTION: 41

The major product of the following reaction is:

Solution:

SN^{2} reaction

QUESTION: 42

The difference in the number of unpaired electrons of a metal ion in its high-spin and low-spin octahedral complexes is two. The metal ion is :

Solution:

Co^{2+} -->d^{7}

hs, n = 3,ls, n = 1

QUESTION: 43

A compound of formula A_{2}B_{3} has the hcp lattice. Which atom forms the hcp lattice and what fraction of tetrahedral voids is occupied by the other atoms :

Solution:

A_{2}B_{3} has HCP lattice

If A form HCP, then 3^{th}/4 of THV must occupied by B to form A_{2}B_{2}

If B form HCP, then 1^{th}/3 of THV must occupied

by A to form A_{2}B_{3}

QUESTION: 44

The reaction that is NOT involved in the ozone layer depletion mechanism in the stratosphere is:

Solution:

QUESTION: 45

The process with negative entropy change is :

Solution:

N_{2}(g) + 3H_{2}(g) ⇔ 2NH_{3}(g) ; Δn_{g} < 0

QUESTION: 46

The major product of the following reaction is:

Solution:

QUESTION: 47

A reaction of cobalt(III ) chloride and ethylenediamine in a 1:2 mole ratio generates two isomeric products A (violet coloured) B (green coloured). A can show optical activity, B is optically inactive. What type of isomers does A and B represent ?

Solution:

[Co(Cn)_{2} Cl_{2}]Cl

cis --> Optically active

rans --> Optically in active

QUESTION: 48

The major product obtained in the following reaction is :

Solution:

QUESTION: 49

Which of the following tests cannot be used for identifying amino acids ?

Solution:

QUESTION: 50

What is the IUPAC name of the following compound ?

Solution:

QUESTION: 51

Which is the most suitable reagent for the following transformation?

Solution:

QUESTION: 52

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

Solution:

QUESTION: 53

In the reaction of oxalate with permaganate in acidic medium, the number of electrons involved in producing one molecule of CO_{2} is :

Solution:

10 e^{–} trans for 10 molecules of CO_{2} so per molecule of CO_{2} transfer of e^{–} is '1'

QUESTION: 54

5.1g NH_{4}SH is introduced in 3.0 L evacuated flask at 327°C. 30% of the solid NH_{4}SH decomposed to NH_{3} and H_{2}S as gases. The K_{p} of the reaction at 327°C is (R = 0.082 L atm mol^{–1}K^{–1}, Molar mass of S = 32 g mol^{/01}, molar mass of N = 14g mol^{–1})

Solution:

NH_{4}SH(s)⇔ NH_{3} (g) H_{2}S(g)

α = 30% = .3

so number of moles at equilibrium

Now use PV = nRT at equilibrium

P_{total} × 3 lit = (.03 + .03) × .082 × 600

P_{total} = .984 atm

At equilibrium

QUESTION: 55

The electrolytes usually used in the electroplating of gold and silver, respectively, are :

Solution:

QUESTION: 56

Elevation in the boiling point for 1 molal solution of glucose is 2 K. The depression in the freezing point of 2 molal solutions of glucose in the same solvent is 2 K. The relation between K_{b} and K_{f} is:

Solution:

QUESTION: 57

An aromatic compound 'A' having molecular formula C_{7}H_{6}O_{2} on treating with aqueous ammonia and heating forms compound 'B'. The compound 'B' on reaction with molecular bromine and potassium hydroxide provides compound 'C' having molecular formula C_{6}H_{7}N. The structure of 'A' is :

Solution:

QUESTION: 58

The ground state energy of hydrogen atom is –13.6 eV. The energy of second excited state He^{+} ion in eV is :

Solution:

QUESTION: 59

For an elementary chemical reaction,

the expression for is :

Solution:

QUESTION: 60

Haemoglobin and gold sol are examples of :

Solution:

Haemoglobin→ positive sol

Ag - sol → negative sol

QUESTION: 61

Let If R(z) and I[z] respectively denote the real and imaginary parts of z, then :

Solution:

QUESTION: 62

Let a_{1},a_{2},a_{3},....a_{10} be in G.P. with a_{i}; > 0 for i = 1,2,...., 10 and S be the set of pairs (r,k), r k∈N (the set of natural numbers) for which

Then the number of elements in S, is :

Solution:

Apply

C_{3} → C_{3} – C_{2}

C_{2} → C_{2} – C_{1}

We get D = 0

QUESTION: 63

The positive value of λ for whic h the co-efficient of x^{2} in the expression is 720, is :

Solution:

QUESTION: 64

The value of is :

Solution:

QUESTION: 65

The value of where [t ] denotes the greatest integer less than or equal to t, is :

Solution:

QUESTION: 66

If the probability of hitting a target by a shooter, in any shot, is 1/3, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than 5/6, is :

Solution:

n_{min} = 5

QUESTION: 67

If me an and standa rd devia tion of 5 observations x_{1}, x_{2}, x_{3},x_{4},x_{5} are 10 and 3, respectively, then the variance of 6 observations x_{1},x_{2}, ....,x_{5} and –50 is equal to :

Solution:

= 507.5

QUESTION: 68

The length of the chord of the parabola x^{2} = 4y having equation x -√2y + 4√2 = 0 is:

Solution:

Solving together we get

Similarly,

QUESTION: 69

Let where b > 0. Then the minimum value of is :

Solution:

|A| = 2(2b^{2} + 2 – b^{2}) – b(2b – b) + 1 (b^{2} – b^{2} – 1)

|A| = 2(b^{2} + 2) – b^{2} – 1

|A| = b^{2} + 3

QUESTION: 70

The ta ngent to the cur ve, passing through the point (1,e) also passes through the point :

Solution:

T : y – e = 3e (x – 1)

y = 3ex – 3e + e

y = (3e)x – 2e

QUESTION: 71

The number of values of θ∈(0,π) for which the system of linear equations

x + 3y + 7z = 0

–x + 4y + 7z = 0

(sin 3θ)x + (cos 2θ) y + 2z = 0

has a non-trivial solution, is :

Solution:

(8 – 7 cos 2θ) – 3(–2 – 7 sin 3θ) + 7 (– cos 2θ – 4 sin 3θ) = 0

14 – 7 cos 2θ + 21 sin 3θ – 7 cos 2θ – 28 sin 3θ = 0

14 – 7 sin 3θ – 14 cos 2θ = 0

14 – 7 (3 sin θ – 4 sin^{3} θ) – 14 (1 – 2 sin^{2} θ) = 0

–21 sin θ+ 28 sin^{3} θ + 28 sin^{2} θ = 0

7 sin θ [–3 + 4 sin^{2} θ + 4 sin θ] = 0

sin θ,

4 sin^{2} θ + 6 sin θ – 2 sin θ – 3 = 0

2 sin θ(2 sin θ + 3) – 1 (2 sin θ + 3) = 0

Hence, 2 solutions in (0, π) Option (4)

QUESTION: 72

If then f'(1/2) is :

Solution:

Differentiate w.r.t. 'x'

f(x) = 2x + 0 – x^{2} f(x)

QUESTION: 73

Let f : (–1,1) → R be a function defined by If K be the set of all points at which f is not differentiable, then K has exactly :

Solution:

f : (–1, 1) → R

Non-derivable at 3 points in (–1, 1) Option (1)

QUESTION: 74

Let where r ≠ ±1. Then S represents :

Solution:

QUESTION: 75

If then K is equal to :

Solution:

∴ K = 2^{25}

QUESTION: 76

Let N be the set of natural numbers and two functions f and g be defined as f,g : N→N such that :

and g(n) = n–(–1)^{n}. The fog is :

Solution:

∴ many one but onto Option (4)

QUESTION: 77

The values of λ such that sum of the squares of the roots of the quadratic equation, x^{2} + (3 – λ) x + 2 = λ has the least value is :

Solution:

α + β = λ– 3

αβ = 2 – λ

α^{2} + β^{2} = (α + β)^{2} – 2αβ = (λ – 3)^{2} – 2(2 – λ)

= λ^{2} + 9 – 6λ – 4 + 2λ

= λ^{2} – 4λ + 5

= (λ – 2)^{2} + 1

∴ λ = 2

QUESTION: 78

Two vertices of a triangle are (0,2) and (4,3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?

Solution:

⇒ 3b – 6 = –4a ⇒ 4a + 3b = 6 ……(ii)

From (i) and (ii)

∴ II^{nd} quadrant.

QUESTION: 79

Two sides of a parallelogram are along the lines, x + y = 3 and x – y + 3 = 0. If its diagonals intersect at (2,4), then one of its vertex is:

Solution:

C ⇒ (4, 5)

Now equation of BC is x – y = –1

and equation of CD is x + y = 9

Solving x + y = 9 and x – y = –3

Point D is (3, 6)

QUESTION: 80

Let and be two given vectors where vectors arenon-collinear. The value of λ for which vectors are collinear, is :

Solution:

QUESTION: 81

The value of is:

Solution:

(Where tanA=20 , tanB=1)

QUESTION: 82

With the usua l notation, in Δ ABC, if ∠A + ∠B = 120^{0}, a = √3 + 1 and √3 - 1, then the ratio ∠A :∠B, is :

Solution:

QUESTION: 83

The plane which bisects the line segment joining the points (–3,–3,4) and (3,7,6) at right angles, passes through which one of the following points?

Solution:

p : 3(x – 0) + 5 (y – 2) + 1 (z – 5) = 0

3x + 5y + z = 15

∴ Option (2)

QUESTION: 84

Consider the following three statements :

P : 5 is a prime number.

Q : 7 is a factor of 192.

R : L.C.M. of 5 and 7 is 35.

Then the truth value of which one of the following statements is true ?

Solution:

It is obvious

∴ Option (4)

QUESTION: 85

On which of the following lines lies the point of intersection of the line, and the plane, x + y + z = 2 ?

Solution:

General point on the given line is

x = 2λ + 4

y = 2λ + 5

z = λ + 3

Solving with plane,

2λ + 4 + 2λ + 5 + λ + 3 = 2

5λ + 12 = 2

5λ = –10

∴ Option (3)

QUESTION: 86

Let f be a differentiable function such that

Solution:

∴ Option (1)

QUESTION: 87

A helicopter is flying along the curve given by y – x^{3/2} = 7, (x ≥ 0). A soldier positioned at the point (1/2,7) wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :

Solution:

Option (3)

QUESTION: 88

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

Solution:

Put x^{3} = t

3x^{2} dx = dt

∴ f(x) = –1 – 4x^{3}

Option (1)

(From the given options (1) is most suitable)

QUESTION: 89

The curve a mongst the family of curves, represented by the differential equation, (x^{2} – y^{2})dx + 2xy dy = 0 which passes through (1,1) is :

Solution:

(x^{2} – y^{2}) dx + 2xy dy = 0

Solving we get,

ln(v^{2} + 1) = –ln x + C

(y^{2} + x^{2}) = Cx

1 + 1 = C ⇒ C = 2

y^{2} + x^{2}= 2x

∴ Option (2)

QUESTION: 90

If the area of an equilateral triangle inscribed in the circle, x^{2} + y^{2} + 10x + 12y + c = 0 is 27√3 sq. units then c is equal to :

Solution:

C = 25

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