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QUESTION: 1

A radioactive nuclei with decay constant 0.5/s is being produced at a constant rate of 100 nuclei/s. If at t=0 there were no nuclei, the time when there are 50 nuclei is:

Solution:

QUESTION: 2

Match the List - I (Phenomenon associated with electromagnetic radiation) with List - II (Part of electromagnetic spectrum) and select the correct code from the choices given below the lists :

Solution:

QUESTION: 3

A parallel plate capacitor is made of two plates of length l, width ω and separated by distance d. A dielectric slab (dielectric constant K) that fits exactly between the plates is held near the edge of the plates. It is pulled into the capacitor by a force where U is the energy of the capacitor when dielectric is inside the capacitor up to distance x (See figure). If the charge on the capacitor is Q then the force on the dielectric when it is near the edge is

Solution:

C = C_{1} + C_{2}

QUESTION: 4

A cone of base radius R and height h is located in a uniform electric field parallel to its base. The electric flux entering the cone is:

Solution:

QUESTION: 5

Three identical bars A, B and C are made of different magnetic materials. When kept in a uniform magnetic field, the field lines around them look as follows :

Make the correspondence of these bars with their material being diamagnetic (D) ferromagnetic (F) and paramagnetic (P)

Solution:

QUESTION: 6

The average mass of rain drops is 3.0 × 10^{–5} kg and their average terminal velocity is 9 m/s. Calculate the energy transferred by rain to each square meter of the surface at a place which receives 100 cm of rain in a year.

Solution:

= 500 × 81

= 40500 J

= 4.05 × 10^{4} J

QUESTION: 7

An ideal monoatomic gas is confinedinacy linder by a spring loaded piston of cross section 8.0 × 10^{–3} m^{2}. Initially, the gas is at 300K and occupies a volume of 2.4 × 10^{–3} m^{3} and the spring is in its relaxed state as shown in figure. The gas is heated by a small heater until the piston moves out slowly by 0.1 m. The force constant of the spring is 8000 N/m and the atmospheric pressure is 1.0 × 10^{5} N/m^{2} The cylinder and the piston are thermally insulated.

The piston and the spring are massless and there is no friction between the piston and the cylinder. The final temperature of the gas will be: (Neglect the heat loss through the lead wires of the heater. The heat capacity of the heater coil is also negligible):

Solution:

A = 8 × 10^{-3} m^{2}

T_{1} = 300 K

V_{1} = 2.4 × 10^{-3} m^{3}

V_{2} = V_{1} + AΔx

= 2.4 × 10^{-3} × 8 × 10^{-3} × 0.1

= 3.2 × 10^{-3} m3

K = 8000 N/m

T_{2} = ?

P_{1} = 10^{5}N/m^{2 }

QUESTION: 8

The angular frequency of the damped oscillator is given by where k is the spring constant, m is the mass of the oscillator and r is the damping constant. If the ratio is 8%, the change in time period compared to the undamped oscillator is approximate as follows:

Solution:

QUESTION: 9

A coil of circular cross-section having 1000 turns and 4 cm^{2} face area is placed with its axis parallel to a magnetic field which decreases by 10^{-2} Wb m^{-2} in 0.01 s. The e.m.f. induced in the coil is:

Solution:

QUESTION: 10

Three straight parallel current carrying conductors are shown in the figure. The force experienced by the middle conductor of length 25 cm is:

Solution:

QUESTION: 11

A body of mass 5kg under the action of constant force has velocity at t = 0 s as and at t = 10 s as :-

Solution:

QUESTION: 12

In the circuit diagrams (A, B, C, and D) shown below, R is a high resistance and S is a resistance the order of galvanometer G. The correct circuit, corresponding to the half deflection method for finding the resistance and figure of merit of the galvanometer, is the circuit labelled as:

Solution:

QUESTION: 13

A tank with a small hole at the bottom has been filled with water and kerosene (specific gravity 0.8). The height of water is 3m and that of kerosene 2m. When the hole is opened the velocity of fluid coming out from it is nearly:

(take g = 10 ms^{-2} and density of water = 10^{3} kg m^{-3})

Solution:

1000 × 10 × 3 + 800 × 10 × 2 = 1/2 × 1000 v^{2}

QUESTION: 14

A photon of wavelength λ is scattered from an electron, which was at rest. The wavelength shift Δλ is three times of λ and the angle of scattering θ is 60°. The angle at which the electron recoiled is φ The value of tan φ is:

(electron speed is much smaller than the speed of light)

Solution:

QUESTION: 15

The Bulk moduli of Ethanol, Mercury and Water are given as 0.9, 25 and 2.2 respectively in units of 10^{9 }Nm^{-2}. For a given value of pressure, the fractional compression in volume is Which of the following statements about for these three liquids is correct? :-

Solution:

QUESTION: 16

A hot body, obeying Newton's law of cooling is cooling down from its peak value 80ºC to an ambient temperature of 30ºC. It takes 5 minutes in cooling down from 80°C to 40°C. How much time will it take to cool down from 62°C to 32°C? (Given ln 2 = 0.693, ln 5 = 1.609)

Solution:

ln 5 = 5c = 1.609

ln 16 = ct = 4 × 0.693

t = 8.6 min

QUESTION: 17

A Zener diode is connected to a battery and a load as shown below :

The currents I, I_{Z} and I_{L} are respectively.

Solution:

QUESTION: 18

An air bubble of radius 0.1 cm is in a liquid having surface tension 0.06 N/m and density 10^{3} kg/m^{3}. The pressure inside the bubble is 1100 Nm^{-2} greater than the atmospheric pressure. At what depth is the bubble below the surface of the liquid? (g = 9.8 ms^{-2}):-

Solution:

= 980

QUESTION: 19

In the circuit shown, current (in A) through the 50 V and 30 V batteries are, respectively:-

Solution:

KVL in loop abgha

20 I_{1} = 50

I_{1} = 2.5 A

KVL in loop abcdefgha

50 – 5I_{2} – 30 – 5I_{2} = 0

I_{2} = 2A

KVL in loop cdefc

30 = 10 (I_{2} + I_{3})

⇒ I_{2} + I_{3} = 3

I_{3} = 3 – 2

= 1A

∴ Current through 50 V battery is = I_{1} + I_{2}

= 2.5 + 2.0

= 4.5 A

current through 30V battery = I_{3} = 1A

QUESTION: 20

During an adiabatic compression, 830J of work is done on 2 moles of a diatomic ideal gas to reduce its volume by 50%. The change in its temperature is nearly: (R = 8.3 JK^{-1}mol^{-1})

Solution:

QUESTION: 21

A small ball of mass m starts at a point A with speed v_{0} and moves along a frictionless track AB as shown. The track BC has coefficient of friction μ. The ball comes to stop at C after traveling a distance L which is:

Solution:

QUESTION: 22

In a compound microscope the focal length of objective lens is 1.2 cm and focal length of eye piece is 3.0 cm. When object is kept at 1.25 cm in front of objective, final image is formed at infinity. Magnifying power of the compound microscope should be :

Solution:

= 200

QUESTION: 23

A thin bar of length L has a mass per unit length λ, that increases linearly with distance from one end. If its total mass is M and its mass per unit length at the lighter end is λ_{0}, then the distance of the centre of mass from the lighter end is :

Solution:

Mass per unit lengh = λ_{0} + kx

substitute'k'

QUESTION: 24

In terms of resistance R and time T, the dimensions of ratio of the permeability μ and permittivity ε is :-

Solution:

QUESTION: 25

The initial speed of a bullet fired from a rifle is 630 m/s. The rifle is fired at the centre of a target 700 m away at the same level as the target. How far above the centre of the target the rifle must be aimed in order to hit the target

Solution:

QUESTION: 26

An object is located in a fixed position in front of a screen. Sharp image is obtained on the screen for two positions of a thin lens separated by 10 cm. The size of the images in two situations are in the ratio 3 : 2. What is the distance between the screen and the object?

Solution:

D = 99 cm

QUESTION: 27

Two monochromatic light beams of intensity 16 and 9 units are interfering. The ratio of intensities of bright and dark parts of the resultant pattern is :

Solution:

QUESTION: 28

From a sphere of mass M and radius R, a smaller sphere of radius R/2 is carved out such that the cavity made in the original sphere is between its centre and the periphery. (See figure). For the configuration in the figure where the distance between the centre of the original sphere and the removed sphere is 3R, the gravitational force between the two spheres is:

Solution:

Due to cavity field at P is

QUESTION: 29

An electromagnetic wave of frequency 1 x 10^{14} hertz is propagating along z-axis. The amplitude of electric field is 4 V/m. If ε_{0} = 8.8 × 10^{-12} C^{2}/N-m^{2}, then average energy density of electric field will be:

Solution:

QUESTION: 30

Two factories are sounding their sirens at 800 Hz. A man goes from one factory to other at a speed of 2 m/s. The velocity of sound is 320 m/s. The number of beats heard by the person in one second will be :

Solution:

QUESTION: 31

The appearance of colour in solid alkali metal halides is generally due to:

Solution:

QUESTION: 32

Complete reduction of benzene-diazonium chloride with Zn/HCl gives:

Solution:

QUESTION: 33

Which of the following statements about Na_{2}O_{2} is not correct ?

Solution:

Na_{2}O_{2} is a peroxide which is occupied all paired electrons with π*2px & π*2py.

QUESTION: 34

In allene (C_{3}H_{4}), the type(s) of hybridization of the carbon atoms is (are):

Solution:

QUESTION: 35

In the reaction of formation of sulphur trioxide by contact process the rate of reaction was measured as The rate of reaction in terms of [SO_{2}] in mol L^{–1s–1} will be

Solution:

= –5 × 10^{–4}

QUESTION: 36

Based on the equation

the wavelength of the light that must be absorbed to excite hydrogen electron from level n = 1 to level n = 2 will be (h = 6.625 × 10^{–34} Js, C = 3 × 10^{8} ms^{–1})

Solution:

= 13.25 × 10^{–8}

= 1.325 × 10^{–7} m

QUESTION: 37

Given :-

Fe^{3+}(aq) + e^{-} → Fe^{2+} (aq); E° = +0.77 V

Al^{3+}(aq) + 3e^{-} → Al (s); E° = -1.66 V

Br_{2}(aq) + 2e^{-} → 2Br^{-} ; E° = + 1.09 V

Considering the electrode potentials, which of the following represents the correct order of reducing power?

Solution:

QUESTION: 38

Consider the following equilibrium White precipitate of AgCl appears on adding which of the following?

Solution:

QUESTION: 39

Tischenko reaction is a modification of

Solution:

QUESTION: 40

Which one of the following does not have a pyramidal shape?

Solution:

In N(SiH_{3})_{3} lp present on nitrogen atom of 2nd shall has greater donating tendency to vacant 3dorbital of 'Si' but not this donating tendency to vacant 3d-orbital of 'Si' but not this donating tendency with P, due to 3^{rd} pd element.

QUESTION: 41

The following reaction

is known as

Solution:

In (SiH_{3}) N has strong back bonding tendency than other gsap.

QUESTION: 42

Chlorobenzne reacts with trichloro acetaldehyde in the presence of H_{2}SO_{4 }

The major product formed is

Solution:

QUESTION: 43

Shapes of certain interhalogen compounds are stated below. Which one of them is not correctly stated?

Solution:

BrF_{5} has square pyramidal shape (sp^{3}d^{2}) with one lone pair at below the basal plane.

QUESTION: 44

Which one of the following statements is not correct?

Solution:

QUESTION: 45

Which of the following series correctly represents relations between the elements from X to Y?

Solution:

e^{l} on moving down the gsaap shell number increases so its radii also increase from "C to Ge".

QUESTION: 46

Which of the following statement s about the depletion of ozone layer is correct?

Solution:

QUESTION: 47

The initial volume of a gas cylinder is 750.0 mL. If the pressure of gas inside the cylinder changes from 840.0 mm Hg to 360.0 mm Hg, the final volume the gas will be

Solution:

P_{1}V_{1} = P_{2}V_{2 }

⇒ 840 × 750 = 360 × V_{2}

= 1750 ml

= 1.75 L

QUESTION: 48

If λ_{0} and λ be threshold wavelength and wavelength of incident light, the velocity of photoelectron ejected from the metal surface is:

Solution:

QUESTION: 49

Which one of the following is used as Antihistamine?

Solution:

QUESTION: 50

The molar heat capacity (C_{p}) of CD_{2}O is 10 cals at 1000 K. The change in entropy associated with cooling of 32 g of CD_{2}O vapour from 1000 K to 100 K at constant pressure will be (D = deuterium, at. mass = 2u)

Solution:

QUESTION: 51

The gas liberated by the electrolysis of Dipotassium succinate solution is

Solution:

QUESTION: 52

Which of the following name formula combinations is not correct?

Solution:

Correct Name of [Mn(CN)_{5}]^{2-} is Pentacyanomagnate (III) ion.

QUESTION: 53

For the reaction, 2N_{2}O_{5} → 4NO_{2} + O_{2}, the rate equation can be expressed in two ways k and k' are related as

Solution:

Now

QUESTION: 54

An organic compound A, C_{5}H_{8}O; reacts with H_{2}O, NH_{3} and CH_{3}COOH as described below:

A is

Solution:

QUESTION: 55

A gaseous compound of nitrogen and hydrogen contains 12.5%(by mass) of hydrogen. The density of the compound relative to hydrogen is 16. The molecular formula of the compound is :

Solution:

Empirical formula = NH_{2}

Since Vapour density = 16

∴ mol. wt. = 32

∵ Molecular formula = n × Emp. formula = 2 × NH_{2}

= N_{2}H_{4}

QUESTION: 56

Assuming that the degree of hydrolysis is small, the pH of 0.1 M solution of sodium acetate (K_{a} = 1.0 × 10^{–5}) will be

Solution:

QUESTION: 57

The reagent needed for converting is

Solution:

QUESTION: 58

Consider the coordination compound, [Co(NH_{3})_{6}]Cl_{3}. In the formation of this complex, the species which acts as the Lewis acid is:

Solution:

Metalcation i.e. Ca^{3+} act as a lewis acid which accept lone pair from ligands of NH_{3}.

QUESTION: 59

The correct order of bond dissociation energy among N_{2}, O_{2}, O_{2}^{– }is shown in which of the following arrangements?

Solution:

Bond energy ∝ Bond order bondorder:-

N_{2} = Nb = 10, Na = 4

Hence the order of B.O.

QUESTION: 60

In some solutions, the concentration of H_{3}O^{+} remains constant even when small amounts of strong acid or strong base are added to them. These solutions are known as:-

Solution:

QUESTION: 61

If X has a binomial distribution, B(n, p) with parameters n and p such that P(X = 2) = P(X = 3), then E(X), the mean of variable X, is

Solution:

P(x = 2) = P (x = 3)

QUESTION: 62

The integral is equal to

Solution:

= θ tan^{2}θ - tanq + θ + C

= tan^{-1} x . x^{2} - x + tan^{-1 }x + C

= - x + (1 + x^{2}) tan^{-1 }x + C

QUESTION: 63

Let f be an odd function defined on the set of real numbers such that for x ≥ 0,

f(x) = 3sinx + 4cosx

Then f(x) at

Solution:

f(–x) = – f(x) as f(x) is odd function

QUESTION: 64

The plane containing the line and parallel to the line passes through the point

Solution:

Normal vector =

point (1,2,3) lies in plane so equation of plane = 5(x – 1) –1(y – 2) –1(z – 3) = 0

5x – y – z = 0

so option [1] is correct

QUESTION: 65

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius = √3 is

Solution:

h^{2} + r^{2 }= 3

r^{2} = 3 – h^{2} ...(1)

∴ V = πr^{2} . 2h

= 2π (r^{2} h)

V = 2π (3h - h^{3})

∴ r^{2} = 3 – h^{2}

r^{2} = 3 – 1 = 2

So V_{max} = 2π (2 × 1)

= 4π

QUESTION: 66

The proposition ~(pv~q)v~ (pvq) is logically equivalent to

Solution:

QUESTION: 67

If the general solution of the differential equation for some function is given by y ln |cx| = x, where c is an arbitrary constant, then (2) is equal to

Solution:

...(1)

is solution of y ln |cx| = x ...(2)

d.w.r. to x

use y' in equation (1)

QUESTION: 68

For the curve y = 3 sinθ cosθ , x = e^{θ}sinθ, 0 ≤ θ ≤ π, the tangent is parallel to x-axis when θ is

Solution:

cos 2θ = 0

Reject (3π/4) because at q = 3π/4

Denomentor cosθ + sinθ = 0

So θ = π/4 ans

QUESTION: 69

is equal to

Solution:

Integrate by parts

= – 9e

QUESTION: 70

Let f(x) = x|x|, g(x) = sinx and h(x) = (gof)(x). Then

Solution:

h' (0) = h' (0^{+}) = h' (0^{–})

so h'(x) is continuous at x = 0

h"(0+) ≠ h"(0–) so h"(x) is not continuous at

x = 0

so h'(x) is not differentiable at x = 0

QUESTION: 71

A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that x∈A is

Solution:

Total non empty subsects = 2^{7} –1 = 127 Let x ∈ S also present in A

So no. of A's containg x = 2^{6}

QUESTION: 72

then k is equal to

Solution:

1. (2 + k) = 5

K = 3

QUESTION: 73

Let P(3secθ, 2tanθ) and Q(3secφ, 2tanφ) where θ + φ = π/2 be two distinct points on the hyperbola Then the ordinate of the point of intersection of the normals at P and Q is

Solution:

p (3sec θ, 2tan θ) Q = (3 sec φ, 2 tan φ) θ + φ = π/2 Q = (3 cosec θ, 2 cot θ)

Equation of normal at p

= 3x cos θ + 2y cot θ = 13

= 3x sin θ cos θ + 2y cos θ = 13 sin θ ...(1)

equation of normal at Q ⇒

= 3x sin θ + 2y tan θ = 13

= 3x sin θ cos θ + 2y sin θ = 13 cos θ ...(2)

(1)-(2) ⇒

2y (cos θ – sin θ) = 13 (sin θ - cos θ)

2y = – 13 ⇒ y = -13/2

QUESTION: 74

In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is

Solution:

Let first term is a & C.R = r

a^{2} r^{4} = 49 ⇒ ar^{2} = 7, -7

also given that a + ar^{2} = 35

if ar^{2} = 7 ⇒ a = 35 - 7 = 28

if ar^{2} = - 7 ⇒ a = 35 + 7 = 42

but if a = 42 then r^{2} = -7/42

which is not possible so

a = 28

QUESTION: 75

Let A {2, 3, 5}, B (– 1, 3, 2) and C(λ, 5, μ) be the vertices of a ΔABC. If the median through A is equally inclined to the coordinate axes, then

Solution:

A = (2, 3, 5)

λ = 7 & µ = 10

QUESTION: 76

The set of all real values of λ for which exactly two common tangents can be drawn to the circles

x^{2} + y^{2} - 4x - 4y + 6 = 0

and x^{2} + y^{2} - 10x - 10y + λ = 0 is the interval

Solution:

C_{1} (2, 2)C_{2} (5, 5)

λ ∈ (18, 42)

QUESTION: 77

If z_{1}, z_{2} and z_{3}, z_{4} are 2 pairs of complex conjugate numbers, then

Solution:

QUESTION: 78

If 2cosθ + sinθ = then 7cosθ + 6sinθ is equal to

Solution:

2 cos θ + sin θ = 1 ...(1)

7 cos θ + 6 sin θ = k (let) ...(2)

QUESTION: 79

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is :

Solution:

Eight digit no divisible by 9 i.e. sum of digits divisible by 9

(i) Total no formed by 1,2,3,4,5,6,7,8 = 81

(ii) Total no formed by 0,2,3,4,5, 6,7,9 = 7×7!

(iii) Total no formed by 1,0,3,4,5,6,9,8 = 7×7!

(iv) Total no formed by 1,2,0,4,5,9,7,8 = 7×7!

(v) Total no formed by 1,2,3,0,5,6,7,8 = 7×7!

8! + 28 × 7 !

= 36 × 7 !

QUESTION: 80

The coefficient of x^{50} in the binomial expansion of

(1 + x)^{1000} + x(1 + x)^{999} + x^{2}(1 + x)^{998} + ......... + x^{1000} is

Solution:

Coefficient of x^{50} e^{n}

= (1 + x)^{1001} – x^{1001}

QUESTION: 81

Let L_{1} be the length of the common chord of the curves x^{2} + y^{2} = 9 and y^{2} = 8x, and L_{2} be the length of the latus rectum of y^{2} = 8x then

Solution:

x^{2} + y^{2} = 9 & y^{2} = 8x

L_{2} = L.R. of y^{2} = 8x ⇒ L_{2} = 8

Solve x^{2} + 8x = 9 ⇒ x = 1, – 9

x = – 9 reject

∴ y^{2} = 8x so y^{2} = 8

L_{1} < L_{2}

QUESTION: 82

The angle of elevation of the top of a vertical tower from a point P on the horizontal ground was observed to be α. After moving a distance 2 metres from P towards the foot of the tower, the angle of elevation changes to β. Then the height (in metres) of the tower is

Solution:

From figure

QUESTION: 83

Two ships A and B are sailing straight away from a fixed point O along routes such that ∠AOB is always 120°. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr)

Solution:

Let at any time t

OA = x OB = y

AB^{2} = x^{2} + y^{2} + xy ...(1)

D.w.R. To . t

...(2)

QUESTION: 84

If α and β are roots of the equation for some k, and α^{2} + β^{2} = 66, then α^{3} + β^{3} is equal to

Solution:

QUESTION: 85

Let for i = 1, 2, 3 p_{i}(x) be a polynomial of degree 2 in x, p_{i}'(x) and p_{i}''(x) be the first and second order derivatives of p_{i}(x) respectively.

and B(x) = [A(x)]^{T}A(x). Then determinant of B(x)

Solution:

Let P_{i} = a_{i} x^{2} + b_{i}x + c_{i} a_{i} ≠ 0

b_{i}, c_{i} ∈ R

use (i) C_{2} → C_{2} – x C_{3 }

So |B| = |A^{T}| |A| = |A|^{2} = constant independent from n

QUESTION: 86

The sum of the first 20 terms common between the series 3 + 7 +11 +15 +.....and 1 + 6 + 11 +16 +....., is :

Solution:

New A.P of common terms having

a = 11 as Ist term & d = 20

= 4020

QUESTION: 87

then a value of

Solution:

QUESTION: 88

Let A be a 3 × 3 matrix such that

Then A^{–1} is

Solution:

∴ AA^{–1} = I

use column transformation and make RHS as I

QUESTION: 89

A stair-case of length l rests against a vertical wall and a floor of a room,. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the stair-case begins to slide on the floor, then the locus of P is :

Solution:

Let any time one end is A (x, 0) & other and B(0, y) so

l^{2} = x^{2} + y^{2} ...(1)

Let P is (h, k) using section formula

use in (1)

Locus of Pt p is ellipse

which equation is

QUESTION: 90

The base of an equilateral triangle is along the line given by 3x + 4y = 9. If a vertex of the triangle is (1, 2), then the length of a side of the triangle is :

Solution:

Let BC is base of equilater triangle ABC with side a and A (1, 2)

AD = a sin 60º

AD is perpendicular distance of PtA from line 3x + 4y – 9 = 0

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