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Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length l and mass m. The rod is pivoted at its centre 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:
A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is K_{1} and that of the outer cylinder is K_{2}. Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is:
A travelling harmonic wave is represented by the equation y (x, t) = 10^{–3} sin (50 t + 2x), where x and y are in meter and t is in seconds. Which of the following is a correct statement about the wave?
The wave is propagating along the
y= a sin(ωt + kx)
⇒ wave is moving along –ve xaxis with speed
A straight rod of length L extends from x = a to x=L + a. The gravitational force is exerted on a point mass 'm' at x = 0, if the mass per unit length of the rod is A + Bx^{2}, is given by:
A light wave is incident normally on a glass slab of refractive index 1.5. If 4% of light gets reflected and the amplitude of the electric field of the incident light is 30V/m, then the amplitude of the electric field for the wave propogating in the glass medium will be:
The output of the given logic circuit is :
In the figure shown, after the switch 'S' is turned from position 'A' to position 'B', the energy dissipated in the circuit in terms of capacitance 'C' and total charge 'Q' is:
A particle of mass m moves in a circular orbit in a central potential field If Bohr's
quantization conditions are applied, radii of possible orbitals and energy levels vary with quantum number n as:
Two electric bulbs, rated at (25 W, 220 V) and (100 W, 220 V), are connected in series across a 220 V voltage source. If the 25 W and 100 W bulbs draw powers P_{1} and P_{2} respectively, then:
A satellite of mass M is in a circular orbit of radius R about the centre of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastically. The speeds of the satellite and the meteorite are the same, just before the collision.
The subsequent motion of the combined body will be :
Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be I. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is:
A passeng er train o f leng th 60m travels a t a speed of 80 km/hr. Another freight train of length 120 m travels at a speed of 30 km/hr.
The ratio of times taken by the passenger train to completely cross the freight train when : (i) they are moving in the same direction, and (ii) in the opposite directions is :
An ide al gas occupies a volu me of 2m^{3} at a pressure of 3 × 10^{6} Pa. The energy of the gas is:
Energy
Considering gas is monoatomic i.e. f = 3 E. = 9 × 10^{6} J
Option(4)
A 100 V carrier wave is made to vary between 160 V and 40 V by a modulating signal. What is the modulation index?
The galvanometer deflection, when key K_{1} is closed but K_{2} is open, equals θ_{0} (see figure).
On closing K_{2} also and adjusting R_{2} to 5Ω, the deflection in galvanometer becomes
Case I
Case II
A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle 60° with ground level. But he finds the aeroplane right vertically above his position.
If is the speed of sound, speed of the plane is :
A proton and an aparticle (with their masses in the ratio of 1:4 and charges in the ratio of 1:2) are accelerated from rest through a potential difference V. If a uniform magnetic field (B) is set up perpendicular to their velocities, the ratio of the radii r_{p} : r_{α} of the circular paths described by them will be :
KE = qΔV
A point source of light, S is pla ce d at a distance L in front of the centre of plane mirror of width d which is hanging vertically on a wall.
A man walks in front of the mirror along a line parallel to the mirror, at a distance 2L as shown below. The distance over which the man can see the image of the light source in the mirror is :
3d
“The least count of the main scale of a screw gauge is 1 mm. The minimum number of divisions on its circular scale required to measure 5 micrometer diameter of a wire is ”
Least count of main scale of a screw gauge = 1mm = 10^{3} m
Pitch of screw gauge = least count of main scale of screw gauge = 10^{3} m
Diameter of wire = 5 μm
It is clear that; 5μm is much smaller than least count of main scale. For minimum number of division of circular scale,
Least count of screw gauge = diameter of wire = 5 μm = 5 × 10^{6} m
Least count of screw gauge = pitch/number of division of circular scale
⇒5 × 10^{3 }= 10^{3}/N
⇒N = 1000/5 = 200
A simple pendulum, made of a string of length l and a bob of mass m, is released from a small angle θ_{0}. It strikes a block of mass M, kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle θ_{1}. Then M is given by :
By momentum conservation
By componendo divided
What is the position and nature of image formed by lens combination shown in figure? (f_{1}, f_{2} are focal lengths)
For first lens
For second lens
In the figure shown, a circuit contains two identical resistors with resistance R = 5W and an inductance with L = 2mH. An ideal battery of 15 V is connected in the circuit. What will be the current through the battery long after the switch is closed?
Ideal inductor will behave like zero resistance long time after switch is closed
Determine the electric di pole mo ment of the system of three charges, placed on the vertices of an equilateral triangle, as shown in the figure:
P_{1} = q(d)
P_{2} = qd
Resultant = 2 P cos30º
The position vector of the centre of mass of a symmetric uniform bar of negligible area of crosssection as shown in figure is :
As sh own i n t he fi gure, tw o i nfi ni tel y lon g, identical wires are bent by 90° and placed in such a way that the segments LP and QM are along the xaxis, while segments PS and QN are parallel to the yaxis. If OP = OQ = 4cm, and the magnitude of the magnetic field at O is 10^{–4} T, and the two wires carry equal currents (see figure), the magnitude of the current in each wire and the direction of the magnetic field at O will be (μ_{0} = 4π × 10^{–7} NA^{– 2}) :
Magnetic field at ‘O’ will be done to ‘PS’ and ‘QN’ only
i.e. B_{0} = B_{PS} + B_{QN} → Both inwards Let current in each wire = i
In a meter bridge, the wire of length 1 m has a nonuniform crosssection such that, the variation of its resistance R with length ℓ is Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point P. What is the length AP?
For the given wire :
where C = constant.
Let resistance of part AP is R_{1} and PB is R_{2}
∴ By balanced WSB concept.
Putting R_{1} = R_{2}
For the given cyclic process CAB as shown for a gas, the work done is :
Since P–V indicator diagram is given, so work done by gas is area under the cyclic diagram.
An ideal battery of 4 V and resistance R are connected in series in the primary circuit of a potentiometer of length 1 m and resistance 5W.The value of R, to give a potential difference of 5 mV across 10 cm of potentiometer wire, is :
Let current flowing in the wire is i.
If resistance of 10 m length of wire is x
then
A part icle A of mass 'm' and charg e 'q ' is accelerated by a potential difference of 50 V.Another particle B of mass '4 m' and charge 'q' is accelerated by a potential difference of 2500 V. The ratio of deBroglie wavelengths is close to:
K.E. acquired by charge = K = qV
There is a uniform spher ically sym metric surface charge density at a distance R_{0} from the origin. The charge distribution is initially at rest and starts expanding because of mutual repulsion. The figure that represents best the speed V(R(t)) of the distribution as a function of its instantaneous radius R (t) is :
At any instant 't'
Total energy of charge distribution is constant
Also the slope of vs curve will go on decreasing
∴ Graph is correctly shown by option(1)
Water samples with BOD values of 4 ppm and 18 ppm, respectively, are
Clean water have BOD value of less than 5 ppm whereas highly polluted water could have BOD value of 17 ppm or more.
Given
Temperature/K
On the basis of data given above, predict which of the following gases shows least adsorption on a definite amount of charcoal?
More easily liquefiable a gas is (i.e. having higher critical temperature), the more readily it will be adsorbed.
∴ Least adsorption is shown by H_{2} (least critical temperature)
The metal dorbitals that are directly facing the ligands in K_{3}[Co(CN)_{6}] are
During splitting in octahederal coordination entities, dx^{2} – y^{2} and orbitals point towards the direction of ligands (i.e. they experience more repulsion and their energy is raised)
A metal on combustion in excess air forms X. X upon hydrolysis with water yields H_{2}O_{2} and O_{2} along with another product. The metal is
The correct order for acid strength of compounds is as follows :
Order of acidic strength is
The hardness of a water sample (in terms of equivalents of CaCO_{3}) containing 10^{–3} M CaSO_{4} is (molar mass of CaSO_{4} = 136 g mol^{–1})
10^{–3} M CaSO_{4} ≌ 10^{3} M CaCO_{3}
10^{–3} M CaCO_{3} means 10^{–3} moles of CaCO_{3} are present in 1L
ie 100 mg of CaCO_{3} is present in 1L solution.
Hardness of water = Number of milligram of CaCO_{3 } per litre of water.
∴ Hardness of water = 100 ppm
In the following reaction
The best combination is
∴ Best combination is HCHO and MeOH
Polyβhydroxybutyratecoβhydroxyvalerate (PHBV) is a copolymer of ___.
∴ Monomers of PHBV are 3Hydroxybutanoic acid and 3Hydroxypentanoic acid.
The molecule that has minimum/no role in the formation of photochemical smog, is
NO, O_{3} and HCHO are involved in the formation photochemical smog.
N_{2} has no role in photochemical smog
The increasing order of reactivity of the following compounds towards reaction with alkyl halides directly is
Reactivity of compounds (nucleophiles) with alkyl halides will depend upon the availability of lone pair of electrons on nitrogen (amines or acid amides)
cannot be prepared by
Reaction (3) gives primary alcohol which is different from tertiary alcohol given by the remaining reactions.
Two solids dissociate as follows
The total pressure when both the solids dissociate simultaneously is
∴ P_{1}(P_{1} + P_{2}) + P_{2}(P_{1} + P_{2}) = x + y
⇒ (P_{1} + P_{2})_{2} = x + y
∴ Total pressure atm at equilibrium
The standard electrode potential E^{o} and its temperature coefficient for a cell are 2 V and – 5×10^{–4} VK^{–1} at 300 K respectively. The cell reaction is
Zn(s) + Cu^{2+}(aq) → Zn^{2+} (aq) + Cu(s)
The standard reaction enthalpy (Δ_{f}H^{o} ) at 300 K in kJ mol^{1} is,
[Use R = 8 JK^{1} mol^{1} and F = 96,000 C mol^{1}1]
so,
Cell reaction :
Zn(s) + Cu^{2+}(aq) → Zn^{2+} (aq) + Cu(s)
Decomposition of X exhibits a rate constant of 0.05 mg/year. How many years are required for the decomposition of 5 mg of X into 2.5 mg?
Rate constant of decomposition of X = 0.05 μg/year From unit of rate constant, it is clear that the decomposition follows zero order kinetics.
For zero order kinetics,
In the HallHeroult process, aluminium is formed at the cathode. The cathode is made out of
In HallHeroult process, steel vessel with carbon lining acts as cathode.
What is the work function of the metal if the light of wavelength 4000 Å generates photoelectrons of velocity 6 × 10^{5} ms^{–1} from it?
(Mass of electron = 9 × 10^{–31} kg
Velocity of light = 3 × 108 ms^{–1}
Planck’s constant = 6.626 × 10^{–34} Js
Charge of electron = 1.6 × 10^{–19} JeV^{–1})
∴ Work function = 3.1  1 = 2.1 eV
Among the following four aromatic compounds, which one will have the lowest melting point?
In general, polarity increases the intermolecular force of attraction and as a result increases the melting point.
In the following reactions, products A and B are
The pair of metal ions that can give a spin only magnetic moment of 3.9 BM for the complex [M(H_{2}O)_{6}]Cl_{2}, is
μ = 3.9 BM
So, the central metal ion has 3 unpaired electrons.
∴ Configuration is either d^{3} or d^{7} as H_{2}O is a weak field ligand.
V^{2+} has d^{3} configuration.
Co^{2+} has d^{7} configuration
In a chemical reaction, the initial concentration of B was 1.5 times of the concentration of A, but the equilibrium concentrations of A and B were found to be equal. The equilibrium constant (K) for the aforesaid chemical reaction is
given
3 – 2x = 2 – x
⇒ x = 1
The major product of the following reaction
DIBALH followed by hydrolysis converts nitrile to aldehyde and ester to aldehyde and alcohol.
For a diatomic ideal gas in a closed system, which of the following plots does not correctly describe the relation between various thermodynamic quantities?
C_{p} and C_{v} for ideal gases are dependant on temperature only. So, C_{p} will not change with pressure.
The volume of gas A is twice than that of gas B. The compressibility factor of gas A is thrice than that of gas B at same temperature. The pressure of the gases for equal number of moles are
Among the following compounds most basic amino acid is
Lysine is the most basic among the given amino acids
Mn_{2}(CO)_{10} is an organometallic compound due to the presence of
It is organometallic compound due to presence of Mn – C bond.
The major product of the following reaction is
Iodine reacts with concentrated HNO_{3} to yield Y along with other products. The oxidation state of iodine in Y, is
Conc. HNO_{3} oxidises I_{2} to iodic acid (HIO_{3}).
The element with Z = 120 (not yet discovered) will be an/a
Element with Z = 120 will belong to alkaline earth metals.
Freezing point of a 4% aqueous solution of X is equal to freezing point of 12% aqueous solution of Y.
If molecular weight of X is A, then molecular weight of Y is
(Since density of solutions are not given therefore assuming molality to be equal to molarity and given % as % W/V)
50 mL of 0.5 M oxalic acid is needed to neutralize 25 mL of sodium hydroxide solution. The amount of NaOH in 50 mL of the given sodium hydroxide solution is
2 × 50 × 0.5 = 25 × M
⇒ M = 2
∴ Moles of NaOH in 50 mL
∴ Weight = 4 grams
For x >1, i f (2x)^{2y} = 4e^{2x–2y}, then is equal to :
(2x)^{2y} = 4e^{2x–2y}
2yℓn2x = ℓn4 + 2x – 2y
The sum of the distinct real values of m, f or which the vec tors, are coplaner, is :
sum of distinct solutions = –1
Let S be the set of all points in (–π,π) at which the function, f(x) = min {sinx, cosx} is not differentiable. Then S is a subset of which of the following?
The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now from an A.P. Then the sum of the original three terms of the given G.P. is
The integral is equal to : (where C is a constant of integration)
Let If then A is equal to :
Let S = {1,2,3, ... ., 100}. The number of nonempty subsets A of S such that the product of elements in A is even is :
S = {1,2, 3100}
= Total non empty subsetssubsets with product of element is odd
If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observation is :
If a variable line, 3x+4y–λ=0 is such that the two circles x^{2} + y^{2} – 2x – 2y + 1 = 0 and x^{2}+y^{2}–18x–2y+78 = 0 are on its opposite sides, then the set of all values of l is the interval :
Centre of circles are opposite side of line
(3 + 4 – λ) (27 + 4 – λ) < 0
(λ – 7) (λ – 31) < 0
λ ∈ (7, 31)
distance from S_{1}
distance from S_{2}
A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of is :
let C_{1} and C_{2} be the centres of the circles x^{2}+y^{2}–2x–2y–2 = 0 and x^{2}+y^{2}–6x–6y+14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC_{1}QC_{2} is :
In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :
If the stra ight line, 2x–3y+17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, b), then β equals :
Let f and g be continuous functions on [0, a] such that f(x) = f(a–x) and g(x)+g(a–x)=4, then is equal to :
The maximum area (in sq. units) of a rectangle having its base on the xaxis and its other two vertices on the parabola, y = 12–x^{2} such that the rectangle lies inside the parabola, is :
f(a) = 2a(12 – a)^{2}
f'(a) = 2(12 – 3a^{2})
maximum at a = 2
maximum area = f(2) = 32
The Boolean expression is equivalent to:
Considering only the principal values of inverse functions, the set
An ordered pair(α,β) for which the system of linear equations (1+α)x + βy+z = 2
αx+(1+β)y+z = 3
αx+βy+2z = 2 has a unique solution is
For unique solution
The area (in sq. units) of the region bounded by the parabola, y = x^{2} + 2 and the lines, y = x + 1, x = 0 and x = 3, is :
Req. area
If λ be the ratio of the roots of the quadratic equation in x, 3m^{2}x^{2}+m(m–4)x+2 = 0, then the least value of m for which , is :
3m^{2}x^{2} + m(m – 4) x + 2 = 0
If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of the following points does not lie on this hyperbola?
If is a purely imaginary numberand z = 2, then a value of α is :
Let P(4, –4) and Q(9, 6) be two points on the parabola, y^{2}= 4x and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of ΔPXQ is maximum. Then this maximum area (in sq. units) is :
y^{2} = 4x
2yy' = 4
the perpendicular distance from the origin to the plane containing the two lines, and is:
The maximum value of for any real value of θ is :
A tetrahe dron has ver tices P(1, 2, 1), Q(2, 1, 3), R(–1,1,2) and O(0, 0, 0). The angle between the faces OPQ and PQR is :
Lety = y(x) be the solution of the differential equation, If 2y(2) = log_{e}4–1, then y(e) is equal to :
Let and Q = [q_{ij}] be two 3×3 matrices such that Q–P_{5} = I_{3}. Then is equal to :
Aliter
Consider three boxes, each containing 10 balls labelled 1,2,....,10. Suppose one ball is randomly drawn from each of the boxes.
Denote by n_{i}, the label of the ball drawn from the i^{th} box, (i = 1, 2, 3). Then, the number of ways in which the balls can be chosen such that n_{1} < n_{2} < n_{3} is :
No. of ways = ^{10}C_{3} = 120
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