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The potential energy of a particle of mass m at a distance r from a fixed point O is given by V (r) = kr^{2}/2,where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radiusR about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum aboutO, which of the following statements is (are) true?
Consider a body of mass 1.0 kg at rest at the origin at time t = 0. A force is applied on the body, where and . The torque acting on the body about the origin at time . Which of the following statements is (are) true?
A uniform capillary tube of inner radius r is dipped vertically into a beaker filled with water. The water
rises to a height h in the capillary tube above the water surface in the beaker. The surface tension of water is σ. The angle of contact between water and the wall of the capillary tube is θ. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?
In the figure below, the switches S_{1} and S_{2} are
closed simultaneously at t = 0 and a current starts to
flow in the circuit. Both the batteries have the same
magnitude of the electromotive force (emf) and the
polarities are as indicated in the figure. Ignore
mutual inductance between the inductors. The
current I in the middle wire reaches its maximum
magnitude I_{max} at time . Which of the
following statements is (are) true?
Two infinitely long straight wires lie in the xyplane along the lines x = ±R. The wire located at x = +R
carries a constant current I_{1} and the wire located at x = –R carries a constant current I_{2}. A circular loop of radius R is suspended with its centre at (0, 0, √3R ) and in a plane parallel to the xyplane. This loop carries a constant current I in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the + ˆj direction. Which of the following statements regarding the magnetic field is (are) true?
(A) If I_{1} = I_{2}, magnetic field due to infinite wires is equal to zero . So there must be a non–zero magnetic field at O due to the current carrying loop.
(B) If I_{1} > 0 & I_{2} < 0 , magnetic field due to straight lines are along positive z–axis and due to loop it
is along negative z–axis.
(C) If I_{1} < 0 & I_{2} > 0 magnetic field due to straight wires are along negative z–axis and due to the loop it is also along negative z–axis.
(D)
One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where V is the volume and T is the temperature). Which of the statements below is (are) true?
Process II is an isothermal expansion
Process IV is an isothermal compression
In isobaric process, volume is directly proportional to temperature.
Two vectors are defined as , where a is a constant and at time t = for the first time, the value of , in seconds, is _____.
Two men are walking along a horizontal straight line in the same direction. The man in front walks at a
speed 1.0 ms ^{−1} and the man behind walks at a speed 2.0 ms^{−1.} A third man is standing at a height 12 m above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency 1430 H_{z}. The speed of sound in air is 330 ms^{−1 }At the instant, when the moving men are 10 m apart, the stationary man is equidistant from them. The frequency of beats in H_{z}, heard by the stationary man at this instant, is __________.
A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is , then the height of the top of the inclined plane, in metres, is __________. Take g = 10 ms^{–2}.
A springblock system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 N m^{−1} and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 kg moving with a speed of 2.0 m s^{−1 }collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is _________.
For collision :
using com
Using
time taken for the block to came to the unstretched position of spring for the first time after the collision
distance between blocks =
Three identical capacitors C_{1} , C_{2} and C_{3} have a capacitance of 1.0 each and they are uncharged initially. They are connected in a circuit as shown in the figure and C_{1} is then filled completely with a dielectric material of relative permittivity . The cell electromotive force (emf) V_{0} = 8V. First the switch S_{1} is closed while the switch S_{2} is kept open. When the capacitor C_{3} is fully charged, S_{1} is opened and S_{2} is closed simultaneously. When all the capacitors reach equilibrium, the charge on C_{3} is found to be The value of =_________.
After S_{1} is closed, C_{3} has charge 8 . when S_{1} opened and S_{2} closed, C_{3} has charge 5 . so remaining 3
resides on C_{1} and C_{2
}
In the xyplane, the region y > 0 has a uniform magnetic field B_{1} kˆ and the region y < 0 has another uniform magnetic field B_{2} kˆ. A positively charged particle is projected from the origin along the positive yaxis with speed as shown in the figure. Neglect gravity in this problem. Let t = T be the time when the particle crosses the xaxis from below for the first time. If B_{2} = 4B_{1} , the average speed of the particle, in ms^{–1} , along the xaxis in the time interval T is __________.
Avg. speed along xaxis
Sunlight of intensity 1.3 kWm^{–2} is incident normally on a thin convex lens of focal length 20 cm. Ignore
the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in kW m^{–2}, at a distance 22 cm from the lens on the other side is __________.
The area of A over which the light falls satisfy :
Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures T_{1}= 300 K and T_{2} = 100 K, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are K_{1} and K_{2} respectively. If the temperature at the junction of the two cylinders in the steady state is 200 K, then K_{1} / K_{2} =__________.
In steady state, heat current in both material is same
Consider the ratio to be determined by measuring a dimensionless quantity a. If the error in the measurement of a is , then what is the error Δr in determining r ?
In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed in the first 1.0s. For x ≪ 1, ln(1 + x) = x up to first power in x . The error , in the determination of the decay constant in s^{–1}, is
The compound(s) which generate(s) N_{2} gas upon thermal decomposition below 300°C is (are)
(a) (N_{2}O can further decompose to N_{2} and O_{2} at temperature above 300^{o}C)
(b)
(c)
(d) Mg_{3}N_{2} does not decompose at any temperature.
The correct statement(s) regarding the binary transition metal carbonyl compounds is (are)
(Atomic numbers: Fe = 26, Ni = 28)
(a) Electronic configuration of central metal atom in both cases is [Ar] 3d^{10}4s^{2}4p^{6} (8 electrons in
outermost shell and 18 valence electrons respectively).
(b) Low spin complex because CO is a strong field ligand.
(c) Metalcarbon bond strengthens when the oxidation state of metal is lowered.
(d) The carbonyl C – O bond becomes stronger when the oxidation state is increased.
B ased on the compounds of group 15 elements, the correct statement(s) is (are)
Bi_{2}O_{5} is more basic than N_{2}O_{5}.
NF_{3} is more covalent than BiF_{3}.
PH_{3} boils at lower temperature than NH_{3}.
N – N single bond is weaker than P – P single bond.
In the following reaction sequence, the correct structure(s) of X is (are)
The reaction(s) leading to the formation of 1,3,5trimethylbenzene is(are)
A reversible cyclic process for an ideal gas is shown below. Here, P, V, and T are pressure, volume and
temperature, respectively. The thermodynamic parameters q, w, H and U are heat, work, enthalpy and
internal energy, respectively.
The correct option(s) is (are)
Among the species given below, the total number of diamagnetic species is ___.
H atom, NO_{2} monomer, O_{2 }^{−} (superoxide), dimeric sulphur in vapour phase, Mn_{3}O_{4}, (NH_{4})_{2}[FeCl_{4}], (NH_{4})_{2}[NiCl_{4}], K_{2}MnO_{4}, K_{2}CrO_{4}
Paramagnetic: H, NO_{2} monomer O_{2 }^{−} superoxide , S_{2} (Vapour)[Mn_{3}O_{4} is mixed oxide of Mn^{+2 }and Mn^{+3}],
(NH_{4})_{2} [FeCl_{4}], (NH_{4})_{2}[NiCl_{4}], K_{2}MnO_{4}
Diamagnetic: K_{2}CrO_{4}
The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by
NiCl_{2}.6H_{2}O to form a stable coordination compound. Assume that both the reactions are 100% complete. If 1584 g of ammonium sulphate and 952 g of NiCl_{2}.6H_{2}O are used in the preparation, the combined weight (in grams) of gypsum and the nickelammonia coordination compound thus produced is ____.
(Atomic weights in g mol^{–1} : H = 1, N = 14, O = 16, S = 32, Cl = 35.5, Ca = 40, Ni = 59)
Consider an ionic solid MX with NaCl structure. Construct a new structure (Z) whose unit cell is
constructed from the unit cell of MX following the sequential instructions given below. Neglect the charge
balance.
(i) Remove all the anions (X) except the central one
(ii) Replace all the face centered cations (M) by anions (X)
(iii) Remove all the corner cations (M)
(iv) Replace the central anion (X) with cation (M)
The value of in Z is _______ .
F or the electrochemical cell,
Mg(s)  Mg^{2+} (aq, 1 M)  Cu^{2+} (aq, 1 M)  Cu(s) the standard emf of the cell is 2.70 V at 300 K. When the concentration of Mg^{2+} is changed to x M, the cell potential changes to 2.67 V at 300 K. The value of x is ______.
(given, = 11500 K V^{−1}, where F is the Faraday constant and R is the gas constant, ln(10) = 2.30)
A closed tank has two compartments A and B, both filled with oxygen (assumed to be ideal gas). The
partition separating the two compartments is fixed and is a perfect heat insulator (Figure 1). If the old
partition is replaced by a new partition which can slide and conduct heat but does NOT allow the gas to
leak across (Figure 2), the volume (in m^{3}) of the compartment A after the system attains equilibrium is
Liquids A and B form ideal solution over the entire range of composition. At temperature T, equimolar
binary solution of liquids A and B has vapour pressure 45 Torr. At the same temperature, a new solution of A and B having mole fractions x_{A }and x_{B}, respectively, has vapour pressure of 22.5 Torr. The value of x_{A}/x_{B }in the new solution is ____.
(given that the vapour pressure of pure liquid A is 20 Torr at temperature T)
The solubility of a salt of weak acid (AB) at pH 3 is Y × 10^{–3} mol L^{−1}. The value of Y is ____.
(Given that the value of solubility product of AB (K_{sp}) = 2 × 10^{–10} and the value of ionization constant of HB (K_{a}) = 1 × 10^{–8})
The plot given below shows P—T curves (where P is the pressure and T is the temperature) for two
solvents X and Y and isomolal solutions of NaCl in these solvents. NaCl completely dissociates in both the
solvents.
On addition of equal number of moles of a nonvolatile solute S in equal amount (in kg) of these solvents, the elevation of boiling point of solvent X is three times that of solvent Y. Solute S is known to undergo dimerization in these solvents. If the degree of dimerization is 0.7 in solvent Y, the degree of dimerization in solvent X is _________.
The compound P undergoes the following reactions and forms the compound R. The compound R is:
For a nonzero complex number z, let arg(z) denote the principal argument with < arg(z) Then, which of the following statement(s) is (are) FALSE ?
In a triangle PQR, let ∠PQR = 30° and the sides PQ and QR have lengths 10√3 and 10, respectively.
Then, which of the following statement(s) is (are) TRUE ?
Let P_{1} : 2x + y – z = 3 and P_{2} : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is (are) TRUE ?
(A) Direction ratios of line of intersection are given by
(B)
(C) Angle between
(D) P3 : x – y + z = 0
Distance of (2, 1, 1) from the plane =
For every twice differentiable function f : R → [–2, 2] with (f(0))^{2} + (f '(0))^{2} = 85, which of the following
statement(s) is (are) TRUE ?
Let f : R → R and g : R → R be two nonconstant differentiable functions. If
and f(1) = g(2) = 1, then which of the following statement(s) is (are) TRUE ?
Let f : [0, ∝) → R be a continuous function such that
for all x ∈[0, ∝). Then, which of the following statement(s) is (are) TRUE ?
The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the
repetition of digits is allowed, is ______ .
Divisible by 4 ⇒ last 2 digits divisible by 4 ⇒ ends in 12, 24, 32, 44 or 52
∴ 5^{3} x 5 = 625
Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ….. , and Y be the set consisting of the first 2018 terms of arithmetic progression 9, 16, 23, ….. . Then, the number of
elements in the set X ∪ Y is _____ .
n(X ∪Y) = n(X) + n(Y) – n(X ∪Y)
1, 6, 11, …. 2018 term
T_{n} = 1 + (n – 1)5 = 5n – 4
U_{K} = 9 + (K – 1)7 = 7K + 2
for common terms
5n – 4 = 7K + 2
The number of real solutions of the equation
lying in the interval
(Here, the inverse trigonometric functions sin^{–1}x and cos^{–1}x assume values in respectively.)
For each positive integer n, let .
For x ∈ R, let [x] be the greatest integer less than or equal to x. If , then the value of [L] is_____ .
Let and be two unit vectors such that . For some x, y ∈ R, let and the vector c is inclined at the same angle α to both and , then the value of 8cos^{2} α is _____ .
Let a, b, c be three non  zero real numbers such that the equation
has two distinct real roots α and ß with . Then, the value of is ____ .
A farmer F_{1} has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring f armer F_{2} takes away the region which lies between the side PQ and a curve of the form y = x^{n} (n > 1). If the area of the region taken away by the farmer F_{2} is exactly 30% of the area of ΔPQR, then the value of n is _____ .
Let E_{1}E_{2} and F_{1}F_{2} be the chords of S passing through the point P_{0}(1, 1) and parallel to the xaxis and the yaxis, respectively. Let G_{1}G_{2} be the chord of S passing through P0 and having slope –1. Let the tangents to S at E_{1} and E_{2} meet at E_{3}, the tangents to S at F_{1} and F_{2} meet at F_{3}, and the tangents to S at G_{1} and G_{2} meet at G_{3}. Then, the points E_{3}, F_{3}, and G_{3} lie on the curve
Clearly they lie on x + y = 4
Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect thecoordinate axes at the points M and N. Then, the midpoint of the line segment MN must lie on the curve
The probability that, on the examination day, the student S_{1} gets the previously allotted seat R_{1 }, and
NONE of the remaining students gets the seat previously allotted to him/her is
For i = 1, 2, 3, 4, let Ti denote the event that the students S_{i }and S_{i+1} do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T_{1} ∩ T_{2} ∩ T_{3} ∩ T_{4} is
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