A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the xaxis. It's kinetic energy K changes with time as dK/dt = γt, where γ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?
(i) force is constant.
(ii) speed is proportional to t
(iii) Force is constant, so it is conservative
Consider a thin square plate floating on a viscous liquid in a large tank. The height h of the liquid in the tank is much less than the width of the tank. The floating plate is pulled horizontally with a constant velocity u_{0}. Which of the following statements is (are) true?
An infinitely long thin nonconducting wire is parallel to the zaxis and carries a uniform line charge density λ. It pierces a thin nonconducting spherical shell of radius R in such a way that the arc PQ subtends an angle 120° at the centre O of the spherical shell, as shown in the figure. The permittivity of free space is ε_{0}. Which of the following statements is (are) true?
Charge inside the spherical shell = λ. 2Rcos 30º = λR √3
So, electric flux =
A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length f, as shown in the figure. Which of the figures shown in the four options qualitatively represent(s) the shape of the image of the bent wire? (These figures are not to scale.)
In a radioactive decay chain, Th nucleus decays to Pb nucleus. Let N_{α} and N_{β} be the number of α and β^{} particles, respectively, emitted in this decay process. Which of the following statements is (are) true?
In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500 Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length 50.7 cm and 83.9 cm. Which of the following statements is (are) true?
A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass m = 0.4 kg is at rest on this surface. An impulse of 1.0 Ns is applied to the block at time t = 0 so that it starts moving along the xaxis with a velocity v(t) = , where v_{0} is a constant and The displacement of the block, in meters, at is _________. Take e^{1} = 0.37.
A ball is projected from the ground at an angle of 45° with the horizontal surface. It reaches a maximum height of 120 m and returns to the ground. Upon hitting the ground for the first time, it loses half of its kinetic energy. Immediately after the bounce, the velocity of the ball makes an angle of 30° with the horizontal surface. The maximum height it reaches after the bounce, in metres, is ___________.
A particle, of mass 10^{3} kg and charge 1.0 C, is initially at rest. At time t = 0, the particle comes under the influence of an electric field , where E_{0} = 1.0 NC^{1} and ω = 10^{3} rad s^{1} . Consider the effect of only the electrical force on the particle. Then the maximum speed, in m s^{1}, attained by the particle at subsequent times is ____________.
A moving coil galvanometer has 50 turns and each turn has an area 2 x 10^{4} m^{2}. The magnetic field produced by the magnet inside the galvanometer is 0.02 T. The torsional constant of the suspension wire is 10^{4} N m rad^{1}. When a current flows through the galvanometer, a fullscale deflection occurs if the coil rotates by 0.2 rad. The resistance of the coil of the galvanometer is 50 Ω. This galvanometer is to be converted into an ammeter capable of measuring current in the range 0  1.0 A. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is __________.
A steel wire of diameter 0.5 mm and Young’s modulus 2 x 10^{11} Nm^{2} carries a load of mass M. The length of the wire with the load is 1.0 m. A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by 1.2 kg, the vernier scale division which coincides with a main scale division is __________. Take g = 10 ms^{2} and is π = 3.2.
One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100 K and the universal gas constant R = 8.0 J mol^{1}K^{1}, the decrease in its internal energy, in Joule, is__________.
In a photoelectric experiment a parallel beam of monochromatic light with power of 200 W is incident on a perfectly absorbing cathode of work function 6.25 eV. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100%. A potential difference of 500 V is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force F = n x 10^{4} N due to the impact of the electrons. The value of n is __________. Mass of the electron m_{e} = 9 x 10^{31 }kg and 1.0 eV = 1.6 x 10^{19} J.
No. of photoelectrons emitted per second
Consider a hydrogenlike ionized atom with atomic number Z with a single electron. In the emission spectrum of this atom, the photon emitted in the n = 2 to n = 1 transition has energy 74.8 eV higher than the photon emitted in the n = 3 to n = 2 transition. The ionization energy of the hydrogen atom is 13.6 eV. The value of Z is __________.
The electric field E is measured at a point P(0, 0, d) generated due to various charge distributions and the dependence of E on d is found to be different for different charge distributions. ListI contains different relations between E and d. ListII describes different electric charge distributions, along with their locations. Match the functions in ListI with the related charge distributions in ListII.
A planet of mass M, has two natural satellites with masses m1 and m2. The radii of their circular orbits are
R1 and R2 respectively. Ignore the gravitational force between the satellites. Define v1, L1, K1 and T1 to be,
respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite
1; and v2, L2, K2 and T2 to be the corresponding quantities of satellite 2. Given m1/m2 = 2 and R1/R2 = 1/4,
match the ratios in ListI to the numbers in ListII.
One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PVdiagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List1 with the corresponding statements in ListII.
Process 1 is adiabatic process, hence Q = 0
Process 2 is isobaric, hence W = P.ΔV = 6 P_{0}V_{0}
Process 3 is isochoric, hence W = 0
Process 4 is isothermal.
In the ListI below, four different paths of a particle are given as functions of time. In these functions, α and β are positive constants of appropriate dimensions and . In each case, the force acting on the particle is either zero or conservative. In ListII, five physical quantities of the particle are mentioned: the linear momentum, the angular momentum about the origin, K is the kinetic energy, U is the potential energy and E is the total energy. Match each path in ListI with those quantities in ListII, which are conserved for that path.
‘P’ is straight line with zero acceleration.
‘Q’ is elliptical path.
‘R’ is circular path
‘S’ is parabolic path
The correct option(s) regarding the complex (en = H_{2}NCH_{2}CH2NH_{2}) is (are)
The correct option(s) to distinguish nitrate salts of Mn2+ and Cu2+ taken separately is (are)
Cu^{2+} shows the characteristic green colour in the flame test.
Ksp of CuS < Ksp of MnS
Aniline reacts with mixed acid (conc. HNO_{3} and conc. H_{2}SO_{4}) at 288 K to give P (51%), Q (47%) and R(2%). The major product(s) of the following reaction sequence is (are)
The Fischer presentation of Dglucose is given below
Dglucose
The correct structures(s) of β  Lglucopyranose is(are)
For a first order reaction A(g) → 2B(g) + C(g) at constant volume and 300 K, the total pressure at the beginning (t = 0) and at time t are Po and P_{t}, respectively. Initially, only A is present with concentration [A]0, and the time required for the partial pressure of A to reach 1/3^{rd} of its initial value. The correct option(s) is (are) (Assume that all these gases behave as ideal gases)
For a reaction, A ⇌ P, the plots of [A] and [P] with time at temperatures T_{1} and T_{2} are given below.
If T_{2} > T_{1}, the correct statement(s) is (are) (Assume ΔH^{θ} and ΔS^{θ} are independent of temperature and ratio
of lnK at T_{1} to lnK at T_{2} is greater than T_{2}/T_{1}. Here H, S, G and K are enthalpy, entropy, Gibbs energy and
equilibrium constant, respectively.)
The total number of compounds having at least one bridging oxo group among the molecules given below
is ____.
N_{2}O_{3}, N_{2}O_{5}, P_{4}O_{6}, P_{4}O_{7}, H_{4}P_{2}O_{5}, H_{5}P_{3}O_{10}, H_{2}S_{2}O_{3}, H_{2}S_{2}O_{5}
Galena (an ore) is partially oxidized by passing air through it at high temperature. After some time, the
passage of air is stopped, but the heating is continued in a closed furnace such that the contents undergo
selfreduction. The weight (in kg) of Pb produced per kg of O_{2} consumed is ____.
(Atomic weights in g mol^{1}: O = 16, S = 32, Pb = 207)
To measure the quantity of MnCl_{2} dissolved in an aqueous solution, it was completely converted to KMnO_{4}
using the reaction, MnCl_{2} + K_{2}S_{2}O_{8} + H_{2}O >KMnO_{4} + H_{2}SO_{4} + HCl (equation not balanced). Few drops
of concentrated HCl were added to this solution and gently warmed. Further, oxalic acid (225 mg) was
added in portions till the colour of the permanganate ion disappeared. The quantity of MnCl_{2} (in mg)
present in the initial solution is ____.
(Atomic weights in g mol−1: Mn = 55, Cl = 35.5)
Number of meq of MnCl_{2} = number of meq of KMnO_{4}
= number of meq of H_{2}C_{2}O_{4}
= 5
Weight of MnCl_{2} taken = = 126 mg
For the given compound X, the total number of optically active stereoisomers is ____
In the following reaction sequence, the amount of D (in g) formed from 10 moles of acetophenone is ____.
(Atomic weights in g mol–1: H = 1, C = 12, N = 14, O = 16, Br = 80. The yield (%) corresponding to the
product in each step is given in the parenthesis)
The surface of copper gets tarnished by the formation of copper oxide. N_{2} gas was passed to prevent the
oxide formation during heating of copper at 1250 K. However, the N_{2} gas contains 1 mole % of water
vapour as impurity. The water vapour oxidises copper as per the reaction given below:
is the minimum partial pressure of H_{2} (in bar) needed to prevent the oxidation at 1250 K. The value of
ln is ____.
(Given: total pressure = 1 bar, R (universal gas constant) = 8 J K^{1} mol^{1}, ln(10) = 2.3. Cu(s) and Cu_{2}O(s)
are mutually immiscible.
At 1250 K: 2Cu(s) + ½ O_{2}(g) > Cu_{2}O(s); ΔG^{θ} = − 78,000 J mol−1
H_{2}(g) + ½ O_{2}(g) > H_{2}O(g); ΔG^{θ} = − 1,78,000 J mol−1; G is the Gibbs energy)
Consider the following reversible reaction,
A(g)+ B(g) ⇌ AB(g)
The activation energy of the backward reaction exceeds that of the forward reaction by 2RT (in J mol−1 ). If the preexponential factor of the forward reaction is 4 times that of the reverse reaction, the absolute value of ΔG^{Ɵ} (in J mol−1 ) for the reaction at 300 K is ____.
(Given; ln(2) = 0.7, RT = 2500 J mol−1 at 300 K and G is the Gibbs energy)
Consider an electrochemical cell: A(s)  A^{n+} (aq, 2 M)  B^{2n+} (aq, 1 M)  B(s). The value of ΔH^{Ɵ} for the cell reaction is twice that of ΔG^{Ɵ} at 300 K. If the emf of the cell is zero, the ΔS^{Ɵ} (in J K^{−1} mol^{−1} ) of the cell reaction per mole of B formed at 300 K is ____. (Given: ln(2) = 0.7, R (universal gas constant) = 8.3 J K^{−1} mol^{−1} . H, S and G are enthalpy, entropy and Gibbs energy, respectively.)
Match each set of hybrid orbitals from LIST–I with complex(es) given in LIST–II.
The correct option is
The desired product X can be prepared by reacting the major product of the reactions in LISTI with one or more appropriate reagents in LISTII.
(Given, order of migratory aptitude: aryl > alkyl > hydrogen)
The correct option is
LISTI contains reactions and LISTII contains major products.
Match each reaction in LISTI with one or more products in LISTII and choose the correct option.
Dilution processes of different aqueous solutions, with water, are given in LISTI. The effects of dilution of the solutions on [H+ ] are given in LISTII.
(Note: Degree of dissociation (α) of weak acid and weak base is << 1; degree of hydrolysis of salt <<1; [H+ ] represents the concentration of H+ ions)
Match each process given in LISTI with one or more effect(s) in LISTII. The correct option is
For any positive integer n, define as
(Here the inverse trigonometric function tan^{1} x assumes values in )
Then, which of the following statement(s) is (are) TRUE?
Let T be the line passing through the points P(2, 7) and Q(2, 5). Let F_{1} be the set of all pairs of circles (S_{1}, S_{2}) such that T is tangent to S_{1} at P and tangent to S_{2} at Q, and also such that S_{1} and S_{2} touch each other at a point, say, M. Let E_{1} be the set representing the locus of M as the pair (S_{1}, S_{2}) varies in F_{1}. Let the set of all straight line segments joining a pair of distinct points of E_{1} and passing through the point R(1, 1) be F_{2}. Let E_{2} be the set of the midpoints of the line segments in the set F_{2}. Then, which of the following statement(s) is (are) TRUE ?
Locus of M is circle center (0, 1) and radius , excluding points P and Q
=> x^{2} + y^{2} – 2y = 39 excluding points P and Q
(–2, 7) is excluded from locus and does not lie in E_{1}
Locus of point in E_{2} is circle with diameter (0, 1) and (1, 1), excluding midpoints of chords passing through P or Q
=> x ^{2} – x + (y – 1)^{2} = 0
does not lie in E_{2} and doesn’t lie in E_{2} since latter is midpoint of chord through P
Let S be the set of all column matrices such that b_{1}, b_{2}, b_{3 }ε R and the system of equations (in real variables)
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each
as planes are non parallel is must represent family of planes for solution to exist
Must be 2x + b + 3z + b_{3} = 0 which gives μ = 1, b_{1} + b_{2} + 3b_{3} = 0 which is not true always
Consider two straight lines, each of which is tangent to both the circle x^{2} + y^{2} = 1/2 and the parabola y^{2} = 4x. Let these lines intersect at the point Q. Consider the ellipse whose center is at the origin O(0, 0) and whose semimajor axis is OQ. If the length of the minor axis of this ellipse is , then which of the following statement(s) is (are) TRUE?
Let s, t, r be nonzero complex numbers and L be the set of solutions z = x + iy (x, y ε R, i = ) of the equation , where .Then, which of the following statement(s) is (are) TRUE ?
Eliminating
(A) L has one element exactly
(B) s = t = i and r = i => L has no element
(C) Adding both equations
Which either represents no points or a straight line In either case, number of points of intersection of given circle and L cannot be more than 2
(D) If α and β both satisfy the given equation then any number of form satisfies the given equation where
Let be a twice differentiable function such that
If , then which of the following statement(s) is (are) TRUE ?
Using L Hospitals rule
The value of the integral
Let P be a matrix of order 3 x 3 such that all the entries in P are from the set {1, 0, 1}. Then, the maximum possible value of the determinant of P is _____ .
If 0 is used maximum can only be 4 If 0 not used then by using only {–1, 1} can only form matrix with maximum determinant value 4
Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If α is the number of oneone functions from X to Y and β is the number of onto functions from Y to X, then the value of is ________________ .
Let f : R > R be a differentiable function with f (0) = 0. If y = f (x) satisfies the differential equation then the value of is ____________ .
Let f : R> R be a differentiable function with f (0) = 1 and satisfying the equation for all x, y ε R. Then, the value of loge(f (4)) is _____ .
Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the midpoint of PQ lies in the plane x + y = 3) lies on the zaxis. Let the distance of P from the xaxis be 5. If R is the image of P in the xyplane, then the length of PR is _______________.
Let Q = (0, 0, z_{1}) then P is image of Q in x + y = 3
P = (3, 3, z_{1})
=> (3)^{2} + z_{1}^{2} = 25 z_{1} =4
Length of PR = 2 z_{1} = 8
Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the xaxis, yaxis and zaxis,
respectively, where O(0, 0, 0) is the origin. Let be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If and , then the value of is __________.
Let where denote binomial coefficients. Then the value of X is ____________.
Let E1 = and E2 = is a real number Here, the inverse trigonometric function sin 1 x assumes values in Let be the function defined by and be the function defined by
The correct option is :
In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls
G_{1}, G_{2}, G_{3}, G_{4}, G_{5}.
(i) Let α_{1} be the total number of ways in which the committee can be formed such that the committee has
5 members, having exactly 3 boys and 2 girls.
(ii) Let α_{2} be the total number of ways in which the committee can be formed such that the committee has
at least 2 members, and having an equal number of boys and girls.
(iii) Let α_{3} be the total number of ways in which the committee can be formed such that the committee has
5 members, at least 2 of them being girls.
(iv) Let α_{4} be the total number of ways in which the committee can be formed such that the committee has
4 members, having atleast 2 girls and such that both M_{1} and G_{1} are NOT in the committee together.
The correct option is :
Let H : , where a > b > 0, be a hyperbola in the xyplane whose conjugate axis LM subtends an angle of 60° at one of its vertices N. Let the area of the triangle LMN be .
The correct option is :
Let and be functions defined by
(i)
(ii) where the inverse trigonometric function tan1x assumes values in
(iii) where, for t ε R, [t] denotes the greatest integer less than or equal to t,
(iv)
The correct option is :
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