An inverted cone is filled with water at a constant rate. The volume of water inside the cone as a function of time is represented by the curve :

 

A spacecraft flies at a constant height R above a planet of radius R. At the instant the spacecraft is over the north pole, the lowest latitude visible from the spacecraft is :

An experiment consists of tossing a coin 20 times. Such an experiment is performed 50 times. The number of heads and the number of tails in each experiment are noted. What is the correlation coefficient between the two ?

Which of these groups of numbers has the smallest mean ?
Group A : 1, 2, 3, 4, 5, 6, 7, 8, 9
Group B : 1, 2, 3, 4, 6, 6, 7, 8, 9
Group C : 1, 2, 2, 4, 5, 6, 7, 8, 9
Group D : 1, 3, 3, 4, 5, 6, 7, 9, 9

Identical balls are tightly arranged in the shape of an equilateral triangle with each side containing n balls. How many balls are there in the arrangement ?

A shopkeeper has a faulty pan balance with a zero offset. When an object is placed in the left. plan it is balanced by a standard 100 g weight. When it is placed in the right pan it is bálanced by a standard 80 g weight. What is the actual weight of the object ?

A and B start from the same point in opposite directions along a circular track simultaneously. Speed of B is 2 / 3 rd that of A. How many times will A and B cross each other before meeting at the starting point ?

Consider a solid cube of side 5 units. After painting, it is cut into cubes of 1 unit. Find the probability that a randomly chosen unit cube has only one side painted .

How many integers in the set {1, 2, 3, .... 100} have exactly 3 divisors ?

The arithmetic and geometric means of two numbers are 65 and 25, respectively. What are these two numbers ?

Shyam spent half of his money and was left with as many as he had rupees before, but with half as many rupees as he had paise before. Which of the following is a possible amount of money he is left with ?

A cylindrical road roller having a diameter of 1.5 m moves at a speed of 3 km / h while levelling a road. How much length of the road will be levelled in 45 minutes ?

An intravenous fluid is given to a child of 7.5 kg, at the rate of 20 drop/minute. The prescribed dose of the fluid is 40 ml per kg of body weight. If the volume of a drop is 0.05 ml, how many hours are needed to complete the dose ?

A cousin is a non - sibling with a common ancestor. If there is exactly one pair of siblings in a group of 5 persons, then the maximum possible number of pairs of cousins in the group. is :

In a tournament with 8 teams, a win fetches 3 points and a draw, 1. After all teams have played three matches each, total number of points earned by all teams put together must lie between

An appropriate diagram to represent the relations between the categories KEYBOARD, HARDWARE, OPERATING SYSTEM and CPU is :

Trade figures and populations in appropriate units in a certain year are given for 7 countries .

If countries are ranked according to the difference in their per capita exports over import, then the best and worst ranking countries are respectively

At least two among three persons A, B and C are truthful. If A calls B a liar and if B calls C a liar, then which of the following is FALSE ?

The maximum area of a right - angled triangle inscribed in a circle of radius r is :

If we replace the mathematical operations in the expression (11 + 4 – 2) ÷ 2 × 6 as given in the table :

then its new, value is :

A particle in one dimension executes oscillatory motion in a potential V (x) = A |x|, where A > 0 is a constant of appropriate dimension. If the time period T of its oscillation depends on the total energy E as Ea, then the value of a is :

The equation of motion of a one - dimensional forced harmonic oscillator in the presence of a dissipative force is described by

The general form of the particular soltion, in terms of constants A, B etc., is :

The vector potential for an almost point like magnetic dipole located at the origin is
where (r, θ, ϕ) denote the spherical polar coordinates and ϕis the unit vector along ϕ f.
A particle of mass m and charge q, moving in the equatorial plane of the dipole, starts at time = t = 0 with an initial speed v₀ and an impact parameter b. Its instantaneous speed at the point of closest approach is :

A particle, thrown with a speed v from the earth's surface, attains a maximum height h(measured from the surface of the earth). If v is half the escape velocity and Rdenotes the radius of earth, then h / R is :

A particle of mass 1GeV/c² and its antiparticle, both moving with the same speed v, produce a new particle X of mass 10 GeV/c² in a headon collision. The minimum value of v required for this process is closest to :

The volume of the region common to the interiors of two infinitely long cylinders defined by x² + y² = 25 and x² + 4 z² = 25 is best approximated by :

The volume integral  is over a region V bounded by a surface (an infinitesimal area element being , where is the outward unit normal). If it changes to  when the vector A is changed to A + A, then I can be expressed as

A generic 3 x 3 real matrix.A has eigenvalues 0, 1 and 6, and I is the 3 x 3 identity matrix. The quantity/quantities that cannot be determined from this information is/are the :

A discrete random variable X takes a value from the set { –1, 0, 1, 2} with the corresponding probabilities p (X) = 3 / 10,2 / 10,2 / 10 and 3 / 10, respectively. The probability distribution q(Y) = (q(0), q(1), q(4)) of the random variable Y = X² is :

The components of the electric field, in a region of space devoid of a y charge or current sources, are given to be where a_i and b_ij are constants independent of the coordinates. The number of independent components of the matrix b_ij is :

A conducting wire in the shape of a circle lies on the (x, y) - plane, with its centre at the origin. A bar magnet moves with a constant velocity towards the wire along the z - axis (as shown in the figure below) .

We take t = 0 to be the instant at which the midpoint of the magnet is at the centre of the wire loop and the induced current to be positive when it is counter - clockwise as viewed by the observer facing the loop and the incoming magnet. In these conventions, the besi schematic representation of the induced curren I (t) as a function of t, is :

In an experiment to measure the charge to mass ratio e / m of the electron by Thomson's method, the values of the deflecting electric field and the accelerating potential are 6 x10⁶ N / C (newton per coulomb) and 150 V, respectively. The magnitude of the magnetic field that leads to zero deflection of the electron beam is closest to :

A monochromatic source emitting radiation with a certain frequency moves with a velocity v away from a stationary observer. A. It is moving towards another observer B (also at rest) along a line joining the two. The frequencies of the radiation recorded by A and B are v_A and v_B, respectively. If the ratio
then the value of v / c is:

The Hamiltonian of a particle of mass m in one - dimension is  where > 0 is. a constant. If E₁ and E₂, respectively. denote the ground stàte energies of the particle for = 1 and = 2 (in appropriate units) the ratio E₂ / E₁ is best approximated by :

A particle of mass m is in a one dimensional infinite potential well of length L, extending from x = 0.to x = L. When it is in the energy eigenstate labelled by n, (n = 1,2,3, ...) the probability of finding it in the interval is 1 / 8. The minimum value of n for which this is possible is :

A two - state system evolves under the action of the  are its two orthonormal states. These states transform to one another under parity, i.e.,  If at time t = 0 the system is in a state of definite parity P = 1,the earliest time t at which the probability of finding the system in a state of parity P = -1 is one, is :

The figures below depict three different wavefunctions of a particle confined to a onedimensional box 

The wavefunctions that correspond to the maximum expectation values (absolute value of the mean position) and respectively, are :

Which of the following two physical quantities cannot be measured simultaneously with arbitrary accuracy for the motion of a quantum particle in three dimensions ?

The ratio C_p / C_V of the specific heats at constant pressure and volume of a monoatomic ideal gas in two dimensions is :

The volume and temperature of a spherical cavity filled with black body radiation are V and 300 K, respectively. If it expands adiabatically to a volume 2 V, its temperature will be closest to :

The total number of phonon modes in a solid of volume where N is the number of primitive cells, w_D is the Debye frequency and density of photon modes is g(w) = AVw² (with A > 0 a constant). If the density of the solid doubles in a phase transition, the Debye temperature Q_D will :

The position of a particle in one dimension changes in discrete steps. With each step it moves to the right, however, the length of the step is drawn from a uniform distribution from the interval where  and w are positive constants. If X denotes the distance from the starting point after N steps, the standard deviation  for large values of N is :

In the LCR circuit shown below, the resistance R = 0.05  the inductance L = 1 H and the capacitance C = 0.04 F.

If the input vin is a square wave of angular frequency 1 rad / s, the output v_out is best approximated by a :

In an experiment, the velocity of a nonrelativistic neutron is determined by measuring the time(~50 ns) it takes to travel from the source to the detector kept at a distance L. Assume that the error in the measurement of L is negligibly small. If we want to estimate the kinetic energy T of the neutron to within 5 % accuracy, i.e., 0.05, the maximum permissible error $|delta t|$ in measuring the time of flight is nearest to :

The door of an X- ray machine room is fitted with a sensor D (0 is open and 1 is closed). It is also equipped with three fire sensors F_1, F₂ and F₃ (each is 0 when disabled and 1 when enabled). The X - ray machine can operate only if the door is closed and at least 2 fire sensors are enabled. The logic circuit to ensure that the machine can be operated is :

If we use the Fourier transform (x, y) = to solve the partial differential equation  in the half - plane the Fourier modes (y) depend on y as y^B and y^B. The values of a and b are : 

The Newton - Raphson method is to be used to determine the reciprocal of the number x = 4. If we start with the initial guess 0.20, then after the first iteration the reciprocal is :

The Legendre polynomials P_n(x), n = 0, 1, 2, ..., satisfying the orthogonality condition .
on the interval [-1,+1] may be defined by the Rodrigues formula
The value of the definite interval

A particle of mass m moves in a potential that is in the coordinates of a non - inertial frame F. The frame F is rotating with respect to an inertial frame with an angular velocity , where  is the unit vector along their common z - axis. The motion of the particle is unstable for all angular frequencies satisfying :

Which instrument is used to measure altitude in aircraft's ?