Five congruent rectangles are drawn inside a big rectangle of perimeter 165 as shown. What is the perimeter of one of the five rectangles?

At one instant, the hour hand and the minute hand of a clock are one over the other in between the markings for 5 and 6 on the dial. At this instant, the tip of the minute hand

There are two buckets A and B. Initially A has 2 litres of water and B is empty. At every hour 1 litre of water is transferred from A to B followed by returning 1/2 litre back to A from B half an hour later. The earliest A will get empty is in

A car is moving at 60 km/h. The instantaneous velocity of the upper most points of its wheels is

Density of a rice grain is 1.5 g/cc and bulk density of rice heap is 0.80 g/cc. If a 1 litre container is completely filled with rice. What will be the approximate volume of pore space in the container?

If D + I + M = 1501
C + I + V + I + L = 157 
L + I + V + I + D = 557 
C + I + V + I + C = 207 
What is V + I + M = ?

Four circles of unit radius each are drawn such that each one touches two others and their centres lie on the vertices of a square. The area of the region enclosed between the circles is

In each of the following groups of words is a hidden number, based on which you should arrange them in descending order. Pick the correct answer.
1. Papers I Xeroxed
2. Wi-Fi veteran
3. Yourself ourselves
4. Breaks even

A shopkeeper purchases a product for Rs. 100 and sells it making a profit of 10%. The customer resells it to the same shopkeeper incurring a loss of 10%. In these dealings the shopkeeper makes

A person walks downhill at 10 km/h, uphill at 6 km/h and on the plane at 7.5 km/h. If the person takes 3 hours to go from a place A to another place B, and 1 hour on the way back, the distance between A and B is

The number of squares in the below figure is

A vessel is partially filled with water. More water is added to it at a rate directly proportional to time  Which of the following graphs depicts correctly the variation of total volume V of water with time t?

A bird leaves its nest and flies away. Its distance x from the nest is plotted as a function of time t. Which of the following plots cannot be right?

Secondary colours are made by a mixture of three primary colours, Red, Green and Blue, in different proportions; each of the primary colours comes in 8 possible levels. Grey corresponds to equal proportions of Red, Green and Blue. How many shades of grey exist in this scheme?

Statement A : The following statement is true.
Statement B : The preceding statement is false.
Choose the correct inference from the following

A living cell has a protoplasm which is water based and demarcated by a lipid bilayer membrane. If a cell is pierced up to 1/5 th of its diameter with a very sharp needle, after taking the needle out

A turtle starts swimming from a point A located on the circumference of a circular pond. After swimming for 4 meiers in a straight line it hits point B on the circumference of the pond. From there it changes direction and swims for 3 meters in a straight line and arrives at point D diametrically opposite to point A. How far is point D from A?

A film projector and microscope give equal magnification. But a film projector is not used to see living cells because

A cubical cardboard box made of 1 cm thick card board has outer side of 29 cm. A tight-fitting cubical box of the same thickness is placed inside it, then another one inside it and so on. How many cubical boxes will be there in the entire set?

The triangle formed by the lines y = x, y = 1 – x and x = 0 in a two dimensional plane is (x and y axes have the same scale)

The Fourier transform of f(x) is

+ yδ" (x) where δ(x) is the Dirac delta - function (and prime denotes derivative), what is f(k) ?

δ

Two masses, m each, are placed at the points 
(x, y) = (a, a) and (– a, –a). Two masses, 2m each, are placed at the points (a, –a) and (–a, a). The principal moments of inertia of the system are

The function  of a complex variable z has

In the scattering of some elementary particles, the scattering cross-section is found to depend on the total energy E and the fundamental constants h (Planck’s constant) and c (the speed of light in vacuum). Using dimensional analysis, the dependence of on these quantities is given by

A particle moves in three-dimensional space in a central potential V(r) = kr⁴, where k is a constant. The angular frequency  for a circular obrit depends on its radius R as

Consider a particle of a mass m moving with a speed v. If T_R denotes the relativistic kinetic energy and TN its non-relativistic approximation, then the value of (T_R – T_N)/T_R for v = 0.01 c, is

The Lagrangian of a system is given by 

where m and k are positive constants. The frequencies of its normal modes are

The solution of the differential equation , with initial condition x = 0 at t = 0 is

A beam of unpolarized light in a medium with dielectric constant ₁ is reflected from a plane interface formed with another medium of dielectric constant ₂ = 3₁. The two media have identical magnetic permeability. If the angle of incidence is 60°, then the reflected light

The ground state energy of a particle of mass m in the potential  are constants  approximately is

Given a uniform magnetic field  (where B₀ is a constant), a possible choice for the magnetic vector potential A is

A Hermitian operator has two normalized eigenstates I1> and I2> with eigenvalues 1 and 2, respectively. The two states   are such that  Which of the following are possible values of  ?

A hollow metallic sphere of radius a, which is kept at a potential V₀, has a charge Q at its centre. The potential at a point outside the sphere, at a distance r from the centre, is

The Hamiltonian for a spin–1/2 particle at rest is given by are Pauli spin matrices and E₀ and a are constants. The eigenvalues of this Hamiltonian are 

Consider a charge Q at the origin of 3-dimensional coordinate system. The flux of the electric field through the curved surface of a cone that has a height h and a circular base of radius R (as shown in the figure) is

Let _n/m denote the eigenstates of a hydrogen atom in the usual notation. The state 

is an eigenstate of

If the parameters y and x are related by y = log (x), then the circuit that can be used to produce an output voltage V₀ varying linearly with x is

For a system of independent non-interacting one-dimensional oscillators, the value of the free energy per oscillator, in the limit T → 0, is

Two data sets A and B consist of 60 and 10 readings of a voltage measured using voltmeters of resolution of 1 mV and 0.5 mV respectively. The uncertainty in the mean voltage obtained from the data sets A and B are U_A and U_B, respectively. If the uncertainty of the mean of the combined data sets is U_AB, then which of the following statements is correct?

The partition function of a system of N Ising spins is are functions of temperature, but are independent of N. If  the free energy per spin in the limt N →  is :

The heat capacity of (the interior of) a refrigerator is 4.2 kJ/K. The minimum work that must be done to lower the internal temperature from 18°C to 17°C when the outside temperature is 27°C is

In the schematic figure given below, the initial values of 4 bit shift registers A and B are 1011 and 0010 respectively. The values at SO_A and SO_B after the pulse T₂ are respectively

The Hamiltonian of a system of N non-interacting spin-1/2 particles is are the components of i^th spin along an external magnetic field B. At a temperature T such that the specific heat per particle is :

If the reverse bias voltage of a silicon varactor is increased by a factor of 2, the corresponding transition capacitance

For a dynamical system governed by the equation 

The Hermite polynomial H_n(x) satisfies the differential equation 

The corresponding generating function  satisfies the equation

A canonical transformation (p, q) → (P, Q) is performed on the Hamiltonian via the generating function  which of the following graphs shows schematically the dependence  of Q(t) on t ?

A distant source, emitting radiation of frequency ω, moves with a velocity Ac/5 in a certain direction with respect to a receiver (as shown in the figure). 

The upper cut-off frequency of the receiver is Let q be the angle as shown. For the receiver to detect the radiation, ø should at least be

ω

A function f (x) satisfies the differential equation  , where w is positive. The Fourier transform  of f, and  the solution of the equation are, respectively

There is an equilateral triangle in the XY plane with its centre at the origin. The distance of its sides from the origin is 3.5 cm. The area of its circumcircle in cm² is

A sphere of iron of radius R/2 fixed to one end of a string was lowered into water in a cylindrical container of base radius R to keep exactly half the sphere dipped. The rise in the level of water in the container will be

A crystal grows by stacking of unit cells of 10 × 20 × 5 nm size as shown in the diagram given below. How many unit cells will make a crystal of 1 cm³ volume? 

What is the value of  to  ?

A solid cylinder of basal area A was held dipped in water in a cylindrical vessel of basal area 2A vertically such that a length h of the cylinder is immersed. The lower tip of the cylinder is at a height h from the base of the vessel. What will be the height of water in the vessel when the cylinder is taken out ?

Of all the triangles that can be inscribed in a semicircle of radius R with the diameter as one side, the biggest one has the area

Choose the largest number

A daily sheet calendar of the year 2013 contains sheets of 10 × 10 cm size. All the sheets of the calendar are spread over the floor of a room of 5m × 7.3 m size. What percentage of the floor will be covered by these sheets?

How many rectangles (which are not squares) are there in the following figure?

Define  What is the value of  Here 1 cm = least common multiple and gcd = greatest common divisor.

A square pyramid is to be made using a wire such that only one strand of wire is used for each edge. Wiat is the minimum number of times that the wire has to be cut in order to make the pyramid?

A crow is flying along a horizontal circle of radius R at a height R above the horizontal ground. Each of a number of men on the ground found that the angular height of the crow was a fixed angle (< 45°) when it was closest to him. Then all these men must be on a circle on the ground with a radius

How many pairs of positive integers have gcd 20 and 1cm 600? (gcd = greatest common divisor; 1cm = least common multiple)

During an evening party, when Ms. Black, Ms. Brown and Ms. White met, Ms. Brown remarked, “It is interesting that our dresses are white, black or brown, but for each of us the name does not match the colour of the dress!”. Ms. White replied, “But your white dress does not suit you!”. Pick the correct answer.

Two integers are picked at random from the first 15 positive integers without replacement. What is the probability that the sum of the two numbers is 20?

Identify the next figure in the sequence

In a customer survey conducted during Monday to Friday, of the customers who asked for child care facilities in super markets, 23% were men and the rest, women. Among them, 19.9% of the women and 8.8% of the men were willing to pay for the facilities.
1. What is the ratio of the men to women customers who wanted childcare facilities?
2. If the survey had been conducted during the weekend instead, how will the result change?
    With the above data,

The map given below shows contour lines which connect points of equal ground surface elevation in a region. Inverted ‘V’ shaped portions of contour lines represent a valley along which a river flows. What is the downstream direction of the river?

During a summer vacation, of 20 friends from a hostel, each wrote a letter to each of all others. The total number of letters written was

Ten grams of ice at 0°C is added to a beaker containing 30 grams of water at 25°C. What is the final temperature of the system when it comes to thermal equilibrium? (The specific heat of water is 1 cal/gm/°C and latent heat of melting of ice is 80 cal/gm).

A vessel has two compartments of volume V₁ and V₂, containing an ideal gas at pressures p₁ and p₂, and temperatures T₁ and T₂ respectively. If the wall separating the compartments is removed, the resulting equilibrium temperature will be

For temperature T₁ > T₂ the qualitative temperature dependence of the probability distribution F(v) of the speed v of a molecule in three dimensions is correctly represented by the following figure

A system of non-interacting spin –1/2 charged particles are placed in an external magnetic field. At low temperature T, the leading behaviour of the excess energy above the ground state energy, depends on T as  (c is a constant)

The acceleration due to gravity g is determined by measuring the time period T and the length L, of a simple pendulum. If the uncertainties in the measurements of T and L are T and L respectively, the fractional error Dg/g in measuring g is best approximated by

A silicon transistor with built-in voltage 0.7V is used in the circuit shown, with V_BB = 9.7V, R_B = 300 kW, V_CC = 12 V and R_C = 2 kW. Which of the following figures correctly represents the load line and the quiescent Q point?

If the analog input to an 8-bit successive approximation ADC is increased from 1.0 V to 2.0 V, then the conversion time will

The insulation resistance R of an insulated cable is measured by connecting it in parallel with a capacitor C, a voltmeter, and battery B as shown. The voltage across the cable dropped form 150 V to 15 V, 1000 seconds after the switch S is closed. If the capacitance of the cable is 5 x F, then its insulation resistance is approximately

The approximation cos1 is valid up to 3 decimal places as long as || is less than (take 180º/  57.29°)

The area of a disc in its rest frame S is equal to 1 (in some units). The disc will appear distorted to an observer O moving with a speed u with respect to S along the plane of the disc. The area of the disc measured in the rest frame of the observer O is (c is the speed of light in vacuum)

A particle of charge e and mass m is located at the midpoint of the line joining two fixed collinear dipoles with unit charges as shown in the figure. (The particle is constrained to move only along the line joining the dipoles.) Assuming that the length of the dipoles is much shorter than their separation, the natural frequency of oscillation of the particle is

A current I is created by a narrow beam of protons moving in vacuum with constant velocity . The direction and magnitude,respectively, of the Poynting vector outside the beam at a radial distance r (much larger than the width of the beam) from the axis, are

If the electric and magnetic fields are unchanged when the vector potential A' changes (in suitable units) according to then the scalar potential F must simultaneously change to

Consider an axially symmetric static charge distribution of the form,  The radial component of the dipole moment due to this charge distribution is 

In a basis in which the z-component S_Z of the spin is diagonal, an electron is in a spin state 
The probabilities that a measurement of S_Z will yield the values  are, respectively

Consider the normalized state |ŷ} of a particle in a one-dimensional harmonic oscillator where |0} and |1} denote the ground and first excited states respectively, and b1 and b2 are real constants. The expectation value of the displacement x in the state will be a minimum when

A muon (µ^–) from cosmic rays is trapped by a proton to form a hydrogen-like atom. Given that a muon is approximately 200 times heavier than an electron, the longest wavelength of the spectral line (in the analogue of the Lyman series) of such an atom will be 

The un-normalized wavefunction of a particle in a spherically symmetric potential is given by  where f(r) is a function of the radial variable r. The eigenvalue of the operator L² (namely the square of the orbital angular momentum) is

Given that 

the value of H4 (0) is

A unit vector  on the xv-plane is at an angle of 120° with respect to Î. The angle between the vectors will be 60° if

With z = x + iy, which of the following functions f (x, y) is not a (complex) analytic function of z?

A planet of mass m and an angular momentum L moves in a circular orbit in a potential, V(r) = –k/r, where k is a constant. If it is slightly perturbed radially, the angular frequency of radial oscillations is

he Lagrangian of a particle of mass m moving in one dimension is given by where b is a positive constant. The coordinate of the particle x(t) at time t is given by : (in the following c₁ and c₂ are constants) 

A uniform cylinder of radius r and length l, and a uniform sphere of radius R are released on an inclined plane when their centres of mass are at the same height. If they roll down without slipping, and if the sphere reaches the bottom of the plane with a speed V, then the speed of the cylinder when it reaches the bottom is

he components of a vector potential  A = (A₀, A₁, A₂, A₃) 
are given by A = k(–xyz, yzt, zxt, xyt)
where k is a constant. The three components of the electric field are

In the Born approximation, the scattering amplitude f(q) for the Yukawa potential   is given by 

If Ψ _n/m denotes the eigenfunction of the Hamiltonian with a potential V = (r) then the expectation value of the operator L_x² + L_y² in the state  is

Ψ

An oscillating current I(f) = I₀ exp (–iwt) flows in the direction of the y-axis through a thin metal sheet of area 1.0 cm2 kept in the xy-plane. The rate of total energy radiated per unit area from the surfaces of the metal sheet at a distance of 100 m is

Consider a two-dimensional infinite square well

Its normalized eigenfunctions are 

where nx, ny = 1, 2, 3, .... If a perturbation 

is applied, then the correction to the energy of the first excited state to order V₀ is

Consider a system of two Ising spins S₁ and S₂ taking values ±1 with interaction energy given by e = –JS₁S₂, when it is in thermal equilibrium at temperature T. For large T, the average energy of the system varies as C/k_BT, with C given by

Consider three particles A, B and C, each with an attribute S that can take two values ±1. Let S_A = 1, S_B = 1 and S_C = –1 at a given instant. In the next instant, each S value can change to – S with probability 1/3. The probability that S_A + S_B + S_C remains unchanged is

Find the missing letter
A    B    C    D
F     I     L    O
K    P    U    Z
P   W    D    ?

Consider a right-angled triangle ABC where  AB = AC = 3. A rectangle APOQ is drawn inside it, as shown, such that the height of the rectangle is twice its width. The rec tangle is moved horizontally by a distance 0.2 as shown schematically in the diagram (not to scale).

What is the value of the ratio 

80 gsm paper is cut into sheets of 200 mm × 300 mm size and assembled in packets of 500 sheets. What will be the weight of a packet? (gsm = g/m²)

Three indentical flat equilateral-triangular plates of side 5 cm each are placed together such that they form a trapezium. The length of the longer of the two parallel sides of this trapezium is

An archer climbs to the top of a 10 m high building and aims at a bird atop a tree 17 m away. The line of sight from the archer to the bird makes an angle of 45° to the horizontal. What is the height of the tree?

Consider the set of numbers {17¹, 17², ..., 17³⁰⁰}. How many of these numbers end with the digit 3 ?

Find the missing number in the triangle.

The time gap between the two instants, one before and one after 12.00 noon, when the angle between the hour hand and the minute hand is 66°, is

A merchant buys equal number of shirts and trousers and pays Rs. 38000. If the cost of 3 shirts is Rs. 800 and that of a trouser is Rs. 1000, then how many shirts were bought?

In the growing year of child m the  height increases as the squareroot of the age while the weight increasesin the direct proportion to the age. The ratio of the weight ti the square of the height in this pharse of growth

In 450 g of pure coffee powder 50 g of chicory is added. A person buys 100 g of this mixture and adds 5 g of chicory to that. What would be the rounded-off percentage of chicory in this final mixture?

Suppose in a box there are 20 red, 30 black, 40 blue and 50 white balls. What is the minimum number of balls to be drawn, without replacement, so that you are certain about getting 4 red, 5 black. 6 blue and 7 white balls ?

Suppose     
x D y = (x – y)²
x O y = (x + y)²
x * y = (x × y)^–1
x · y = x × y
+, – and × have their usual meanings. What is the value of {(197 o 315) – (197  315)} • (197 * 315)?

Students in group A obtained the following marks : 40, 80, 70, 50, 60, 90, 30. Students in group B obtained 40, 80, 35, 70, 85, 45, 50, 75, 60 marks. Define dispersion (D) = (maximum marks – minimum marks), and relative dispersion (RD) = 
Then,

The following diagram shows 2 perpendicularly inter-grown prismatic crystals (twins) of identical shape and size. What is the volume of the object shown (units are arbitrary)?

If A × B = 24, B × C = 32, C × D = 48 then A  × D

Suppose n is a positive integer.  Then ( n² + 1) (2n + 1)

There is a train of length 500 m, in which a man is standing at the rear end. At the instant, the rear end crosses a stationary observer on a platform, the man starts walking from the rear to the front and the front to the rear of the train at a constant speed of 3 km/hr. The speed of the train is 80 km/hr. The distance of the man from the observer at the end of 30 minutes is

A rectangular area of sides 9 and 6 units is to be covered by square tiles of sides 1, 2 and 5 units. The minimum number of tiles needed for this is

If all horses are donkeys, some donkeys are monkeys, and some monkeys are men, then which statement must be true?

One gram of salt is dissolved in water ttjat is filled to a height of 5 cm in a beaker of diameter 10 cm. The accuracy of length measurement is 0.01 cm while that of mass measurement is 0.01 mg. When measuring the concentration C, the fractional error  C/C is

A system can have three energy levels : E = 0, ±. The level E = 0 is doubly degenerate, while the others are non-degenerate. The average energy at inverse temperature  is

For a particular thermodynamic system the entropy S is related to the internal energy U and volume V by  S = c U^3/4 V^1/4
where c is a constant. The Gibbs potential  G = U – TS + pV  for this system is

An op amp-based voltage follower

A particle of mass m in three dimensions is in the potential 
Its ground state energy is

Which of the graphs below gives the correct qualitative behaviour of the energy density E_T(y) of blackbody radiation of wavelength  at two temperatures T₁ and T₂(T₁ < T₂)?

Given that the uncertainty p_r in the ground state
of the hydrogen atom is

An RC network produces a phase-shift of 30°. How many such RC networks should be cascaded together and connected to a Common Emitter amplifier so that the final circuit hebave1 as an oscillator?

he free energy F of a system depends on a thermodynamic variable as F = –a² + b⁶
with a, b > 0. The value of , when the system is in thermodynamic equilibrium, is

The inner shield of a triaxial conductor is driven by an (ideal) op-amp follower circuit as shown. The effective capacitance between the signal-carrying conductor and ground is

Consider a system of two non-interacting identical fermions, each of mass m in an infinite square well potential of width a. (Take the potential inside the well to be zero and ignore spin). The composite wavefunction for the system with total energy
is

A particle of mass m in the potential  is in an eigenstate of energy . The corresponding unnormalized eigenfunction is

A particle of mass m and coordinate q has the Lagrangian 
where  is a constant. The Hamiltonian for the system is given by

If and C is the circle of unit radius in the plane defined by z = 1, with the centre on the z-axis, then the value of the integral is

Given 
for |t| < 1, the value of P₅(–1) is

A charged particle is at a distance d from an infinite conducting plane maintained at zero potential. When releasedfrom rest, the particle reaches a speed u at a distance d/2 from the plane. At what distance from the plane will the particle reach the speed 2u ?

Consider the matrixThe eigenvalues of M are

Consider the differential equation  with the initial conditions x(0) = 0 and x(0) = 1. The solution x(t) attains its maximum value when t is

A light source is switched on and off at a constant frequency f. An observer moving with a velocity u with respect to the light source will observe the frequency of the switching to be

If C is the contour defined by | z | = , the value of the integral 0 is

The time period of a simple pendulum under the influence of the acceleration due to gravity g is T. The bob is subjected to an additional acceleration of magnitude g in the horizontal direction. Assuming small oscillations, the mean position and time period of oscillation, respectively, of the bob will be

Consider an electromagnetic wave at the interface between two homogeneous dielectric media of dielectric constants  and  Assuming >  and no charges on the surface, the electric field vector E' and the displacement vector D' in the two media satisfy the following inequalities

If the electrostatic potential in spherical polar co-ordinates is where  and r₀ are constants, then the charge density at a distance r = r₀ will be

A current i_p flows through the primary coil of a transformer. The graph of i_p (t) as a function of time t is shown in the figure below.

A time-dependent current (where K is a constant) is switched on at t = 0 in an infinite current-carrying wire. The magnetic vector potential at a perpendicular distance a from the wire is given (for time t > a/c) by

The pressure of a nonrelativistic free Fermi gas in three-dimensions depends, at T = 0, on the density of fermions n as

A double slit interference experiment uses a laser emitting light of two adjacent frequencies v₁ and v₂(v₁ < v₂). The minimum path difference between the interfering beams for which the interference pattern disappears is

The recently-discovered Higgs boson at the LHC experiment has a decay mode into a photon and a Z boson. If the rest masses of the Higgs and Z boson are 125 GeV/c² and 90 GeV/c² respectively, and the decaying Higgs particle is at rest, the energy of the photon will approximately be

A permanently deformed even-even nucleus with J^p = 2+ has rotational energy 93 keV. The energy of the next excited state is

How much does the total angular momentum quantum number J change in the transition of Cr (3d⁶) atom as it ionizes to Cr²⁺ (3d⁴)?