A real valued function f (x) satisfies the functional equation *f(x – y) = f (x)f (y) – f(a – x)f(a + y)* where *a* is a given constant and *f(0) = 1, f(2a – x)* is equal to

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

Let *f* : (–1, 1) B, be a function defined by f(x) = , then *f* is both one-one and onto when B is the interval

Let *R =* {(3, 3) (6, 6) (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set *A* = {3, 6, 9, 12}. What is the relation then?

What is the domain of the function f(x) = ?

Let *R* = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set *A* = {1, 2, 3, 4}. What is the relation R here?

If *f : R R* satisfies *f (x + y) = f (x) + f (y)*, for all and *f (1) = 7*, then is

Domain of definition of the function

A function *f* from the set of natural numbers to integers defined by

The graph of the function *y = f (x)* is symmetrical about the line *x = 2*, then

If f : R S, defined by *f(x)* = *sin x -* *cos x + 1*, is onto, then what will be the interval of *S?*

What is the range of the function f (x) = ^{7 - x}0_{x - 3} ?

The function is

If x1, x2, x3 and y1, y2, y3 are both in G.P. with the same common ratio, then the points (x1, y1), (x2, y2) and (x3, y3)

What is the domain of ?

What is the period of ?

Which one is not periodic?

Let T be the rth term of an A.P. whose first term r

is a and common difference is d. If for some positive integers m, n, m 1 n, Tm = 1 , and Tn = 1 , then

a – d equals

If the system of linear equations x + 2ay + az = 0, x + 3by + bz = 0, x + 4cy + cz = 0 has a nonzero solution, then a, b, c

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression is equals

Let R1 and R2 respectively be the maximum ranges up and down on an inclined plane and R be the maximum range on the horizontal plane. Then, R1, R,R2 arein

Let f (x) be a polynomial function of second degree. If f (1) = f (–1) and a, b, c are in A.P., then f '(a), f '(b) and f '(c) are in

Fifth term of an GP is 2, then the product of its 9 terms is

Sum of infinite number of terms in GP is 20 and sum of their square is 100. The common ratio of GP is

If 1, log9(31 – x + 2), log3[4 × 3x – 1] are in AP. then x equals

A screw gauge gives the following reading when used to measure the diameter of a wire.

Main scale reading : 0 mm

Circular scale reading : 52 divisions

Given that 1 mm on main scale corresponds to 100 divisions of the circular scale.

What will be the diameter of wire from the above data?

A thermally insulated vessel contains an ideal gas of molecular mass *M* and the ratio of specific heats It is moving with speed *v* and is suddenly brought to rest. Assuming no heat is lost to the surroundings, its temperature increases by

**Direction:** The question has a paragraph followed by two statements, Statement-1 and Statement-2. Of the given four alternatives after the statements, choose the one that describes the statements.

A thin air film is formed by putting the convex surface of a plane-convex lens over a plane glass plate. With monochromatic light, this film gives an interference pattern due to light reflected from the top (convex) surface and the bottom (glass plate) surface of the film.

**Statement-1:** When light reflects from the air-glass plate interface, the reflected wave suffers a phase change of p.

**Statement-2:** The centre of the interference pattern is dark.

Two particles are executing simple harmonic motion of the same amplitude *A* and frequency w along the *x-axis*. Their mean position is separated by distance *X _{0} (X_{0} > A)*. If the maximum separation between them is (

*X*), the phase difference between their motion is

_{0}+ A

Let the *x–z* plane be the boundary between two transparent media. Medium 1 in *z ≥ 0* has a refractive index of 2 and medium with *z < 0* has a refractive index of . A ray of light in medium 1 given by the vector is incident on the plane of separation. What will be the angle of refraction in medium 2?

A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, what will be the angular speed of the disc?

Two bodies of masses *m* and *4m* are placed at a distance *r*. What will be the gravitational potential at a point on the line joining them where the gravitational field is zero?

A fully charged capacitor *C* with initial charge *q _{0}* is connected to a coil of self-inductance

*L*at

*t = 0*. What is the time at which the energy is stored equally between the electric and the magnetic fields?

Work done in increasing the size of a soap bubble from a radius of 3 cm to 5 cm is nearly.

[Surface tension of soap solution = 0.03 N m^{-1}]

Two identical charged spheres suspended from a common point by two massless strings of length *l* are initially a distance *d(d < < l)* apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the charges approach each other with a velocity *v*. Then as a function of distance x between them

Three perfect gases at absolute temperatures *T _{1}, T_{2}* and

*T*are mixed. The masses of molecules are

_{3}*m*and

_{1}, m_{2}*m*and the number of molecules are

_{3}*n*and

_{1}, n_{2}*n*respectively. Assuming no loss of energy, the final temperature of the mixture is

_{3}

If a wire is stretched to make it 0.1% longer, its resistance will

A car is fitted with a convex side-view mirror of focal length 20 cm. A second car 2.8 m behind the first car is overtaking the first car at a relative speed of 15 m s^{-1}. The speed of the image of the second car as seen in the mirror of the first one is

The electrostatic potential inside a charged spherical ball is given by where *r* is the distance from the centre; *a, b* are constants. Then the charge density inside the ball is

An object moving with a speed of *6.25 m s ^{-1}*, is decelerated at a rate given by , where

*v*is the instantaneous speed. What would be the time taken by the object to come to rest?

A Carnot engine operating between temperatures *T _{1}* and

*T*has efficiency 1/6. When

_{2}*T*is lowered by 62 K, its efficiency increases to 1/3. Then what will be

_{2}*T*and

_{1}*T*

_{2 }?

A current *I* flows in an infinitely long wire with a cross-section in the form of a semicircular ring of radius *R*. What is the magnitude of the magnetic induction along its axis?

A boat is moving due east in a region where the earth’s magnetic field is *5.0 × 10 ^{-5} N A^{-1}m^{-1}*due north and horizontal. The boat carries a vertical aerial

*2 m*long. If the speed of the boat is

*1.50 m s*, what will be the magnitude of the induced emf in the wire of aerial?

^{-1}

A Carnot engine operating between temperatures *T _{1}* and

*T*has efficiency 1/6. When

_{2}*T*is lowered by 62 K, its efficiency increases to 1/3. Then what will be

_{2}*T*and

_{1}*T*

_{2 }?

A current *I* flows in an infinitely long wire with a cross-section in the form of a semicircular ring of radius *R*. What is the magnitude of the magnetic induction along its axis?

A resistor *R* and 2 capacitor in series is connected through a switch to *200 V* direct supply. Across the capacitor is a neon bulb that lights up at *120 V*. Calculate the value of *R* to make the bulb light up 5 s after the switch has been closed. (log_{10}2.5 = 0.4)

The transverse displacement *y(x,t)* of a wave on a string is given by

This represents a

What is the energy required for the electron excitation in Li^{++ }from the first to the third Bohr orbit?

A mass *M*, attached to a horizontal spring, executes SHM with an amplitude *A _{1}*. When the mass

*M*passes through its mean position then a smaller mass

*m*is placed over it and both of them move together with amplitude

*A*.

_{2}What is the ratio of ?

The half-life of a radioactive substance is 20 minutes. What will be the approximate time interval *(t _{2} – t_{1})* between the time

*t*when 2/3 of it has decayed and time

_{2}*t*when 1/3 of it had decayed?

_{1}