Let *z, w* be complex numbers such that *z + = 0* and *zw = * Then arg z equals

Let *z _{1}* and

*z*be two roots of the equation

_{2}*z*,

^{2}+ az + b = 0*z*being complex further, assume that the origin,

*z*and

_{1}*z*form an equilateral triangle, then what will be

_{2}*a*?

^{2}

The radius of the circle, having centre at (2, 1), whose one of the chord is a diameter of the circle x2 +y2 –2x–6y+6=0

A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is-

Consider a family of circles which are passing through the point (–1, 1) and are tangent to x-axis. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval-

Three distinct points A, B and C are given in the 2–dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point (–1, 0) is equal to 1/3 . Then the circumcentre of the triangle ABC is at the point :-

The point diametrically opposite to the point (1, 0) on the circle x2 + y2 + 2x + 4y – 3 = 0 is-

A and B are two fixed points and P moves such that PA = nPB where n != 1. Show that locus of P is a circle and for different values of n all the circles have a common radical axis.

The circle x2 + y2 – 6x – 10y + k = 0 does not touch or intersect the coordinate axes and the point (1, 4) is inside the circle. Find the range of the value of k.

Find the equation of the image of the circle x2 + y2 + 16x – 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0?

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?