Question & Answer

If f(y) = e^y, g(y) = y ; y > 0 and , then

Let f(x) be a function satisfying f'(x) = f (x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = x² . Then what is the value of the integral  ?

What is the area of the region bounded by the curves y = |x – 1| and y = 3 – |x|?

The value of  is

The area bounded by the curves y = lnx, y = ln |x|, y = |lnx| and y = |ln |x|| is

What is the value of ?

If y = f(x) makes +ve intercept of 2 and 0 unit x and y and encloses an area of 3/4 square unit with the axes then  is

What is the value of ?

Suppose the radius of earth is half of its present value (Mass remaining unchanged). The value of g at a height equal to the final radius is 10x m/s2. Find the value of x.

Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are)

Two spherical planets P and Q have the same uniform density , masses MP and MQ, and surface areas A and 4A, respectively. A spherical planet R also has uniform density  and its mass is (MP + MQ). The escape velocities from the planets P, Q and R, are VP, VQ and VR, respectively. ThenTwo spherical planets P and Q have the same uniform density, masses MP and MQ, and surface areas A and 4A, respectively. A spherical planet R also has uniform density  and its mass is (MP + MQ). The escape velocities from the planets P, Q and R, are VP, VQ and VR, respectively. Then

Suppose the radius of earth is half of its present value (Mass remaining unchanged). The value of g at a height equal to the final radius is 10x m/s2. Find the value of x.

Two bodies, each of mass M, are kept fixed with a separation 2L. A particle of mass m is projected from the midpoint of the line joining their centres, perpendicular to the line. The gravitational constant is G. The correct statement(s) is (are)

Two spherical planets P and Q have the same uniform density , masses MP and MQ, and surface areas A and 4A, respectively. A spherical planet R also has uniform density  and its mass is (MP + MQ). The escape velocities from the planets P, Q and R, are VP, VQ and VR, respectively. ThenTwo spherical planets P and Q have the same uniform density, masses MP and MQ, and surface areas A and 4A, respectively. A spherical planet R also has uniform density  and its mass is (MP + MQ). The escape velocities from the planets P, Q and R, are VP, VQ and VR, respectively. Then

The weight of an object on the surface of the Earth is 40 N. Its weight at a height equal to the radius of the Earth is

The ratio between masses of two planets is 3 : 5 and the ratio between their radii is 5 : 3. The ratio between
their acceleration due to gravity will be

A point mass when enters d depth below the earth’s surface such that the g reduces by 1% then the value of d is [Take Re = 6400 km]

If the radius of earth were to shrink by one percent and mass remains same, then acceleration due to gravity on the surface of the earth would

 If the change in value of g at a height h above the surface of the earth is same as that at a depth d below it, when both h and d are much smaller than the radius of the earth, then

A planet is at R distance from sun and its time period of revolution is T. What will be its new time period of revolution if it is brought 0.5 R distance closer to sun?

The time period of a satellite of the earth is 10 hours. If the separation between the earth and the satellite is increased to 4 times the previous value, then what will be the new time period of the satellite?

Which of the following quantities related to planets remains same during their motion in elliptical orbits as seen from the sun?

 equals

What is the value ?

Suppose the cubic x³ – px + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds?

Let 

Then which one of the following is true?

How many real solutions does the equation x⁷ + 14x⁵ + 16x³ + 30x – 560 = 0 have?

If p and q are positive real numbers such that p² + q² = 1, then the maximum value of (p + q) is

The function f : R / {0}  R given by

can be made continuous at x = 0 by defining f(0) as

Let f : R  R be a function defined by f(x) = min {x + 1, |x| + 1}. Then which of the following is true ?

The function f(x) = tan^-1(sin x + cos x) is an increasing function in

A value of c for which conclusion of Mean Value Theorem holds for the function f(x) = log_e x on the interval [1, 3] is

If x^m∙ yⁿ = (x + y)^m+n, then dy/dx is

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having a fence are of the same length x. What will be the maximum area enclosed by the park?

The set of points where   is differentiable, is

What will be the angle between the tangents to the curve y = x² – 5x + 6 at the points (2, 0) and (3, 0)?

The function g(x) = x/2 + 2/x has a local minimum at

If x is real, what is the maximum value of  ?

A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³/min. When the thickness of ice is 5 cm, then what will be the rate at which the thickness of ice decreases?

Let  and  be the distinct roots of ax² + bx + c = 0, then  is equal to

What is the value of ?

The real number x, when added to its inverse gives the minimum value of the sum at x equal to

If  then f(x) is

If the function f(x) = 2x³ – 9ax² + 12a²x + 1, where a > 0, attains its maximum and minumum at p and q respectively such that p² = q, then a equals

Let  f(x) is continuous in  then what will be  ?

If  then what are the values of a and b?

Let f(a) = g(a) = k and their nth derivatives fⁿ(a), gⁿ(a) exist and are not equal for some n.

Further if  then what will be the value of k?

What is the value of ?

The normal to the curve x = a(cos+  sin), y = a(sin – cos) at any point  is such that