List of top Questions asked in KEAM

If \( f(x) = \lfloor x \rfloor \), where \( \lfloor x \rfloor \) denotes the greatest integer function, and if the domain of \( f \) is \( \{-3.01, 2.99\} \), then the range of \( f \) is

A projectile is projected with a velocity of \( 20 \, {ms}^{-1} \) at an angle of 45° to the horizontal. After some time its velocity vector makes an angle of 30° to the horizontal. Its speed at this instant (in \( {ms}^{-1} \)) is:

The area of the region bounded by the curve \( y = 3x^2 \) and the x-axis, between \( x = -1 \) and \( x = 1 \), is:

The value of \( \int_{-1}^{1} x^2 \sin x \, dx \) is equal to:

The order and degree of the following differential equation: \( \frac{d^2 y}{dx^2} - 2x = \sqrt{y} + \frac{dy}{dx} \), respectively, are:

The value of \( \int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\tan x + \sin x}{1 + \cos^2 x} \, dx \) is equal to:

The solution of the differential equation \( x + y \frac{dy}{dx} = 0 \), given that at \( x = 0 \), \( y = 5 \), is:

The integral \( \int \frac{1 + x^2 + x^4}{(1 - x^3)(1 + x^3)} \, dx \) is equal to:

A train starts from X towards Y at 3 pm (time \( t = 0 \)) with velocity \( v(t) = 10t + 25 \) km per hour, where \( t \) is measured in hours.
Then the distance covered by the train at 5 pm (in km) is:

If \( f(x) = x|x| \), then \( f'(-1) + f'(1) \) is equal to:

In moving a body of mass \( m \) down a smooth incline of inclination \( \theta \) with velocity \( v \), the power required is (g = acceleration due to gravity):

The general solution of the differential equation \( (x + y)^2 \frac{dy}{dx} = 1 \) is:

The value of \( \int_0^1 x(1 - x)^{10} \, dx \) is equal to:

If the voltage across a bulb rated 220V – 60W drops by 1.5% of its rated value, the percentage drop in the rated value of the power is:

If \( n \) small identical liquid drops, each having terminal velocity \( v \), merge together, then the terminal velocity of the bigger drop is:

In the travelling plane wave equation given by \( y = A \sin \omega \left( \frac{x}{v} - t \right) \), where \( \omega \) is the angular velocity and \( v \) is the linear velocity.
The dimension of \( \omega t \) is:

The limiting molar conductances of NaCl, HCl and CH\(_3\)COONa at 300 K are 126.4, 425.9 and 91.0 S cm\(^2\) mol\(^-1\) respectively. The limiting molar conductance of acetic acid at 300 K is:

The function \( f(x) = (x - 4)^2 (1 + x)^3 \) attains a local extremum at the point:

A proton and a photon have the same energy. Then the de-Broglie wavelength of proton \( \lambda_p \) and wavelength of photon \( \lambda_0 \) are related by:

The resistance of a conductivity cell filled with 0.02 M KCl solution is 520 ohm at 298 K. The conductivity of the solution at 298 K is (Cell constant = 130 cm\(^{-1}\)):

The length of the second's hand in a watch is 1 cm. The magnitude of the change in the velocity of its tip in 30 seconds (in cm/s) is:

If the complex numbers \( (2 + i)x + (1 - i)y + 2i - 3 \) and \( x + (-1 + 2i)y + 1 + i \) are equal, then \( (x, y) \) is