List of top Questions asked in KEAM

When a cricketer catches a ball in 30 s, the force required is 2.5 N. The force required to catch that ball in 50 s is:
If the initial speed of the car moving at constant acceleration is halved, then the stopping distance \( S \) becomes:
The angle between \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \) is:
If the time period \( T \) of a satellite revolving close to the earth is given as \( T = 2\pi R^a g^b \), then the value of \( a \) and \( b \) are respectively (where \( R \) is the radius of the earth):
If

\[ \int x e^{-x} \, dx = M e^{-x} + C, \quad \text{where } C \text{ is an arbitrary constant, then } M \text{ is equal to:} \]
When \( y = vx \), the differential equation \[ \frac{dy}{dx} = \frac{y}{x} + \frac{f\left( \frac{y}{x} \right)}{f'\left( \frac{y}{x} \right)} \] reduces to:
The value of \[ \int_0^{\frac{\pi}{2}} \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} \, dx \] is equal to:
The area bounded by the curves \( y = x^2 \) and \( y = 2x \) in the first quadrant, is equal to:
The integral

\[ \int \frac{dx}{x^8 \left( 1 + x^7 \right)^{2/3}} \] is equal to:

The limit: \[ \lim_{x \to 0} \frac{\sin \left( \pi \sin^2 x \right)}{x^2} \] is equal to:

Let \( \left\lfloor x \right\rfloor \) be the greatest integer less than or equal to \( x \). Then \[ \lim_{x \to 0^-} \frac{x \left( \left\lfloor x \right\rfloor + |x| \right)}{|x|} \] is equal to:
The integrating factor of

\[ (1 + 2e^{-x}) \frac{dy}{dx} - 2e^{-x} y = 1 + e^{-x} \] is:
The solution of \( \frac{dy}{\cos y} = dx \) is:
Evaluate the integral \[ \int_{-500}^{500} \ln \left( \frac{1000 + x}{1000 - x} \right) dx \]
If \[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ? \]
The integral

\[ \int \frac{\sec x}{(\sec x + \tan x)^2} \, dx \] is:
The solution of \( \frac{e^y}{dx} = x + 2 \) is:
The integral \( \int (x^4 - 8x^2 + 16x)(4x^3 - 16x + 16) \, dx \) is:
The value of \( \int_{-4}^{-2} \left[ (x+3)^3 + 2 + (x+3)\cos(x+3) \right] \, dx \) is equal to:
Find the value of \[ \left| \left( \frac{1+i}{\sqrt{2}} \right)^{2024} \right|. \]
If \( x = 5 \tan t \) and \( y = 5 \sec t \), then \( \frac{dy}{dx} \) at \( t = \frac{\pi}{3} \) is:
If \( \left\lfloor x^2 \right\rfloor \) is the greatest integer less than or equal to \( x^2 \), then \[ \int_0^{\sqrt{2}} \left\lfloor x^2 \right\rfloor \, dx = \]
The solution of \( (y \cos y + \sin y) \, dy = (2x \log x + x) \, dx \) is:
The area enclosed by the curve \[ x = 3 \cos \theta, \quad y = 5 \sin \theta, \quad 0 \leq \theta \leq 2\pi, \] is equal to:
The coefficient of \( x^3 \) in the expansion of \[ \frac{1}{(1 + 2x)^{-10}} \] is: