List of top Mathematics Questions asked in KEAM

Let \( a, b, c \) be positive numbers. If \( a + b + c \geq K \left[ (a + b)(b + c)(c + a) \right]^{1/3} \), then the maximum value of \( K \) is:
Let \( z = x + iy \), where \( y>0 \). If \( z + \overline{z} = 6 \) and \( |z| + | \overline{z} | = 10 \), then \( z = \)
If \( z = 1 + i \), then the maximum value of \( |z + 12 + 9i| \) is
The eccentricity of the ellipse \[ p x^2 + 5y^2 = 80, \quad \text{where } p > 5, \] is \( \frac{\sqrt{3}}{2} \). Then the value of \( p \) is
The line joining the points $ (2,2,2) $ and $ (6,6,6) $ meets the line $$ \frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 5}{-1} $$ at the point 
The angle between the lines $$ \frac{x-1}{2} = \frac{2y+3}{4} = \frac{z+5}{-2} \quad \text{and} \quad \frac{x-3}{4} = \frac{y+1}{-4} = \frac{z+3}{-4} $$ is equal to 
If the lines \( \frac{x-1}{2} = \frac{y-2}{\alpha} = \frac{z-3}{2} \) and \( \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{-2} \) are perpendicular, then the value of \( \alpha \) is:
Evaluate the limit: $$ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{x^2 - 3x - 4} \right) $$ is equal to 
The value of \[ \int_0^{\frac{\pi}{2}} \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} \, dx \] is equal to:
The integral

\[ \int \frac{\sec x}{(\sec x + \tan x)^2} \, dx \] is:
If \( \left\lfloor x^2 \right\rfloor \) is the greatest integer less than or equal to \( x^2 \), then \[ \int_0^{\sqrt{2}} \left\lfloor x^2 \right\rfloor \, dx = \]
If \[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ? \]

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

The integral

\[ \int \frac{dx}{x^8 \left( 1 + x^7 \right)^{2/3}} \] is equal to:
The solution of \( \frac{dy}{\cos y} = dx \) is:
The integral \( \int (x^4 - 8x^2 + 16x)(4x^3 - 16x + 16) \, dx \) is:
If \( x = 5 \tan t \) and \( y = 5 \sec t \), then \( \frac{dy}{dx} \) at \( t = \frac{\pi}{3} \) is:
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 \( \int_{-4}^{-2} \left[ (x+3)^3 + 2 + (x+3)\cos(x+3) \right] \, dx \) is equal to:
If

\[ \int x e^{-x} \, dx = M e^{-x} + C, \quad \text{where } C \text{ is an arbitrary constant, then } M \text{ is equal to:} \]
The solution of \( \frac{e^y}{dx} = x + 2 \) is:
The solution of \( (y \cos y + \sin y) \, dy = (2x \log x + x) \, dx \) is:
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 area enclosed by the curve \[ x = 3 \cos \theta, \quad y = 5 \sin \theta, \quad 0 \leq \theta \leq 2\pi, \] is equal to: