List of top Questions asked in KEAM

Let \( x \text{ and } y \) be two positive real numbers. Then
\[ \left( x + \frac{1}{x} \right) \left( y + \frac{1}{y} \right) \] is greater than or equal to:
 

If \( |x| \leq 1 \), then \( \sin \left( 2 \sin^{-1} x + \cos^{-1} x \right) \) is equal to:
Find the value of \[ \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right). \] The answer is:
Find the value of \[ \sin \left( 2 \sin^{-1} \left( \frac{1}{2} \right) \right). \] The answer is:
If \[ \sin^{-1} x + \cos^{-1} y = 0, \] then \( x^2 + y^2 \) is equal to:

If \( a \text{ and } b \) are A.M. and G.M. of \( x \text{ and } y \) respectively, then \( x^2 + y^2 \) is equal to: 
 

Simplify the following expression: \[ \cos 18^\circ \cos 42^\circ \cos 78^\circ. \] The simplified form is:
If \[ \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2}, \] then \( x + y + z = \):
If \( \sec(\alpha + \beta) = \frac{\sqrt{7}}{\sqrt{3}} \), then \( \sin(\alpha + \beta) + \tan(\alpha + \beta) \) is equal to:
Simplify the following expression: \[ (\sec A - \cos A)(\tan A - \cot A) \] The simplified form is:
Simplify the following expression: \[ \frac{\sin 7x + \sin 5x}{\cos 7x + \cos 5x} \] The simplified form is:
cos \( A \cos 2A \) is equal to:
If \( \left( 1 + \cos \frac{\pi}{8} \right) \left( 1 + \cos \frac{7\pi}{8} \right) \) =
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 integral \( \int (x^4 - 8x^2 + 16x)(4x^3 - 16x + 16) \, dx \) is:
If \[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ? \]
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 value of \[ \int_0^{\frac{\pi}{2}} \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} \, dx \] is equal to:
The value of \( \int_{-4}^{-2} \left[ (x+3)^3 + 2 + (x+3)\cos(x+3) \right] \, dx \) is equal to:
The area bounded by the curves \( y = x^2 \) and \( y = 2x \) in the first quadrant, is equal to:
The integrating factor of

\[ (1 + 2e^{-x}) \frac{dy}{dx} - 2e^{-x} y = 1 + e^{-x} \] is:
Evaluate the integral \[ \int_{-500}^{500} \ln \left( \frac{1000 + x}{1000 - x} \right) dx \]
The solution of \( (y \cos y + \sin y) \, dy = (2x \log x + x) \, dx \) is:

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: