List of top Questions asked in KEAM

Evaluate the following statement: \[ f(x) = \sin(|x|) - |x| \text{ is not differentiable at } x = \underline{\hspace{2cm}} \]
Find the minimum of the function \[ f(x) = |x+2| \]
Evaluate the following integral: \[ \int e^x \left[\frac{1}{1+x(1+x)^2}\right] dx \]
Find the integral: \[ \int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx \]
Equation of line through (0, 0, 1) and (1, 1, 0)
Evaluate: \[ \sec \left( \cos^{-1} \left( \frac{2024}{2025} \right) \right) \]
Find the probability of getting at most 2 heads when 4 coins are tossed.
Coefficient of \( x^9 \) in the expansion of \( \left( 4 - \frac{x^2}{9} \right)^{12} \)
If \( n \) cells of emf \( E \) and internal resistance \( r \) are connected in parallel, what is the equivalent emf and internal resistance of the combination?
Water flows through a pipe with velocity \( V_1 = 3 \, \text{m/s} \) where the area of the pipe is \( A_1 \). What is the velocity \( V_2 \), where the diameter of the pipe is half of that at area \( A_1 \)?
Find the range of \( f(x) = \log_e(4x^2 - 4x + 1) \)
Evaluate the integral: \[ \int \frac{\cos \theta}{2 - \sin^2 \theta} \, d\theta \]
Which of the following contains 2 primary –OH groups in its structure?
Molecule with 2\( \sigma \) and 2\( \pi \) bonds:
Evaluate the following integral: \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( x^4 + x^3 + x \right) \cos x \, dx \]
If \[ \frac{x}{4} + \frac{x}{3} < 13, \text{ then } x \in \]
10 g of (90 percent pure) \( \text{CaCO}_3 \), treated with excess of HCl, gives what mass of \( \text{CO}_2 \)?
Integrating factor of \[ \frac{dy}{dx} - 2y = 2x - 3 \]
When a rotating disc suddenly shrinks in radius, what happens to the angular velocity \( \omega \)?
Solution set of \( -12x > 38 \text{ for } x \in \mathbb{N} \) \[ -12x > 38 \]
Evaluate the limit: \[ \lim_{x \to 2} \frac{(x^3 - 8) \sin(x - 2)}{x^2 - 4x + 4} \]
The most basic oxide
Find the limit: \[ \lim_{x \to 0} \frac{\sin[x]}{[x]}, \text{ where } [x] \text{ represents greatest integer function} \]

Find the mean deviation of the following data: 

Pressure of pure benzene is 0.75. When 0.2g of a non-volatile solute is added to 39g of benzene, pressure changes to 0.745. The molar mass of solute is?