List of top Questions asked in KEAM

Evaluate \( \lim_{x \to 1} \frac{\sqrt{x+3} \cdot \sqrt{x-1}}{x-1} \)

Given the numbers 4, 7, \( x \), 13, 16, with a mean of 10, find the mean deviation about the mean.

Given \( (3\cos x - 2\sec x)^2 = 9\cos^2 x + 4\tan^2 x + k \), find \( k \).

Max of \( Z = 7x + 10y \) subject to \( x + y \geq 3 \), \( x + 2y \geq 4 \), \( x, y \geq 0 \).

The coefficient of \( x^3 \) in \( (2+x)^n \) is 160. Find the coefficient of \( x^6 \) in \( (2-x^2)^n \).

Find \(\int \frac{x \cos 2x}{\cos x - \sin x} \, dx\)

If 2 vectors \( 4\hat{i} + \ell \hat{j} - 6 \hat{k} \) and \( -6 \hat{i} + 12 \hat{j} + 9 \hat{k} \) are collinear, find \(\ell\).

Evaluate \( \lim_{x \to 0} \frac{x - \tan(3x)}{\sin(2x)} \)

Arithmetic mean & Geometric mean of 2 numbers \( a \) & \( b \) in the ratio 5:3. Find \( \frac{a^2 + b^2}{ab} \).

Given \( y = \frac{1}{1 + \tan x} \), find \( f^{-1}(x) \), where \( 0<x<\frac{\pi}{2} \).

If \( (2 - x)^9 = a_0 + a_1x + \dots + a_9x^9 \), find \( a_1 + a_2 + \dots + a_8 \)

If the area of the circle \( x^2 + y^2 + 8x - 6y + c = 0 \).

Given that \( \vec{a} = 2\hat{i} - \lambda \hat{j} + 5\hat{k} \), \( \vec{b} = \mu \hat{i} + 7\hat{j} + 3\hat{k} \), and the midpoint of \( \overrightarrow{AB} = 3\hat{i} + 2\hat{j} + 4\hat{k} \), find \( \lambda + \mu \).

If \( n(B) = 61 \), \( n(A \cup B) = 99 \), \( n(A \cap B) = 28 \), find \( n(A') \).

Maximum of \( f(x) = \alpha - 4x - x^2 \), find \(\alpha = ?\)

Given \( y = 4e^{-x} - 2e^{-2x} - e^{-3x} \), find \( y'' \).

Find \(\int_{0}^{\frac{\pi}{2}} \sqrt{\cos x \sin 2x} \, dx\)

The distance of the point (10, 10, 10) from the Z-axis.

Find the integral \( \int 2 \, dy = (y + \cos x) \, dx \)

Solve \( 6(2x+3) + x>53 - 2x \).

Find \( \alpha \) if \[ \lim_{x \to 0} \frac{1 - \sec^2(\alpha x)}{\alpha x^2} = -3 \]

Find minimum value of \( \sin x \sin \left( x + \frac{\pi}{3} \right) \)

Find \(\int_{-1}^{1} \frac{\log(1 + |x|)}{1 + |x|} \, dx\)

If \( \alpha = \frac{\pi}{4} \), find \( (\sin \alpha + \sin \beta)^2 + (\cos \alpha + \cos \beta)^2 \).

If \( y = \frac{1 + \tan^2 x}{1 - \tan^2 x} \), find \( y' \left( \frac{\pi}{8} \right) \), where \( 0<x<\frac{\pi}{4} \).