List of top Questions asked in KEAM- 2025

In the graphical method of a linear programming problem, the optimal solution lies

Among the following pairs of vectors, if the resultant of two vectors can never have magnitude 4 units, the magnitudes of the vectors are

\( [L^0 M^0 T^{-1}] \) is the dimensional formula for

The area of the region bounded by \( y = x^{5/2} \) and \( y = x \) (in square units) is

If \( y'(x) = 2y \), \( y(x) \ge 0 \) and \( y(0) = e^2 \), then \( y(x) = \)

\( \int_{0}^{1} \frac{3^{2x}}{3^x + 1}\,dx = \)

The value of \( \int_{\pi/10}^{2\pi/5} \frac{\cot^3 x}{1+\cot^3 x}\,dx \) is equal to

If \( \int_{-\sqrt{3}}^{1} (-6x^2 + 18)\,dx = \alpha + \beta\sqrt{3} \), then the value of \( \alpha + \beta \) is equal to

\( \int \frac{2x^2 + 4x + 3}{x^2 + x + 1} \, dx = \)

\( \int \frac{\sin^{-1}x}{\sqrt{1-x^2}} \, dx = \)

\( \int e^x(x^2-2)\cos(e^x(x^2-2x)) dx = \)

\( \int x^7 (x^8 + 1)^{-3/4} \, dx = \)

\( \int e^x \sec x (1+\tan x) dx = \)

Let \( f(x) = x^2 + ax + \beta \). If \( f \) has a local minimum at \( (2,6) \), then \( f(0) \) is equal to

The function \( f(x) = 2x^3 - 3x^2 - 36x + 28 \) is increasing in

The surface area of a solid hemisphere is increasing at the rate of \( 8 \, \text{cm}^2/\text{sec} \) (retaining its shape). Then the rate of change of its volume (in \( \text{cm}^3/\text{sec} \)), when the radius is \( 5 \,\text{cm} \), is

The function \( f(x) = x^2(x-2) \) is strictly decreasing in

If \( y = \tan^{-1}(x^2 - x) \), then \( \frac{dy}{dx} = \)

Let \( f(x) = (\cos^2 x)(a + \cos x) \). If \( f'\left(\frac{\pi}{3}\right) = 0 \), then the value of \( a \) is equal to

\( \lim_{x \to 0} \frac{\sqrt{\cos 2x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \) is equal to

Let \( f(x) = |\sin 3x| - |\cos 3x| \), where \( \frac{\pi}{6} \le x \le \frac{\pi}{3} \). Then the value of \( f\left(\frac{\pi}{4}\right) \) is

Let \( h(x) = f(\sqrt{g(x)}) \). If \( f'(3) = 6 \), \( g'(3) = 3 \) and \( g(3) = 9 \), then the value of \( h'(3) \) is equal to

If \( f(x) = |x^2 + x - 6| \) is not differentiable at \( x = a \) and \( x = b \), then \( a^2 + b^2 = \)

If \( f(x) = \frac{3^x}{3^x + \sqrt{3}} \), then \( f(x) + f(1-x) \) is equal to

The standard deviation of the numbers \( -3, 0, 3, 8 \) is