List of top Mathematics Questions asked in KEAM

The value of the integral \( \int_{2}^{4} \left( \frac{\log t}{t} \right) \, dt \) is equal to:
The solution of the differential equation \( (kx - y^2) dy = (x^2 - ky) dx \) is:
\( \int \frac{\sqrt{5 + x^2}}{x^4} \, dx \) is equal to:
\( \int \frac{(1 + x) e^x{\sin^2(x e^x)} dx \) is equal to:}
\( \int \frac{xe^x}{(1+x)^2 \, dx \) is equal to:}
\( \int e^x (\sin x + 2 \cos x) \sin x \, dx \) is equal to:
\( \int \sqrt{1 + \cos x} \, dx \) is equal to:
\( \int \frac{\sqrt{x^2 - 1}}{x} \, dx \) is equal to:
The value of \( \int_{0}^{1} \frac{dx}{e^x + e} \) is equal to:
The minimum value of \( \sin x + \cos x \) is:
\( \int \frac{1}{x^2 (x^4 + 1)^{3/4 dx \) is equal to:}
The slope of the normal to the curve \( y = x^2 - \frac{1}{x^2} \) at \( (-1, 0) \) is:

The function \[ f(x)= \begin{cases} 2x^2-1, & \text{if } 1 \leq x \leq 4 \\ 151-30x, & \text{if } 4 < x \leq 5 \end{cases} \] is not suitable to apply Rolle's theorem since:

The radius of a cylinder is increasing at the rate of 5 cm/min so that its volume is constant. When its radius is 5 cm and height is 3 cm the rate of decreasing of its height is:
The slope of the tangent to the curve \( y^2 e^{xy} = 9e^{-3}x^2 \) at \( (-1, 3) \) is:
If \( y = \sin^{-1} \left( 2x \sqrt{1 - x^2} \right) \), \( -\frac{1}{\sqrt{2}} \le x \le \frac{1}{\sqrt{2}} \), then \( \frac{dy}{dx} \) is equal to:
The function \( f(x) = 2x^3 - 15x^2 + 36x + 6 \) is strictly decreasing in the interval:
A straight line parallel to the line \( 2x - y + 5 = 0 \) is also a tangent to the curve \( y^2 = 4x + 5 \). Then the point of contact is:
If \( x = a \cos^3 \theta \) and \( y = a \sin^3 \theta \), then \( 1 + \left( \frac{dy}{dx} \right)^2 \) is:
If \( y = f(x^2 + 2) \) and \( f'(3) = 5 \), then \( \frac{dy}{dx} \) at \( x = 1 \) is:
Let \( f(x) = x^2 + bx + 7 \). If \( f'(5) = 2f'\left(\frac{7}{2}\right) \), then the value of \( b \) is:
If \( x = \sin t \) and \( y = \tan t \), then \( \frac{dy}{dx} = \)

Let \[ f(x)= \begin{cases} ax+3, & x \leq 2 \\ a^2x-1, & x > 2 \end{cases} \] Then the values of \(a\) for which \(f\) is continuous for all \(x\) are:

Let \( R \) be the set of all real numbers. Let \( f: R \to R \) be a function such that \( |f(x) - f(y)|^2 \le |x - y|^3, \forall x, y \in R \). Then \( f'(x) = \)
Let \( f(x) = \int_{1^{x} \sin^2 \left( \frac{t}{2} \right) dt \). Then the value of \( \lim_{x \to 0} \frac{f(\pi + x) - f(\pi)}{x} \) is equal to:}