List of top Questions asked in KEAM

Find the TRUE statement of the algebraic operations of scalar and vector quantities.
The dimensional formula of the gravitational constant is \( M^a L^b T^c \), the values of \( a \), \( b \), and \( c \) are respectively:
If the position of the particle is \( \mathbf{r} = 3 \hat{i} + 2 t^2 \hat{j} \), then the magnitude of its velocity at \( t = 5 \) second in \( {ms}^{-1} \) is:
The degree of the differential equation \( (y^m)^2 + (\sin y')^4 + y = 0 \) is:
The general solution of \( \frac{dy}{dx} + y = 5 \) is:
The differential equation \( \frac{dy}{dx} + x = A \) (where A is constant) represents:
Find the area of the smaller segment cut-off from the circle \( x^2 + y^2 = 25 \) by \( x = 3 \):
Find the area bounded by the curves \( y = 2x \) and \( y = x^2 \):
Evaluate \( \int_{-\pi/2}^{\pi/2} \sin^9 x \cos^2 x \, dx \):
Evaluate \( \int_0^1 \log \left( \frac{1}{x - 1} \right) \, dx \):
Evaluate \( \int \frac{dx}{x^2 (x^4 + 1)^{3/4}} \):
Evaluate \( \int x e^{x^2} \, dx \):
Evaluate \( \int x \cos x \, dx \):
The integral \( \int \frac{dx}{1 + e^x} \) is:
The limit \( \lim_{x \to 10} \frac{x - 10}{\sqrt{x + 6} - 4} \) is equal to:
The maximum value of \( y = 12 - |x - 12| \) in the range \( -11 \leq x \leq 11 \) is:
The function \( f(x) = 2x^3 + 9x^2 + 12x - 1 \) is decreasing in the interval:
If \( y = \frac{x^2}{x - 1} \), then \( \frac{dy}{dx} \) at \( x = -1 \) is:
If \( y = \tan^{-1} \left( \frac{\cos x - \sin x}{\cos x + \sin x} \right) \), \( \frac{-\pi}{2}<x<\frac{\pi}{2} \), then \( \frac{dy}{dx} \) is:
If \( f(x) = \sin^{-1}(\cos x) \), then \( \frac{d^2 y}{dx^2} \) at \( x = \frac{\pi}{4} \) is:
If \( f(x) = \cos x - \sin x \), and \( x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \), then \( f' \left( \frac{\pi}{3} \right) \) is equal to:
Let \( f(x) = x - \lfloor x \rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function and \( x \in (-1, 2) \). The number of points at which the function is not continuous is:
If \( f(x) = \left\{ \begin{array}{ll} mx + 1, & \text{when } x \leq \frac{\pi}{2} \\ \sin x + n, & \text{when } x > \frac{\pi}{2} \end{array} \right. \) is continuous at \( x = \frac{\pi}{2} \), then the values of \( m \) and \( n \) are:
Evaluate \( \lim_{x \to 0} \frac{\sin 2x + \sin 5x}{\sin 4x + \sin 6x} \):
If \( g(x) = -\sqrt{25 - x^2} \), then \( g'(1) \) is: