List of top Mathematics Questions asked in KEAM

Let $A(-1,\,2)$, $B(1,\,-2)$ and $C(-2,\,-2)$ be vertices of the triangle $ABC$. The equation of the line passing through $C$ and parallel to $AB$ is

\(x = \frac{1 + \cos 2\theta}{\tan \theta - \sec \theta}\) and \(y = \frac{\tan \theta + \sec \theta}{\sec^2 \theta}\), then \(\frac{y}{x} =\)

If \(\cos^{-1} x - \sin^{-1} x = \frac{\pi}{6}\), then \(x\) is equal to

\(\sin 60^{\circ} - \sin 80^{\circ} + \sin 100^{\circ} - \sin 120^{\circ} =\)

The general solution of the differential equation \((1 + y)dx - (1 - x)dy = 0\) is

The value of \(\int_{0}^{\pi/3}\frac{\tan t}{\cos t} dt\) is equal to

Consider the linear programming problem:
Maximize: \(z = \alpha x + 6y\)
Subject to the constraints
\(3x + 2y \leq 60\)
\(x + 2y \leq 40\)
\(x, y \geq 0\)
If every point in the line segment joining (20, 0) and (10, 15) is optimal solution of the L.P.P, then the value of \(\alpha\) is equal to

The integrating factor of the differential equation \((1 + x^{2})dy = (1 - 2xy)dx\) is

The area of the region bounded by \(\frac{x^{2}}{16} + \frac{y^{2}}{25} = 1\) and the line segment joining (0,5) and (4,0) in the first quadrant is

The value of \(\int_{-1}^{1}|x - 3|dx\) is equal to

\(\int \frac{\sin 4\theta}{\sin 2\theta} \, d\theta =\)

\(\int \frac{x}{5-x^2} \, dx =\)

\(\int \frac{e^{x}}{e^{-x} + 3e^{x}} dx =\)

The function \(f(x) = 2\cos x - x + 3\) is

If \(e^{x}(y + 2\sqrt{1 + x}) = 5\), then \(-\frac{dy}{dx}\) at \((0,3)\) is

If \(y = \sec (\tan^{-1} x)\), then \(\frac{dy}{dx}\) at \(x = \sqrt{3}\) is equal to

If \([x]\) denotes the greatest integer less than or equal to \(x\) for \(x \in \mathbb{R}\), then the value of \(\lim_{x \to 0^{+}} \left[ 2[x] - \frac{x}{|x|} \right]\) is equal to

$\displaystyle\lim_{x \to 0}\frac{1 - \cos 4x}{\tan^2 2x}$ is equal to

\(\lim_{x \to 11} \frac{x - 11}{\sqrt{x^2 + 48} - 13} =\)

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

The integrating factor of the differential equation \( \sin x\, dy = \frac{1}{2}(\sin2x + 2y\cos x)\,dx \) is

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

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

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

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