List of top Mathematics Questions asked in JEE Advanced

Let $ \alpha $ and $ \beta $ be the real numbers such that $$ \lim_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int_{0}^{x} \frac{1}{1 - t^2} \, dt + \beta x \cos x \right) = 2. $$ Then the value of $ \alpha + \beta $ is __________.
For any two points $ M $ and $ N $ in the $ XY $-plane, let $ \overrightarrow{MN} $ denote the vector from $ M $ to $ N $, and $ \vec{0} $ denote the zero vector. Let $ P, Q $, and $ R $ be three distinct points in the $ XY $-plane. Let $ S $ be a point inside the triangle $ \Delta PQR $ such that $$ \overrightarrow{SP} + 5\overrightarrow{SQ} + 6\overrightarrow{SR} = \vec{0}. $$ Let $ E $ and $ F $ be the mid-points of the sides $ PR $ and $ QR $, respectively. Then the value of $$ \frac{\text{length of the line segment } EF}{\text{length of the line segment } ES} $$ is __________.