Let \( M = \sum_{i=1}^{4} X_i X_i^T \), where \[ X_1^T = [1 \ -1 \ 1 \ 0], X_2^T = [1 \ 1 \ 0 \ 1], X_3^T = [1 \ 3 \ 1 \ 0] \, \text{and} \, X_4^T = [1 \ 1 \ 1 \ 0]. \] Then the rank of \( M \) equals ...............
Let \( f : \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f^\prime \) is continuous on \( \mathbb{R} \) with \( f^\prime(3) = 18 \). Define \[ g_n(x) = n \left( f \left( x + \frac{5}{n} \right) - f \left( x - \frac{2}{n} \right) \right). \] Then \( \lim_{n \to \infty} g_n(3) \) equals ...............
Let \( f : \mathbb{R} \to \mathbb{R} \) be a differentiable function with \( \lim_{x \to \infty} f(x) = \infty \) and \(\lim_{x \to \infty} f'(x) = 2. \) Then \[ \lim_{x \to \infty} \left( 1 + \frac{f(x)}{x^2} \right)^x \] equals ..............
The value of \[ \int_0^\frac{\pi}{2} \left( \int_0^x e^{\sin y} \sin x \, dy \right) dx \] equals .............