The value of the surface integral

where S is the external surface of the sphere x² + y² + z² = R² is

Let f(z) be an analytic function, where z = x + iy . If the real part of f(z) is cosh x cos y , and the imaginary part of f(z) is zero for y = 0 , then f(z) is

Consider the system of linear equations

x + 2y + z = 5

2x + ay + 4z = 12

2x + 4y + 6z = b

The values of a and b such that there exists a non-trivial null space and the system admits infinite solutions are

Let $f(.)$ be a twice differentiable function from $ \mathbb{R}^{2} \rightarrow \mathbb{R}$. If $P, \mathbf{x}_{0} \in \mathbb{R}^{2}$ where $\vert \vert P\vert \vert$ is sufficiently small (here $\vert \vert . \vert \vert$ is the Euclidean norm or distance function), then $f (\mathbf{x}_{0} + p) = f(\mathbf{x}_{0}) + \nabla f(\mathbf{x}_{0})^{T}p + \dfrac{1}{2} p^{T} \nabla^{2}f(\psi)p$ where $\psi \in \mathbb{R}^{2}$ is a point on the line segment joining $\mathbf{x}_{0}$ and $\mathbf{x}_{0} + p$. If $\mathbf{x}_{0}$ is a strict local minimum of $f (\mathbf{x})$, then which one of the following statements is TRUE?