As shown in the figure below, to concentric conducting spherical shells, centred at r = 0 and having radii r = c and r = d are maintained at potentials such that the potential V(r) at r = c is V₁ and V(r) at r = d is V₂. Assume that V(r) depends only on r, where r is the radial distance. The expression for V(r) in the region between r = c and r = d is

Consider a 3 $$\times$$ 3 matrix A whose (i, j)-th element, a_i,j = (i $$-$$ j)³. Then the matrix A will be

e⁴ denotes the exponential of a square matrix A. Suppose $$\lambda$$ is an eigen value and v is the corresponding eigen-vector of matrix A.

Consider the following two statements:

Statement 1 : e^$$\lambda$$ is an eigen value of e^A.

Statement 2 : v is an eigen-vector of e^A.

Which one of the following options is correct?

Let $$f(x) = \int\limits_0^x {{e^t}(t - 1)(t - 2)dt} $$. Then f(x) decreases in the interval.