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.

Consider a matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 2} \cr 0 & 1 & 1 \cr } } \right]$$. The matrix A satisfies the equation 6A^$$-$$1 = A² + cA + dI, where c and d are scalars and I is the identify matrix. Then (c + d) is equal to

Let $$f(x,y,z) = 4{x^2} + 7xy + 3x{z^2}$$. The direction in which the function f(x, y, z) increases most rapidly at point P = (1, 0, 2) is