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

Let R be a region in the first quadrant of the xy plane enclosed by a closed curve C considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region R?

Let $$\overrightarrow E (x,y,z) = 2{x^2}\widehat i + 5y\widehat j + 3z\widehat k$$. The value of $$\mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5pt V} {(\overrightarrow \nabla \,.\,\overrightarrow E )dV} $$, where V is the volume enclosed by the unit cube defined by 0 $$\le$$ x $$\le$$ 1, 0 $$\le$$ y $$\le$$ 1, and 0 $$\le$$ z $$\le$$ 1, is