A pair of six-faced dice is rolled twice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is _________ (round off to three decimal places).
Consider the polynomial f(x) = x3 $$-$$ 6x2 + 11x $$-$$ 6 on the domain S, given by 1 $$\le$$ x $$\le$$ 3. The first and second derivatives are f'(x) and f''(x).
Consider the following statements :
I. The given polynomial is zero at the boundary points x = 1 and x = 3.
II. There exists one local maxima of f(x) within the domain S.
III. The second derivative f''(x) > 0 throughout the domains S.
IV. There exists one local minima f(x) within the domain S.
The components of pure shear strain in a sheared are given in the matrix form:
$$\varepsilon = \left[ {\matrix{ 1 & 1 \cr 1 & { - 1} \cr } } \right]$$
Here, Trace ($$\varepsilon $$) = 0. Given, P = Trace ($$\varepsilon$$8) and Q = Trace ($$\varepsilon $$11).
The numerical value of (P + Q) is ___________. (in integer)
The function f(x, y) satisfies the Laplace equation
$$\Delta$$2f(x, y) = 0
on a circular domain of radius r = 1 with its center at point P with coordinates x = 0, y = 0. The value of this function on the circular boundary of this domain is equal to 3. The numerical value of f/(0, 0) is :