What is the minimum number of multiplications involved in computing the matrix product $$PQR?$$ Matrix $$P$$ has $$4$$ rows and $$2$$ columns, matrix $$Q$$ has $$2$$ rows and $$4$$ columns and matrix $$R$$ has $$4$$ rows and $$1$$ column ______.

The solution $$\int\limits_0^{\pi /4} {{{\cos }^4}3\theta {{\sin }^3}\,6\theta d\theta \,\,} $$ is :

Find the value of $$\lambda $$ such that the function $$f(x)$$ is a valid probability density function ________.

There is no value of $$x$$ that can simultaneously satisfy both the given equations. Therefore, find the 'least squares error' solution to the two equations, i.e., find the value of $$x$$ that minimizes the sum of squares of the errors in the two equations
$$2x=3$$
$$4x=1$$