Problem 1: The value of the integral \(\int_{0}^{1} \int_{0}^{x^2} xy \, dy \, dx\) is:
Solution:
First, perform the inner integral with respect to y:
\(\int_{0}^{x^2} xy \, dy = x \left[ \frac{y^2}{2} \right]_{0}^{x^2} = \frac{x^5}{2}\)
Now perform the outer integral:
\(\int_{0}^{1} \frac{x^5}{2} \, dx = \frac{1}{2} \left[ \frac{x^6}{6} \right]_{0}^{1} = \frac{1}{12}\)
Answer: \(\boxed{\dfrac{1}{12}}\)
Problem 2: The rank of the matrix \(\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}\) is:
The given matrix is in upper triangular form with all diagonal elements non-zero.
For such matrices, the rank is equal to the number of non-zero rows, which is 3 in this case.
Answer: \(\boxed{3}\)