Consider the data of $f(x)$ given in the table.
| $i$ | $0$ | $1$ | $2$ |
|---|---|---|---|
| $x_i$ | $1$ | $2$ | $3$ |
| $f(x_i)$ | $0$ | $0.3010$ | $0.4771$ |
The value of $f(1.5)$ estimated using second-order Newton’s interpolation formula is ________________ (rounded off to 2 decimal places).
What are the eigenvalues of the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ ?
A vector field $\vec{p}$ and a scalar field $r$ are given by:
$\vec{p} = (2x^2 - 3xy + z^2) \hat{i} + (2y^2 - 3yz + x^2) \hat{j} + (2z^2 - 3xz + x^2) \hat{k}$
$r = 6x^2 + 4y^2 - z^2 - 9xyz - 2xy + 3xz - yz$
Consider the statements P and Q:
P: Curl of the gradient of the scalar field $r$ is a null vector.
Q: Divergence of curl of the vector field $\vec{p}$ is zero.
Which one of the following options is CORRECT?
The return period of a large earthquake for a given region is 200 years. Assuming that earthquake occurrence follows Poisson’s distribution, the probability that it will be exceeded at least once in 50 years is ______________ % (rounded off to the nearest integer).