Given Information
We are given with some statements that we have to check if they are True or False

Step-1: (a)
Since the identity matrix I is symmetric so, \[\begin{array}{l} Q = {{\bf{x}}^T}I{\bf{x}}\\ = {{\bf{x}}^T}{\bf{x}}\\ = \langle {\bf{x}},{\bf{x}}\rangle \\ = {\left| {\left| {\bf{x}} \right|} \right|^2} \end{array}\] It proves that the expression $\| \mathbf { x } \| ^ { 2 }$ is a Quadratic form.

The statement is False



Step-2: (b)
By the principal Axes theorem, there is an orthogonal change of variable $\mathbf { x } = P y$ which transforms $\mathbf { x } ^ { T } A \mathbf { x }$ into a quadratic form $\mathbf { y } ^ { T }$ Dy with no cross product terms. Hence,

The statement is True



Step-3: (c)
From a geometric view of Principle axes, if $\mathrm { A }$ is $2 \times 2$ symmetric matrix then the set of $\mathrm { x }$ such that $\mathrm { x } ^ { T } A \mathrm { x } = c$ (constant) corresponds to a circle, ellipse, a hyperbola, two intersecting lines, a single point or contains no points at all. Hence, it can take any of the form from a circle, ellipse, a hyperbola, two intersecting lines, a single point or contains no points at all. Thus

The statement is False



Step-4: (d)
A quadratic form Q(x) is indefinite if assumes both positive and negative values. So, the quadratic form Q is indefinite if it is both positive semidefinite and negative semidefinite. Hence

The statement is False



Step-5: (e)
If $Q ( \mathrm { x } ) = \mathrm { x } ^ { T } A \mathrm { x } < 0 ,$ for all $\mathrm { x } \neq 0 ,$ then $\mathrm { Q }$ is negative definite. The Q is negative definite if and only if the eigenvalues of are all negative.

The statement is False