Verify whether the following statements are true or false with justif…

Previous year question for Engineering Mathematics-II.

Question

Verify whether the following statements are true or false with justifications in each case: $(1 \times 10 = 10)$

(i) The series $\sum \frac{1}{n^2}$ is divergent.

(ii) The function $f(x) = 1 + |x|$ is an odd function in $(-2, 2)$.

(iii) The point $x = 0$ is an ordinary point of the differential equation $xy'' + 2y' + y = 0$.

(iv) The general solution of $r - 6s + 9t = 0$ is $z = f(y + 3x) + x \cdot g(y + 3x)$.

(v) The complete solution of $DD'Z = e^x$ is $z = f(y) + g(y) + xe^x$.

(vi) If $A = \begin{bmatrix} 2 & 4 \\ -1 & -2 \end{bmatrix}$, then $A^2 = 0$.

(vii) A subset of a linearly independent set of vectors is linearly independent.

(viii) The mapping $T: R \to R^2$ defined as $T(x) = (x, x^2)$ is a linear transformation, where $R$ is the set of all reals.

(ix) If $S$ and $T$ are linear transformations on $R^2$ such that the composite map $SoT = O$, then $ToS = O$.

(x) The eigen values for the matrix $\begin{bmatrix} 1 & 2 & 0 \\ 2 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ are $1, -1, -1$.