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 justification in each case:

i) The series $\sum \sin\frac{1}{n}$ is convergent.

ii) The function $f(x) = x + x^2$ is even is $(-2, 2)$.

iii) The function $f(x) = \cos x$ is analytic for all real $x$.

iv) One power series solution of the Legendre's differential equation of order 2 is a second degree polynomial.

v) $x^2 p^2 + y^2 q^2 = z^2$ is a linear p.d.e. of first order.

vi) C.F. of $(D^2 - DD')z = 0$ is $z = f(x) + g(x+y)$.

vii) The vectors $(1, 0, 0), (0, 2, 0)$ and $(0, 0, -3)$ are linearly independent.

viii) Eigen values of a square matrix are real numbers.

ix) The map $T: R^2 \to R$ defined as $T(x,y) = xy$ is linear, where R denotes set of all reals.

x) There is a unique basis set of vectors for a vector space.