(a) Consider the relation scheme $$R(A, B, C)$$ with the following functional dependencies:
$$\eqalign{ & A,B \to C \cr & \,\,\,\,\,\,C \to A \cr} $$
Show that the scheme $$R$$ is the Third Normal Form $$(3NF)$$ but not in Boyce-Code Normal Form $$(BCNF).$$

(b) Determine the minimal keys of relation $$R.$$

$$\mathop {Lim}\limits_{x \to \infty } {{{x^3} - \cos x} \over {{x^2} + {{\left( {\sin x} \right)}^2}}} = \_\_\_\_\_\_.$$

The rank of the following (n + 1) x (n + 1) matrix, where a is a real number is $$$\left[ {\matrix{ 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr . & . & . & . & . & . & . \cr . & . & . & . & . & . & . \cr . & . & . & . & . & . & . \cr 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr } } \right]$$$

Let A be the set of all nonsingular matrices over real numbers and let * be the matrix multiplication operator. Then