(d) : $R$ is reflexive : $3x + \alpha x$ is a multiple of 7 .\\ $\Rightarrow (3 + \alpha)x$ is a multiple of $7 \Rightarrow \alpha = 7k + 4$ or $\alpha = 7k - 3$\\ $R$ is symmetric : $3x + \alpha y$ is a multiple of 7\\ $\Rightarrow \ 3y + \alpha x$ is a multiple of 7\\ i.e., $3x + \alpha y = 7\lambda \Rightarrow 3y + \alpha x = 7\mu$\\ i.e., $(3 + \alpha)(x + y) = 7(\lambda + \mu) \Rightarrow \alpha = 7k + 4$ or $7k - 3$ \end{enumerate} Similarly $R$ is transitive\\ i.e., $3x + \alpha y = 7\lambda,3y + \alpha z = 7\mu \Rightarrow 3x + \alpha z = 7v$\\ $\Rightarrow \ \alpha = 7k + 4$ or $7k - 3$\\ So, $R$ is equivalence relation iff 4 is the remainder when $\alpha$ is divided by 7 . \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \setcounter{enumi}{17}