1. Given relations on $A=\{a,b,c\}$:

$R_1=\{(c,a),(b,b),(a,c),(c,c),(b,c),(a,a)\}$ $R_2=\{(a,b),(b,a),(c,c),(c,a),(a,a),(b,b),(a,c)\}$

We check symmetry and transitivity for each relation.

2. Check $R_1$ for symmetry

A relation is symmetric if whenever $(x,y)\in R$, then $(y,x)\in R$.

Elements of $R_1$:

• $(c,a)$ and its reverse $(a,c)$ is present.
• $(b,b)$ reverse is itself, present.
• $(a,c)$ and its reverse $(c,a)$ is present.
• $(c,c)$ reverse is itself, present.
• $(b,c)$ would require $(c,b)$, but $(c,b)\notin R_1$.
• $(a,a)$ reverse is itself, present.

So $R_1$ is not symmetric.

3. Check $R_1$ for transitivity

A relation is transitive if whenever $(x,y)\in R$ and $(y,z)\in R$, then $(x,z)\in R$.

We test key compositions:

• $(c,a)$ and $(a,c)$ imply $(c,c)$, which is present.
• $(c,a)$ and $(a,a)$ imply $(c,a)$, present.
• $(a,c)$ and $(c,a)$ imply $(a,a)$, present.
• $(a,c)$ and $(c,c)$ imply $(a,c)$, present.
• $(b,b)$ and $(b,c)$ imply $(b,c)$, present.
• $(b,c)$ and $(c,a)$ imply $(b,a)$ must be present.

But $(b,a)\notin R_1$.

Hence $R_1$ is not transitive.

Therefore, statement B is false.

4. Check $R_2$ for symmetry

Elements of $R_2$:

• $(a,b)$ and reverse $(b,a)$ are both present.
• $(c,c)$ reverse is itself.
• $(c,a)$ requires $(a,c)$, which is present.
• $(a,a)$ reverse is itself.
• $(b,b)$ reverse is itself.
• $(a,c)$ requires $(c,a)$, which is present.

Thus $R_2$ is symmetric.

5. Check $R_2$ for transitivity

We look for a counterexample.

Since $(b,a)\in R_2$ and $(a,c)\in R_2$, transitivity would require:

$(b,c)\in R_2$

But $(b,c)\notin R_2$.

So $R_2$ is not transitive.

Therefore, statement C is true.

6. Check options

• A: both $R_1$ and $R_2$ are not symmetric. ❌ False, because $R_2$ is symmetric.
• B: $R_1$ is not symmetric but it is transitive. ❌ False, $R_1$ is not transitive.
• C: $R_2$ is symmetric but it is not transitive. ✅ True.
• D: both $R_1$ and $R_2$ are transitive. ❌ False.

7. Final answer

The correct option is:

$\boxed{C}$

This matches the stored correct answer.