Given that,
- Bank $\text{A:}$
- Interest rate $ = 6 \% \; \text{per annum (CI)}$
- Time $ = \text{half yearly}$
- Bank $\text{B:}$
- Interest rate $ = x \% \; \text{per annum (SI)}$
- Bank $\text{C:}$
- Interest rate $ = 2x \% \; \text{per annum (SI)}$
Let Raju invested $₹ \; P$ in bank $\text{B}$ for $t$ years, hence Rupa invested $₹ \; 10000$ in bank $\text{C}$ for $2t$ years.
we know that, $\boxed{A = P \left(1 + \frac{R}{100} \right)^{t}}$
$\qquad \qquad \boxed{\text{CI = A – P}}$
Now, $\text{SI = CI}$
$\Rightarrow \frac{P \times x \times t}{100} = P \left(1 + \frac{3}{100} \right)^{2} – P$
$\Rightarrow \frac{xt}{100} = \left(\frac{103}{100} \right)^{2} – 1$
$\Rightarrow \frac{xt}{100} = 1.0609 – 1$
$\Rightarrow \frac{xt}{100} = 0.0609$
$\Rightarrow \boxed{xt = 6.09}$
Now, we can calculate $\text{SI}.$
$\text{SI} = \frac{10000 \times 2t \times 2x}{100}$
$\Rightarrow \text{SI} = 400 xt$
$\Rightarrow \text{SI} = 400 \times 6.09 \quad [\because xt = 6.09]$
$\Rightarrow \boxed{\text{SI} = ₹ \; 2436}$
$\therefore$ The interest accrued, in $\text{INR},$ to Rupa is $₹ \; 2436.$
Correct Answer $: \text{B}$