Jake takes out a bank loan of R600 000 to pay for his new car. He repays the loan with monthly instalments of R9 000, starting one month after the granting of the loan. The interest rate is 13% per annum, compounded quarterly.

The given nominal interest rate is 13% p.a. compounded quarterly.
Nominal rate ($i_{nom,q}$) = 0.13.
Number of compounding periods per year ($m_q$) = 4.
The formula for the effective annual interest rate ($i_{eff}$) is:

Let the nominal interest rate compounded monthly be $i_{nom,m}$.
Number of compounding periods per year ($m_m$) = 12.
The formula relating effective annual rate to nominal monthly rate is:

Take the 12th root of both sides:

Convert to a percentage and round to two decimal places:

Thus, the effective interest rate is $\boxed{\text{12.86\% p.a. compounded monthly}}$.

Loan amount ($P$) = R600 000.
Monthly payment ($X$) = R9 000.
The nominal interest rate compounded monthly is $12.864\%$ (from 3.1, using the unrounded value for accuracy).
Monthly interest rate ($i$) = $\frac{0.12864}{12} = 0.01072$.

The formula is:

Substitute the known values:

Divide by 9 000 and multiply by 0.01072:

Rearrange to solve for $(1.01072)^{-n}$:

Take the natural logarithm of both sides:

The number of full monthly repayments Jake would pay is $\boxed{\text{117}}$.

Number of full payments ($k$) = 117.
Loan amount ($P$) = R600 000.
Monthly payment ($X$) = R9 000.
Monthly interest rate ($i$) = 0.01072.
The formula for the outstanding balance ($B_k$) after $k$ payments is:

Calculate $(1.01072)^{117} \approx 3.46879800$.

Final payment = $B_{117}(1+i)$

The value of Jake's final payment is $\boxed{\text{R8 685.99}}$.

Total payments = (Number of full payments $\times$ Monthly payment) + Final payment
Total payments = $(117 \times \text{R}9\,000) + \text{R}8\,685.99$
Total payments = $\text{R}1\,053\,000 + \text{R}8\,685.99$
Total payments = $\text{R}1\,061\,685.99$
The car cost Jake a total of $\boxed{\text{R1 061 685.99}}$.