(a) Annuity due present value: $PV=a\dfrac{1-(1+i)^{-n}}{i}(1+i)\approx\text{\u20b9}1{,}68{,}709$. (b) $r=\dfrac{431}{\sqrt{598\times338}}\approx0.96$.

A 5-mark either/or from the Financial Mathematics / Correlation units of the Tamil Nadu HSC Class-11 Business Mathematics & Statistics syllabus. Both alternatives are solved.

(a) Present value of the camera (annuity due).

The instalments are paid "right from the date of purchase," i.e. at the beginning of each year, so this is an annuity due.

Step 1 — Formula.

$PV=a\left[\frac{1-(1+i)^{-n}}{i}\right](1+i),$

with $a=36000,\ i=\dfrac{16}{100}=0.16,\ n=7,\ (1.16)^7=2.828.$

Step 2 — Compute $(1.16)^{-7}$. $(1.16)^{-7}=\dfrac{1}{2.828}=0.35361.$

Step 3 — Ordinary-annuity factor.

$\frac{1-0.35361}{0.16}=\frac{0.64639}{0.16}=4.03996.$

Step 4 — Adjust for annuity due and multiply.

$PV=36000\times4.03996\times1.16=36000\times4.68635\approx168709.$

(b) Karl Pearson's coefficient of correlation.

Step 1 — Means. $\sum X=162,\ \sum Y=171,\ N=9$, so $\bar X=18,\ \bar Y=19.$

Step 2 — Deviations $x=X-18,\ y=Y-19$ and products.