Hans, Sohan and Kishore were partners in a firm sharing profits and losses in the ratio of 3 : 2 : 1. The firm closes its books on 31st March every year. On 1st August, 2024, Kishore died. The partnership deed provided that on the death of a partner, his executors will be entitled for:
(i) Balance in his capital account less drawings.
(ii) Interest on capital @ 12% p.a.
(iii) His share of goodwill.
(iv) His share in the profits of the firm till the date of his death calculated on the basis of average profit of the previous four years.
The following information was obtained from the books of the firm on the date of Kishore’s death:
(a) Capital on 1st April, 2024 = 4,00,000, Drawings = 90,000
(b) Goodwill of the firm = 60,000
(c) Profits for last 4 years: 2,00,000, 2,20,000, 1,20,000, 1,80,000
Step 1: Adjusted Capital Balance
4,00,000 - 90,000 = 3,10,000
Step 2: Interest on Capital (12% for 4 months)
4,00,000 × (12/100) × (4/12) = 16,000
Step 3: Kishore’s Share of Goodwill (1/6)
60,000 × (1/6) = 10,000
Step 4: Kishore’s Share in Profit till Death (1/6 of Average Profit)
Average Profit = (2,00,000 + 2,20,000 + 1,20,000 + 1,80,000) / 4 = 1,80,000
Profit till 4 months = 1,80,000 × (4/12) = 60,000
Kishore’s Share = 60,000 × (1/6) = 10,000
Kishore’s Capital Account
| Particulars | Amount | Particulars | Amount |
| To Drawings A/c | 90,000 | By Balance b/d | 4,00,000 |
| By Interest on Capital A/c | 16,000 | ||
| By Goodwill A/c | 10,000 | ||
| By Profit and Loss Suspense A/c | 10,000 | ||
| To Executor's A/c | 3,46,000 | 4,36,000 |

A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following: 
(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).