Step 1: Rishab's share in profits = \( \frac{1}{4} \) \(\Rightarrow\) Old partners' combined share = \( \frac{3}{4} \)
Old ratio of Bharat : Ishu = 4 : 1
Step 2: Firm's goodwill = 4,00,000 \(\Rightarrow\) Rishab's share = \( \frac{1}{4} \times 4,00,000 = 1,00,000 \)
But Rishab brought only 60,000 in cash for goodwill. Remaining 40,000 is not brought in.
Step 3: Goodwill already appearing in books = 50,000 (to be written off in old ratio 4 : 1)
Journal Entries:
1. For bringing capital and goodwill premium: Bank A/c Dr. & 2,60,000
To Rishab’s Capital A/c & 2,00,000
To Premium for Goodwill A/c & 60,000
2. For distributing goodwill premium among old partners in sacrificing ratio (same as old ratio 4 : 1): \[ \text{Bharat = } \frac{4}{5} \times 60,000 = 48,000 \text{Ishu = } \frac{1}{5} \times 60,000 = 12,000 \] Premium for Goodwill A/c Dr. & 60,000
To Bharat’s Capital A/c & 48,000
To Ishu’s Capital A/c & 12,000
3. For writing off existing goodwill of 50,000 in old ratio 4 : 1: \[ \text{Bharat = } \frac{4}{5} \times 50,000 = 40,000 \text{Ishu = } \frac{1}{5} \times 50,000 = 10,000 \] Bharat’s Capital A/c Dr. & 40,000
Ishu’s Capital A/c Dr. & 10,000
To Goodwill A/c & 50,000

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 \(P,O,Q\).