The closing capital of Keshav is Rs.55,000. To calculate the opening capital, adjustments are made for profit earned during the year and drawings made at the beginning of each quarter.
Step 1: Calculate total drawings:
Drawings made at the beginning of each quarter = Rs.1,500. Total drawings for the year: \[ Rs.1,500 \times 4 = Rs.6,000. \]
Step 2: Adjust for profit:
The profit for the year is Rs.15,000.
Step 3: Calculate opening capital:
Opening capital is calculated as: \[ {Opening Capital} = {Closing Capital} - {Profit} + {Drawings}. \] Substituting the values: \[ {Opening Capital} = Rs.55,000 - Rs.15,000 + Rs.6,000 = Rs.43,000. \]
Conclusion:
The opening capital of Keshav was \( Rs.43,000 \).
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 \(P,O,Q\).