To solve the problem, we need to identify the correct objective of Intramural tournaments, which are competitions held within a single institution among its own students.
1. Understanding Intramural Tournaments:
Intramural tournaments are organized within the same institution such as a school or college. They are designed to encourage participation, physical activity, and social bonding among students of the same campus.
2. Evaluating the Options:
- (A) To achieve high performance at the highest level of the tournament: This aligns more with extramural or elite competitions, not intramurals.
- (B) To develop the feeling of integration with other institutions: This applies to inter-school or extramural events.
- (C) To provide opportunities for choosing a career in sports: Intramurals are for recreational and participatory purposes, not career-oriented.
- (D) To promote health and recreation at the institution: This is the core purpose of intramural tournaments. They focus on fitness, enjoyment, and internal bonding among students.
Final Answer:
(D) To promote health and recreation at the institution.
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 \(P,O,Q\).