NIMCET 2022 — Mathematics PYQ

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Question

NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

Suppose that the temperature at a point (x, y) on a metal plate is T(x, y) = 4x^2 - 4xy + y^2. An ant, walking on the plate, traverses a circle of radius 5 centred at the origin. What is the highest temperature encountered by the ant?

Choose the correct answer:

Explanation

Method 1 — Parametrise the circle (trig).**
On the circle $x^{2} + y^{2} = 25$ put $x = 5 cos t, y = 5 sin t$ . Then

$ T = 4 x^{2} - 4 x y + y^{2} = 25 (4 cos^{2} t - 4 cos t sin t + sin^{2} t) . $

Use $cos^{2} t = \frac{1 + c o s 2 t}{2}, sin^{2} t = \frac{1 - c o s 2 t}{2}, cos t sin t = \frac{s i n 2 t}{2}$ . So

$ 4 cos^{2} - 4 cos sin + sin^{2} = 2 (1 + cos 2 t) - 2 sin 2 t + \frac{1 - c o s 2 t}{2} = 2.5 + 1.5 cos 2 t - 2 sin 2 t . $

The expression $1.5 cos 2 t - 2 sin 2 t$ has amplitude $1. 5^{2} + 2^{2} = 2.25 + 4 = 2.5$ . Thus its maximum is $+ 2.5$ , so the bracketed term max = $2.5 + 2.5 = 5$ . Hence

$ T_{m a x} = 25 \cdot 5 = 125. $

**Method 2 — Quadratic form / eigenvalues.
Write $T = [x y] (4 - 2 am p; - 2 am p; 1) (x y)$ .
On $x^{2} + y^{2} = 25$ the maximum of $T$ is $25$ times the largest eigenvalue of the matrix. The matrix has trace $5$ and determinant $0$ , so eigenvalues are $\frac{5 \pm 5}{2} = 5, 0$ . Largest eigenvalue $= 5$ . Thus $T_{m a x} = 25 \cdot 5 = 125$ .

$125 (option a)$