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Consider the region enclosed by the parabola y = x² - 3 and the straight line y = kx + 2.
Statement-1: If k = 0, the enclosed area is smaller.
Statement-2: The area enclosed by y = x² - 3 and y = kx + 2 equals √(k² + 20).
- Statement-1 is correct, Statement-2 is correct, and Statement-2 correctly explains Statement-1
- Statement-1 is correct, Statement-2 is correct, but Statement-2 does not explain Statement-1
- Statement-1 is incorrect, Statement-2 is correct
- Statement-1 is correct, Statement-2 is incorrect
Correct answer: Statement-1 is correct, Statement-2 is incorrect
Solution
Intersection of y=x^2-3 and y=kx+2 gives x^2-kx-5=0 with root gap sqrt(k^2+20); the enclosed area = (1/6)(k^2+20)^{3/2}, which is minimized at k=0. So Statement-1 (k=0 gives smaller area) is correct, but Statement-2's formula sqrt(k^2+20) is wrong. Correct option: Statement-1 correct, Statement-2 incorrect.
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