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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
If $A = \cos^2\theta + \sin^4\theta$, then for all values of $\theta$:
Solution
$A = \cos^2\theta + \sin^4\theta$Let $\sin^2\theta = t$, $0 \le t \le 1$
Then
$A = (1 - t) + t^2 = t^2 - t + 1$
This is a quadratic in $t$.
Minimum value at $t = \dfrac{1}{2}$:
$A_{\min} = \left(\dfrac{1}{2}\right)^2 - \dfrac{1}{2} + 1 = \dfrac{3}{4}$
Maximum value at endpoints $t=0$ or $t=1$:
$A_{\max} = 1$
Thus:
${\dfrac{3}{4} \le A \le 1}$
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