For 0 ≤ x ≤ 2 π , 2 cosec 2 x 2 y 2 − y + 1 ≤ 2 is

2 y 2 − y + 1 = 2 ( y − 1 ) 2 + 1 ≥ 2 1 and 2 cosec 2 x ≥ 2 , so the left side is at least 2 ⋅ 2 1 = 2 . Equality needs cosec 2 x = 1 and y = 1 . ⇒ x = 2 π , 2 3 π , y = 1

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Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

MathematicsTrigonometric Ratios & IdentitiesMiscellaneous/MixedMedium2 minQB

MathematicsMediummultiple choice

For $0 \leq x \leq 2 π, 2^{cosec^{2} x} \frac{y ^{2}}{2} - y + 1 \leq 2$ is

Options:(select one or more)

Answer:

A, B, C

Solution:

$\frac{y ^{2}}{2} - y + 1 = \frac{( y - 1 ) ^{2} + 1}{2} \geq \frac{1}{2}$ and $2^{cosec^{2} x} \geq 2$ , so the left side is at least $2 \cdot \frac{1}{2} = 2$ . Equality needs $cosec^{2} x = 1$ and $y = 1$ . $\Rightarrow x = \frac{π}{2}, \frac{3 π}{2}, y = 1$

Stream:JEESubject:MathematicsTopic:Trigonometric Ratios & IdentitiesSubtopic:Miscellaneous/Mixed

⏱ 2mℹ️ Source: QB

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