If number of terms in expansion of and then minimum value of x^2 + 3y^2 is

JEE MainMathematicsEasy / ModerateBinomial Theorem

If number of terms in expansion of
$(a + b + c)^{n} + (a + b - c)^{n}\ are\ 4$ and
$\sin^{- 1}\left( \frac{\sqrt{x}}{n} \right) + \sin^{- 1}\sqrt{1 - \frac{x}{n^{2}}} + \tan^{- 1}(y) = \frac{2\pi}{3}$ then
minimum value of x^2 + 3y^2 is

Show answer and solution

(a + b + c)^{n} + (a + b - c)^{n} = 2\left\lbrack {}^{n}C_{0}(a + b)^{n} + nC_{2}(a + b)^{n - 2}C^{2} + \ldots. \right\rbrack

(n + 1) + (n - 1) = 4 \Rightarrow n = 2

0 \leq \frac{\sqrt{x}}{2} \leq 1,0 \leq 1 - \frac{x}{4} \leq 1 \Rightarrow 0 \leq x \leq 4

\frac{\pi}{2} + \tan^{- 1}y = \frac{2\pi}{3} \Rightarrow y = \frac{1}{\sqrt{3}} \Rightarrow 2y^{2} = 1

∴ minimum value is 0 +1=1

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx

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