a+3b+4+4a−43b−23ab=

a+3b+4+4a−43b−23ab=(a)2+(−3b)2+(2)2+2×2×a+2×2(−3b)−2a3b=(a+2−3b)2=a−3b+2=2+a−3bHence option (1) is correct.

a+3b+4+4a−43b−23ab=

a+3b+4+4a−43b−23ab=(a)2+(−3b)2+(2)2+2×2×a+2×2(−3b)−2a3b=(a+2−3b)2=a−3b+2=2+a−3bHence option (1) is correct.

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$\sqrt{a + 3 b + 4 + 4 \sqrt{a} - 4 \sqrt{3 b} - 2 \sqrt{3 ab}} =$

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$2 + \sqrt{a} - \sqrt{3 b}$

$\sqrt{2 a} - \sqrt{b} + 2$

$\sqrt{a} + 2 - \sqrt{b}$

$\sqrt{a} - 2 - \sqrt{b}$

answer is A.

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Detailed Solution

$\sqrt{a + 3 b + 4 + 4 \sqrt{a} - 4 \sqrt{3 b} - 2 \sqrt{3 a b}}$
$= \sqrt{(\sqrt{a})^{2} + (- \sqrt{3 b})^{2} + (2)^{2} + 2 \times 2 \times \sqrt{a} + 2 \times 2 (- \sqrt{3} b) - 2 \sqrt{a} \sqrt{3 b}}$

$\begin{matrix} = \sqrt{(\sqrt{a} + 2 - \sqrt{3 b})^{2}} \\ = \sqrt{a} - \sqrt{3 b} + 2 \\ = 2 + \sqrt{a} - \sqrt{3 b} \end{matrix}$
Hence option (1) is correct.