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प्रश्न
Evaluate the
(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]
उत्तर
(ix) \[(\sqrt{3} + \sqrt{2} )^6 - (\sqrt{3} - \sqrt{2} )^6 \]
\[ = 2[^{6}{}{C}_1 (\sqrt{3} )^5 (\sqrt{2} )^1 + 6{6}{}{C}_3 (\sqrt{3} )^3 (\sqrt{2} )^3 + ^{6}{}{C}_5 (\sqrt{3} )^1 (\sqrt{2} )^5 ]\]
\[= 2[6 \times 9\sqrt{3} \times \sqrt{2} + 20 \times 3\sqrt{3} \times 2\sqrt{2} + 6 \times \sqrt{3} \times 4\sqrt{2}]\]
\[ = 2[\sqrt{6}(54 + 120 + 24)]\]
\[ = 396\sqrt{6}\]
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संबंधित प्रश्न
Using binomial theorem, write down the expansions :
(iii) \[\left( x - \frac{1}{x} \right)^6\]
\[= ^{5}{}{C}_0 (2x )^5 (3y )^0 +^{5}{}{C}_1 (2x )^4 (3y )^1 + ^{5}{}{C}_2 (2x )^3 (3y )^2 + ^{5}{}{C}_3 (2x )^2 (3y )^3 + ^{5}{}{C}_4 (2x )^1 (3y )^4 +^{5}{}{C}_5 (2x )^0 (3y )^5\]
\[= 32 x^5 + 5 \times 16 x^4 \times 3y + 10 \times 8 x^3 \times 9 y^2 + 10 \times 4 x^2 \times 27 y^3 + 5 \times 2x \times 81 y^4 + 243 y^5 \]
\[ = 32 x^5 + 240 x^4 y + 720 x^3 y^2 + 1080 x^2 y^3 + 810x y^4 + 243 y^5 \]
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(i) \[\left( 2x + 3y \right)^5\]
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