Gujarat Boardગુજરાતી માધ્યમધોરણ 11 સાયન્સગણિતદ્રિપદી પ્રમેય1 Mark
MCQ
જો ${{\left( 2+\sqrt{3} \right)}^{4}}+{{\left( 2-\sqrt{3} \right)}^{4}}=x+y\sqrt{3}$ તો $y=........$
✓
$0$
B
$56$
C
$112$
D
$97$
✓
Answer
Correct option: A.
$0$
A
$(2+\sqrt{3})^4=\binom{4}{0}\ 2^4+\binom{4}{1}\ 2^3\sqrt{3}+\binom{4}{2}\ 2^2(\sqrt{3})^2+\binom{4}{3}\ 2(\sqrt{3})^3+\binom{4}{4}(\sqrt{3})^4$
$=16+32\sqrt{3}+72+24\sqrt{3}+9$
તેવી જ રીતે
$(2-\sqrt{3})^4=16-32\sqrt{3}+72-24\sqrt{3}+9$
તો $(2+\sqrt{3})^4+(2-\sqrt{3})^4$
$=32+144+18=194=x+y\sqrt{3}$
$x=144$ અને $y=0$
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