A) 3
B) 4
C) 2
D) 5
Correct Answer: A
Solution :
[a] \[\left[ {{\left( \frac{1-\sqrt{3x+1}}{2} \right)}^{4}}-{{\left( \frac{1-\sqrt{3x+1}}{2} \right)}^{7}} \right]\] \[=\,\,\,\frac{1}{\sqrt{3x+1}}\,\left[ \frac{{{\left( 1+\sqrt{3x+1} \right)}^{7}}-{{\left( 1-\sqrt{3x+1} \right)}^{7}}}{{{2}^{7}}} \right]\] \[=\frac{1}{\sqrt{3x+1}}\left[ \frac{2\left\{ {{\,}^{7}}{{C}_{1}}(\sqrt{3x+1})+{{\,}^{7}}{{C}_{3}}{{(\sqrt{3x+1})}^{3}}+{{\,}^{7}}{{C}_{5}}{{(\sqrt{3x+1})}^{5}}+{{\,}^{7}}{{C}_{7}}{{(\sqrt{3x+1})}^{7}} \right\}}{{{2}^{7}}} \right]\] \[=\frac{1}{{{2}^{6}}}\left[ ^{7}{{C}_{1}}+{{\,}^{7}}{{C}_{3}}(3x+1)+{{\,}^{7}}{{C}_{5}}{{(3x+1)}^{2}}+{{\,}^{7}}{{C}_{7}}{{(3x+1)}^{3}} \right]\] Clearly above is a polynomial of degree 3 in x.
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