A) \[9{{G}^{2}}-\left( 7{{A}^{2}}-{{G}^{2}} \right)\left( {{G}^{6}}+2 \right)=0\]
B) \[6{{G}^{6}}-\left( 7{{A}^{2}}-{{G}^{2}} \right)\left( {{G}^{6}}+2 \right)=0\]
C) \[9{{G}^{6}}-\left( 4{{A}^{2}}-{{G}^{2}} \right)\left( {{G}^{6}}+2 \right)=0\]
D) \[6{{G}^{6}}-(4{{A}^{2}}-{{G}^{2}})\left( {{G}^{6}}+2 \right)=0\]
Correct Answer: C
Solution :
[c] Let \[y=f\left( x \right)={{x}^{3}}-2{{x}^{-3}}\text{+}10\] \[\therefore {{y}_{1}}-{{y}_{2}}=\left( {{x}_{1}}^{3}-{{x}_{2}}^{3} \right)+2\left( {{x}_{2}}^{-3}-{{x}_{1}}^{3} \right)\] \[=({{x}_{1}}^{3}-{{x}_{2}}^{3})+2\left( \frac{x_{1}^{3}-x_{2}^{3}}{{{({{x}_{1}}{{x}_{2}})}^{3}}} \right)\] \[\Rightarrow \frac{{{y}_{1}}-{{y}_{2}}}{{{x}_{1}}-{{x}_{2}}}=({{x}_{1}}^{2}-{{x}_{2}}^{2}+{{x}_{1}}{{x}_{2}})\left( 1+\frac{2}{{{({{x}_{1}}{{x}_{2}})}^{3}}} \right)\] \[=\frac{[{{({{x}_{1}}+{{x}_{2}})}^{2}}-{{x}_{1}}{{x}_{2}}][{{({{x}_{1}}{{x}_{2}})}^{3}}+2]}{{{({{x}_{1}}{{x}_{2}})}^{3}}}\]\[=\frac{(4{{A}^{2}}-{{G}^{2}})({{G}^{6}}+2)}{{{G}^{6}}}=9\] \[\Rightarrow 9{{G}^{6}}-\left( 4{{A}^{2}}-{{G}^{2}} \right)\left( {{G}^{6}}+2 \right)=0\]
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