• # question_answer If ${{a}^{2}}+{{b}^{2}}=7$ and ${{a}^{3}}+{{b}^{3}}=10,$ then the maximum value of $|a+b|$ is A) 4                                  B) 5       C) 6                                  D) 3

[b]  We have, ${{a}^{2}}+{{b}^{2}}=7$ and ${{a}^{3}}+{{b}^{3}}=10$ $\Rightarrow$${{a}^{3}}+{{b}^{3}}=(a+b)({{a}^{2}}+{{b}^{2}}-ab)$$\Rightarrow$$10=(a+b)(7-ab)$ $\Rightarrow$$10=(a+b)\left( 7-\frac{{{(a+b)}^{2}}-7}{2} \right)$ $[\because {{a}^{2}}+{{b}^{2}}={{(a+b)}^{2}}-2ab]$ $\Rightarrow$$20=(a+b)(21-{{(a+b)}^{2}})$$\Rightarrow$${{(a+b)}^{3}}-21(a+b)+20=0$
 Let $a+b=x$ $\therefore$${{x}^{3}}-21x+20=0$ $(x-1)(x-4)(x+5)=0$ $\therefore$$x=1,4,-\,5$ $\left| a+b \right|=1,4,5$ $\therefore$Maximum value of $\left| a+b \right|=5$