KVPY Sample Paper KVPY Stream-SX Model Paper-5

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

Correct Answer: B

Solution :

[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 \[\left| a+b \right|=5\]