Question

# Find the relation between a and b by simplifying the expression (3a+2b)2+(2a+3b)2. Also, find the value of expression when a2+b2=5 and ab=2.

A
113
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B
13(a2+b2)+24ab
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C
24(a2+b2)+13ab
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D
146
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Solution

## The correct option is B 13(a2+b2)+24abGiven the expression (3a+2b)2+(2a+3b)2 Let (3a+2b)=x and (2a+3b)=y ∴ the expression becomes x2+y2 ⇒x2+y2=(x+y)2−2xy [∵(a+b)2=a2+b2+2ab] Now, re-substituting value of x and y in the above expression we get, ⇒x2+y2=[(3a+2b)+(2a+3b)]2−2(3a+2b)(2a+3b) ⇒x2+y2=[3a+2b+2a+3b]2−2[6a2+13ab+6b2] ⇒x2+y2=[5(a+b)]2−2[6(a2+b2)+13ab] ⇒x2+y2=25(a2+b2+2ab)−12(a2+b2)−26ab ⇒x2+y2=25(a2+b2)+50ab−12(a2+b2)−26ab ⇒x2+y2=13(a2+b2)+24ab So, this is the required relation between a and b. ​Now,Putting value of a2+b2=5 and ab=2 we get (3a+2b)2+(2a+3b)2=13(a2+b2)+24ab=13×5+24×2 ⇒13(a2+b2)+24ab=65+48=113.​​​​​​

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