Question : If $m-n=2$ and $mn=15,(m,n>0)$ , then the value of $(m^2-n^2)(m^3-n^3)$ is:
Option 1: 1856
Option 2: 1658
Option 3: 1586
Option 4: 1568
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Correct Answer: 1568
Solution :
Given:
$m-n=2$ and $mn=15,(m,n>0)$
$(m-n)^2=2^2$
⇒ $m^2+n^2-2mn=4$
⇒ $m^2+n^2=4+(2×15)=34$
⇒ $m^2+n^2+30=34+30$ [adding 30 to both sides]
⇒ $m^2+n^2+2mn=64$ [as, $mn=15$]
⇒ $(m+n)^2=8^2$
⇒ $m+n=8$
Now, $(m^2-n^2)(m^3-n^3)$
= $(m+n)(m-n) (m-n)(m^2+mn+n^2)$
= $8×2×2×(34+15)$
= $1568$
Hence, the correct answer is 1568.
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