Question : If $2x+3y = 4$ and $4x^{2}+9y^{2}= 64$, then what is the value of $8x^{3}+27y^{3}$:
Option 1: 253
Option 2: 235
Option 3: 352
Option 4: 325
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Correct Answer: 352
Solution :
Given: $2x+3y=4$ and $4x^2+9x^2=64$
To find: $8x^3+27y^3$
Now,
$2x+3y=4$
Squaring both sides, we get:
⇒ $4x^2+9y^2+12xy=16$
Putting the values, we get:
⇒ $64+12xy=16$
⇒ $12xy=-48$
⇒ $xy=-4$
Now again,
$2x+3y=4$
Cubing both sides, we get:
⇒ $8x^3+27y^3+3×2x×3y(2x+3y)=64$
Putting the values, we get:
⇒ $8x^3+27y^3+18×(-4)×4=64$
$\therefore8x^3+27y^3=64+288= 352$
Hence, the correct answer is 352.
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