Question : The sum of the cubes of two given numbers is 10234, while the sum of the two given numbers is 34. What is the positive difference between the cubes of the two given numbers?
Option 1: 3484
Option 2: 3488
Option 3: 3356
Option 4: 8602
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Correct Answer: 3484
Solution : $(a + b)^3= a^3+ b^3+ 3ab(a + b)$ $⇒(34)^3=10234+3ab(34)$ $⇒39304−10234 = 3ab(34)$ $⇒29070 = 102\times ab$ $⇒ab = \frac{29070}{102} = 285$ We know, $(a - b)^2= (a + b)^2 - 4ab$ $⇒(a - b)^2= 34^2 - 4 \times (285)$ $⇒(a - b)^2= 1156 -1140 = 16$ $⇒(a - b) = 4$ ........ (i) Now $(a+b) = 34$ ........(i) Solving both the equations (i) and (ii) we get, $a = 19$ and $b = 15$ So, $a^3-b^3 = 19^3 - 15^3 = 3484$ Hence, the correct answer is 3484.
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