Question : The square of the difference between two given natural numbers is 324, while the product of these two given numbers is 144. Find the positive difference between the squares of these two given numbers.
Option 1: 630
Option 2: 540
Option 3: 450
Option 4: 360
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Correct Answer: 540
Solution : Let $a$ and $b$ be the two numbers. So, $ab$ = 144 and $(a-b)^2 = 324$ ⇒ $ab$ = 144 and $(a-b) = 18 $ We know that $(a+b)^2 = (a-b)^2 + 4ab$ ⇒ $(a+b)^2 = 324 + 576$ ⇒ $(a+b)^2 = 900$ ⇒ $(a+b) = 30$ Now, $a^2-b^2$ = $(a-b)×(a-b)$ ⇒ $a^2-b^2 = 18 × 30$ $\therefore a^2-b^2 = 540$ Hence, the correct answer is 540.
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Question : The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the positive difference between the cubes of these two given numbers.
Option 1: 4104
Option 2: 5616
Option 3: 2160
Option 4: 5626
Question : The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the sum of the cubes of these two given numbers.
Option 1: 6048
Option 3: 6024
Option 4: 5832
Question : The difference between the cubes of two given natural numbers is 6272, while the positive difference between the two given numbers is 8. What is the product of the two given numbers?
Option 1: 160
Option 2: 240
Option 3: 200
Option 4: 320
Question : The cube of the sum of two given numbers is 1728, while the product of the two given numbers is 32. Find the positive difference between the cubes of the two given numbers.
Option 1: 448
Option 2: 576
Option 3: 480
Option 4: 512
Question : The difference between the cubes of two given natural numbers is 6272, while the positive difference between the two given numbers is 8. What is the sum of the cubes of the two given numbers?
Option 1: 9728
Option 2: 9684
Option 3: 8000
Option 4: 9600
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