SSC CGL 2023 PYQ — Question 51

The correct answer is option 1 i.e. 9728.

Let the two numbers be x and y, such that x > y.

According to the given question,

x³ - y³ = 6272

x - y = 8

Since, a³ - b³ = (a - b)(a² + ab + b²)

⇒ x³ - y³ = (x - y)(x² + xy + y²)

⇒ 6272 = 8(x² + xy + y²)

⇒ x² + xy + y² = 784   .....(i)

Also, (a - b)² = a² - 2ab + b²

⇒ (x - y)² = x² - 2xy + y²

⇒ 8² = x² - 2xy + y²

⇒ x² + y² = 64 + 2xy    ....(ii)

Putting the value from equation (ii) in equation (i),

⇒ 64 + 2xy + xy = 784

⇒ 3xy = 720

⇒ xy = 240

Also, x² + y² = 64 + 2(240)

⇒ x² + y² = 544

We know that, (x + y)² = x² + 2xy + y² 

⇒ (x + y)² = 544 + 2(240)

⇒ (x + y)² = 1024

⇒ x + y = 32

Now, a³ + b³ = (a + b)(a² - ab + b²)

⇒ x³ + y³ = (x + y)(x² - xy + y²)

⇒ x³ + y³ = 32(544 - 240)

⇒ x³ + y³ = 32 × 304

⇒ x³ + y³ = 9728

Hence the sum of the cubes of the two numbers is 9728.