Question

If the sum of two numbers is 11 and the sum of their squares is 65, then the sum of their cubes will be:

• A. 407
• B. 576
• C. 615
• D. 355

Answer 1

Answer:

A. 407

Given:

The sum of two numbers = 11

The sum of their squares = 65

Formula:

$(x+y)^2=x^2+y^2+2xy$

$x^3+y^3=(x+y)[(x+y)^2-3xy]$

Calculation:

Let two numbers be x and y.

x + y = 11

$x^2+y^2=65$

⇒ (x + y)² = x² + y² + 2xy

⇒ 11² = 65 + 2xy

⇒ 2xy = 121 - 65

⇒ $xy = \frac{56}{2}$

⇒ xy = 28

x³ + y³ = (x + y) [(x + y)² - 3xy]

⇒ x³ + y³ = 11 (11² - $3\times{28}$)

⇒ x³ + y³ = $11\times{(121 - 84)}$

⇒ x³ + y³ = $11\times{37}$

∴ x³ + y³ = 407

Shortcut Trick

Let two numbers are x and y.

x + y = 11

$x^2+y^2=65$

Put x = 7 and y = 4, then the condition is satisfied.

$x^3+y^3$

⇒ $7^3+4^3$

⇒ 343 + 64

⇒ 407