Question

For what value of p, is the system of equations:

p³x + (p + 1)³y = (p + 2)³

px + (p + 1)y = p + 2

x + y = 1

consistent?

विकल्प

• p = 0

• p = 1

• p = – 1

• For all p > 1

Answer 1

p = – 1

Explanation:

The given system of equations are:

p³x + (p + 1)³y = (p + 2)³  ......(i)

px + (p + 1)y = (p + 2)  ......(ii)

x + y = 1   ......(iii)

This system is consistent, if values of x and y from first two equation satisfy the third equation.

⇒ `|(p^3, (p + 1)^3, (p + 2)^3),(p, (p + 1), (p + 2)),(1, 1, 1)|` = 0

⇒ `|(p^3, (p + 1)^3 - p^3, (p + 2)^3 - p^3),(p, 1, 2),(1, 0, 0)|` = 0

⇒ 2(p + 1)³ – 2p³ – (p + 2)³ + p³ = 0

⇒ 2(p³ + 1 + 3p² + 3p) – 2p³ – (p³ + 8 + 12p + 6p²) + p³ = 0

⇒ 2p³ + 2 + 6p² + 6p – 2p³ – p³ – 8 – 12p – 6p² + p³ = 0 

⇒ – 6 – 6p = 0

⇒ p = – 1