The value of 'a' for which the equations x3 + ax + 1 = 0 and x4 + ax2 + 1=0 have a common root is

Question

The value of 'a' for which the equations x3 + ax + 1 = 0 and x4 + ax2 + 1=0 have a common root is (A) 2 (B) -2 (C) 0 (D) none of these

Tardigrade Admin · Accepted Answer

Let α be a common root of x3 + ax + 1 = 0 and x4 + ax2 + 1 = 0 ∴ α3 + a α + 1 = 0 and α4 + a α2 + 1 = 0 i.e. α4 + a α2 + α = 0 and α4 + a α2 + 1=0 ∴ α - 1 = 0 ⇒ α = 1 ∴ 1 + a ⋅ 1 + 1 = 0 ⇒ a = -2

Q. The value of $^{'} a^{'}$ for which the equations $x^{3} + a x + 1 = 0$ and $x^{4} + a x^{2} + 1 = 0$ have a common root is

$2$

$- 2$

$0$

none of these

Let $α$ be a common root of
$x^{3} + a x + 1 = 0$ and $x^{4} + a x^{2} + 1 = 0$
$∴ α^{3} + a α + 1 = 0$ and $α^{4} + a α^{2} + 1 = 0$
$i . e . α^{4} + a α^{2} + α = 0$ and $α^{4} + a α^{2} + 1 = 0$
$∴ α - 1 = 0$
$\Rightarrow α = 1$
$∴ 1 + a \cdot 1 + 1 = 0$
$\Rightarrow a = - 2$

Q. The value of ′a′ for which the equations x3+ax+1=0 and x4+ax2+1=0 have a common root is

2

−2

0

none of these

Let α be a common root of x3+ax+1=0 and x4+ax2+1=0 ∴α3+aα+1=0 and α4+aα2+1=0 i.e.α4+aα2+α=0 and α4+aα2+1=0 ∴α−1=0 ⇒α=1 ∴1+a⋅1+1=0 ⇒a=−2

Q. The value of $^{'} a^{'}$ for which the equations $x^{3} + a x + 1 = 0$ and $x^{4} + a x^{2} + 1 = 0$ have a common root is

$2$

$- 2$

$0$

Let $α$ be a common root of
$x^{3} + a x + 1 = 0$ and $x^{4} + a x^{2} + 1 = 0$
$∴ α^{3} + a α + 1 = 0$ and $α^{4} + a α^{2} + 1 = 0$
$i . e . α^{4} + a α^{2} + α = 0$ and $α^{4} + a α^{2} + 1 = 0$
$∴ α - 1 = 0$
$\Rightarrow α = 1$
$∴ 1 + a \cdot 1 + 1 = 0$
$\Rightarrow a = - 2$