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Cramer's Rule

If the equati...

Question

If the equations $x^{2} + 2 λ x + λ^{2} + 1 = 0, λ ϵ R a n d a x^{2} + b x + c = 0$ where a, b, c are lengths of sides of triangle have a common root, then the possible range of values of $λ$ is

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Solution

The correct option is A

$(x + λ)^{2} + 1 = 0$ has clearly imaginary roots

So, both roots of the equations are common

$∴ \frac{a}{1} = \frac{b}{2 λ} = \frac{c}{λ^{2} + 1} = k (s a y) T h e n a = k, b = 2 λ k, c = (λ^{2} + 1) k$

As a, b, c are sides of triangle

a + b > c $\Rightarrow 2 λ + 1$ > $λ^{2} + 1 \Rightarrow λ^{2} - 2 λ$ < 0

$\Rightarrow λ ϵ (0, 2)$

The other conditions also imply same relation.

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If the equations x2+2λx+λ2+1=0, λ ϵ R and ax2+bx+c=0 where a, b, c are lengths of sides of triangle have a common root, then the possible range of values of λ is

The correct option is A (x+λ)2+1=0 has clearly imaginary roots So, both roots of the equations are common ∴a1=b2λ=cλ2+1=k(say)Then a=k, b=2λk,c=(λ2+1)k As a, b, c are sides of triangle a + b > c ⇒ 2λ+1 > λ2+1⇒ λ2−2λ < 0 ⇒λ ϵ(0,2) The other conditions also imply same relation.

If the equations $x^{2} + 2 λ x + λ^{2} + 1 = 0, λ ϵ R a n d a x^{2} + b x + c = 0$ where a, b, c are lengths of sides of triangle have a common root, then the possible range of values of $λ$ is