Question

If equation $x^2 - \lambda x + \mu = 0$ has roots $\lambda$ and $\mu$ then:

• A. $|\lambda| + |\mu| = 2$
• B. $\lambda - \mu = 10$

Answer 1

No option is universally correct.

Answer 2

The roots of the equation $x^2 - \lambda x + \mu = 0$ are given as $\lambda$ and $\mu$.

Using Vieta's formulas:

1. Sum of roots: $\lambda + \mu = -(-\lambda)/1 \implies \lambda + \mu = \lambda$. This simplifies to $\mu = 0$.
2. Product of roots: $\lambda \cdot \mu = \mu/1 \implies \lambda \mu = \mu$. Substituting $\mu=0$ into this equation yields $\lambda \cdot 0 = 0$, which is $0=0$. This equation is always true and does not constrain $\lambda$.

Thus, the only necessary condition derived from the problem statement is $\mu=0$. The equation becomes $x^2 - \lambda x = 0$, with roots $0$ and $\lambda$. This is consistent.

Now, we check the options:

(A) $|\lambda| + |\mu| = 2 \implies |\lambda| + |0| = 2 \implies |\lambda| = 2$. This is only true if $\lambda = 2$ or $\lambda = -2$. It is not true for all possible values of $\lambda$ (e.g., if $\lambda=5$).

(B) $\lambda - \mu = 10 \implies \lambda - 0 = 10 \implies \lambda = 10$. This is only true if $\lambda = 10$. It is not true for all possible values of $\lambda$ (e.g., if $\lambda=2$).

Since neither option is true for all values of $\lambda$ that satisfy the initial condition (with $\mu=0$), neither (A) nor (B) is a necessary consequence.