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(A) Given,$3$ is a root of the equation ${x^2} + ax + 3 = 0$.Substituting $x = 3$ into the equation: ${3^2} + a(3) + 3 = 0$.$9 + 3a + 3 = 0$ implies $

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$3$

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(A) Given,$3$ is a root of the equation ${x^2} + ax + 3 = 0$.Substituting $x = 3$ into the equation: ${3^2} + a(3) + 3 = 0$.$9 + 3a + 3 = 0$ implies $3a = -12$,so $a = -4$.Now,consider the equation ${x^2} + ax + b = 0$. Substituting $a = -4$,we get ${x^2} - 4x + b = 0$.Let the roots of this equation be $\alpha$ and $3\alpha$ (since one root is three times the other).From the sum of roots formula,$\alpha + 3\alpha = -(-4)/1 = 4$.$4\alpha = 4$,which gives $\alpha = 1$.The roots are $1$ and $3(1) = 3$.From the product of roots formula,$\alpha \cdot 3\alpha = b/1$.$1 \cdot 3 = b$,so $b = 3$.

If one of the roots of the equation ${x^2} + ax + 3 = 0$ is $3$ and one of the roots of the equation ${x^2} + ax + b = 0$ is three times the other root,then the value of $b$ is equal to

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