If the roots of the equation x^3 - 9x^2 + alp | vedclass

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(D) Let the roots of the cubic equation be $(a - d)$,$a$,and $(a + d)$.According to the properties of roots of a cubic equation $x^3 - px^2 + qx - r =

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$23$

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(D) Let the roots of the cubic equation be $(a - d)$,$a$,and $(a + d)$.According to the properties of roots of a cubic equation $x^3 - px^2 + qx - r = 0$,the sum of the roots is equal to the coefficient of $x^2$ with a changed sign.$(a - d) + a + (a + d) = 9$$3a = 9 \implies a = 3$.Since $a = 3$ is a root of the equation,it must satisfy the equation $x^3 - 9x^2 + \alpha x - 15 = 0$.Substituting $x = 3$ into the equation:$(3)^3 - 9(3)^2 + \alpha(3) - 15 = 0$$27 - 81 + 3\alpha - 15 = 0$$3\alpha - 69 = 0$$3\alpha = 69$$\alpha = 23$.

If the roots of the equation $x^3 - 9x^2 + \alpha x - 15 = 0$ are in $A.P.$,then $\alpha$ is:

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