NIMCET 2009 — Mathematics PYQ

Since the roots of the cubic equation are in Geometric Progression (G.P.), let the roots be:

where $a$ is the middle term and $r$ is the common ratio.

2. Use Vieta's Formulas

For a cubic equation of the form $A{x}^3+B{x}^2+Cx+D=0$, the product of the roots is given by $-\frac{D}{A}$.

In our equation $x^3-6x^2+kx+64=0$:

• $A=1$

• $B=-6$

• $C=k$

• $D=64$

Product of the roots:

3. Substitute the Root into the Equation

Since $a=-4$ is a root of the equation, it must satisfy the equation $x^3-6x^2+kx+64=0$. Substitute $x=-4$:

The positive and negative 64 cancel each other out:

Correct Option:

(d) -24