Question

If $\alpha,\beta,\gamma$ are the roots of \[ x^3-10x^2+7x+8=0, \] match List-I with List-II and choose the correct option.

• A. A-V, B-III, C-I, D-II
• B. A-V, B-III, C-II, D-IV
• C. A-V, B-III, C-II, D-I
• D. A-V, B-II, C-III, D-I

Hint:

For cubic root problems, write down all three Vieta relations before starting calculations.

Answer 1

The Correct Option is C

Step 1: Concept
Use Vieta's formulas for cubic equations.

Step 2: Meaning
For \[ x^3-10x^2+7x+8=0, \] \[ \alpha+\beta+\gamma=10, \] \[ \alpha\beta+\beta\gamma+\gamma\alpha=7, \] \[ \alpha\beta\gamma=-8. \]

Step 3: Analysis
We obtain \[ A=\alpha+\beta+\gamma=10 \Rightarrow V. \] \[ B=\alpha^2+\beta^2+\gamma^2 =(\alpha+\beta+\gamma)^2 -2(\alpha\beta+\beta\gamma+\gamma\alpha) =100-14=86 \Rightarrow III. \] \[ C=\frac1\alpha+\frac1\beta+\frac1\gamma = \frac{\alpha\beta+\beta\gamma+\gamma\alpha} {\alpha\beta\gamma} = \frac{7}{-8} =-\frac78 \Rightarrow II. \] \[ D= \frac{\alpha}{\beta\gamma} +\frac{\beta}{\gamma\alpha} +\frac{\gamma}{\alpha\beta} = \frac{\alpha^2+\beta^2+\gamma^2} {\alpha\beta\gamma} = \frac{86}{-8} = -\frac{43}{4} \Rightarrow I. \]

Step 4: Conclusion
Thus the correct matching is \[ A-V,\quad B-III,\quad C-II,\quad D-I. \]

Final Answer: (C)