EquationsPYQ May 25Question 4007 of 155
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For equation x3−6x2+5x+12=0\displaystyle x^3 - 6x^2 + 5x + 12 = 0, the product of two roots is 12. Which of the following is correct set of roots of the equation?

Options

A1, -3, -4
B1, 6, 2
C-1, 3, 4
D-1, -6, -2
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Correct Answer

✅ Option c — -1, 3, 4

All Options:

  • A1, -3, -4
  • B1, 6, 2
  • C-1, 3, 4
  • D-1, -6, -2

Detailed Solution & Explanation

Let the roots of the cubic equation x3−6x2+5x+12=0\displaystyle x^3 - 6x^2 + 5x + 12 = 0 be α\displaystyle \alpha, β\displaystyle \beta, and γ\displaystyle \gamma.
For any general cubic equation ax3+bx2+cx+d=0\displaystyle ax^3 + bx^2 + cx + d = 0, the relations between the roots and the coefficients are:
1. Sum of the roots:
α+β+γ=−ba\alpha + \beta + \gamma = -\frac{b}{a}
2. Product of the roots:
αβγ=−da\alpha\beta\gamma = -\frac{d}{a}
For the given equation x3−6x2+5x+12=0\displaystyle x^3 - 6x^2 + 5x + 12 = 0, the coefficients are:
a=1,b=−6,c=5,d=12a = 1, \quad b = -6, \quad c = 5, \quad d = 12
Using the relations, we get:
α+β+γ=−−61=6— (Equation 1)\alpha + \beta + \gamma = -\frac{-6}{1} = 6 \quad \text{--- (Equation 1)}
αβγ=−121=−12— (Equation 2)\alpha\beta\gamma = -\frac{12}{1} = -12 \quad \text{--- (Equation 2)}
We are given that the product of two roots is 12\displaystyle 12. Let αβ=12\displaystyle \alpha\beta = 12.
Substitute αβ=12\displaystyle \alpha\beta = 12 into Equation 2:
12×γ=−12  ⟹  γ=−112 \times \gamma = -12 \implies \gamma = -1
So, one of the roots is −1\displaystyle -1.
Now substitute γ=−1\displaystyle \gamma = -1 into Equation 1 to find the sum of the remaining two roots:
α+β−1=6  ⟹  α+β=7\alpha + \beta - 1 = 6 \implies \alpha + \beta = 7
We now have the sum of two roots (α+β=7\displaystyle \alpha + \beta = 7) and their product (αβ=12\displaystyle \alpha\beta = 12). We can form a quadratic equation to find these two roots:
t2−(α+β)t+αβ=0t^2 - (\alpha + \beta)t + \alpha\beta = 0
t2−7t+12=0t^2 - 7t + 12 = 0
Factorizing this quadratic equation:
(t−3)(t−4)=0(t - 3)(t - 4) = 0
This gives t=3\displaystyle t = 3 or t=4\displaystyle t = 4. Thus, the other two roots are 3\displaystyle 3 and 4\displaystyle 4.
Therefore, the set of roots of the equation is {−1,3,4}\displaystyle \{-1, 3, 4\}.
Hence, **Option C** is the correct answer.

About This Chapter: Equations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Linear, Quadratic and Cubic Equations

This chapter covers Linear, Quadratic and Cubic Equations and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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