EquationsMCQPYQ Dec 23Question 1108 of 221
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The sol. of cubic eq. x3−23x2+142x−120=0\displaystyle x^3 - 23x^2 + 142x - 120 = 0 is given by the triplet:

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — (1,−10,12)\displaystyle (1, -10, 12)

All Options:

  • A$(1, 10, 12)$
  • B(1,−10,12)\displaystyle (1, -10, 12)
  • C(−1,10,−12)\displaystyle (-1, 10, -12)
  • D(1,10,−12)\displaystyle (1, 10, -12)

Detailed Solution & Explanation

Let us solve the cubic equation x3−23x2+142x−120=0\displaystyle x^3 - 23x^2 + 142x - 120 = 0.

Using Vieta's relations for a cubic equation x3−Sx2+P2x−P3=0\displaystyle x^3 - Sx^2 + P_2x - P_3 = 0 with roots α,β,γ\displaystyle α\displaystyle \alpha, β\displaystyle \beta, γ\displaystyle \gamma:
1. Sum of roots: α+β+γ=23\displaystyle α+β+γ=23\displaystyle \alpha + \beta + \gamma = 23
2. Product of roots: αβγ=120\displaystyle α\displaystyle \alpha β\displaystyle \beta γ=120\displaystyle \gamma = 120
3. Sum of products in pairs: αβ+βγ+γα=142\displaystyle α\displaystyle \alphaβ+β\displaystyle \beta + \betaγ+γ\displaystyle \gamma + \gammaα=142\displaystyle \alpha = 142

Let us test the triplet $(1, 10, 12)$ from Option (a):
1. Sum =1+10+12=23\displaystyle = 1 + 10 + 12 = 23 (Satisfied)
2. Product =1×10×12=120\displaystyle = 1 ×\displaystyle \times 10 ×\displaystyle \times 12 = 120 (Satisfied)
3. Sum of products in pairs =1(10)+10(12)+12(1)=10+120+12=142\displaystyle = 1(10) + 10(12) + 12(1) = 10 + 120 + 12 = 142 (Satisfied)

Thus, the roots are indeed $(1, 10, 12)$, which corresponds to Option (a). (Note: The official key marks this as Option (b) due to a typographical error where $10$ was printed as $-10$).

Hence, the correct option is **Option (a)**.

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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