EquationsMCQPYQ June 24Question 1110 of 221
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The equation x3−3x2−4x+12=0\displaystyle x^3 - 3x^2 - 4x + 12 = 0 has three real roots. They are:

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

✅ Option a — −2,2,3\displaystyle -2, 2, 3

All Options:

  • A−2,2,3\displaystyle -2, 2, 3
  • B−2,2,−3\displaystyle -2, 2, -3
  • C$2, -2, -3$
  • D$2, -2, 3$

Detailed Solution & Explanation

Let us solve the cubic equation x3−3x2−4x+12=0\displaystyle x^3 - 3x^2 - 4x + 12 = 0 by factoring by grouping:
x3−3x2−4x+12=0x^3 - 3x^2 - 4x + 12 = 0
x2(x−3)−4(x−3)=0x^2(x - 3) - 4(x - 3) = 0
(x2−4)(x−3)=0(x^2 - 4)(x - 3) = 0
(x−2)(x+2)(x−3)=0(x - 2)(x + 2)(x - 3) = 0

This gives the three roots:
x−2=0  ⟹  x=2\displaystyle x - 2 = 0 \implies x = 2
x+2=0  ⟹  x=−2\displaystyle x + 2 = 0 \implies x = -2
x−3=0  ⟹  x=3\displaystyle x - 3 = 0 \implies x = 3

Thus, the roots are −2,2,3\displaystyle -2, 2, 3, which are listed in both Option (a) and Option (d). Since the order of roots does not affect the set of solutions, both options represent the mathematically correct set of roots, with Option (a) being the officially designated answer.

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.

More Questions from Equations

If 3x+y+2x−y=−1\displaystyle 3x+y+2x−y\displaystyle \frac{3}{x+y} + \frac{2}{x-y} = -1 and 1x+y−1x−y=43\displaystyle 1x+y−1x−y=43\displaystyle \frac{1}{x+y} - \frac{1}{x-y} = \frac{4}{3} then (x,y)\displaystyle (x, y) is:

2x+510+3x+1015=5\displaystyle \frac{2x+5}{10} + \frac{3x+10}{15} = 5, find x\displaystyle x

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The cost of 2\displaystyle 2 oranges and 3\displaystyle 3 apples is $28$. If the cost of an apple is doubled then the cost of $3orangesand\displaystyle oranges and5$ apples is $75$. The original cost of 7\displaystyle 7 oranges and 4\displaystyle 4 apples (in Rs) is:

In a multiple choice question paper consisting of $100questionsof\displaystyle questions of1\displaystyle mark each, a candidate gets $60\% marks. If the candidate attempted all questions and there was a penalty of $0.25$ marks for wrong answers is:

The values of x\displaystyle x and y\displaystyle y satisfying the equations 3x+y+2x−y=3\displaystyle 3x+y+2x−y=3\displaystyle \frac{3}{x+y} + \frac{2}{x-y} = 3 and 2x+y+3x−y=233\displaystyle 2x+y+3x−y=233\displaystyle \frac{2}{x+y} + \frac{3}{x-y} = \frac{23}{3} given by

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