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The roots of the equation x2+(2p−1)x+p2=0\displaystyle x^2 + (2p-1)x + p^2 = 0 are real if

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

✅ Option d — p≤1/4\displaystyle p \le 1/4

All Options:

  • Ap≥1\displaystyle p \ge 1
  • Bp≤4\displaystyle p \le 4
  • Cp≥1/4\displaystyle p \ge 1/4
  • Dp≤1/4\displaystyle p \le 1/4

Detailed Solution & Explanation

Given the quadratic equation:
x2+(2p−1)x+p2=0x^2 + (2p-1)x + p^2 = 0
Here, the coefficients are:
a=1\displaystyle a = 1, b=2p−1\displaystyle b = 2p-1, and c=p2\displaystyle c = p^2.

For the roots of a quadratic equation to be real, the discriminant D\displaystyle D must be greater than or equal to zero (D≥0\displaystyle D ≥\displaystyle \ge 0):
D=b2−4ac≥0D = b^2 - 4ac ≥\displaystyle \ge 0
Substitute the coefficients into the discriminant formula:
(2p−1)2−4(1)(p2)≥0(2p-1)^2 - 4(1)(p^2) ≥\displaystyle \ge 0
Expand and simplify the inequality:
(4p2−4p+1)−4p2≥0(4p^2 - 4p + 1) - 4p^2 ≥\displaystyle \ge 0
−4p+1≥0-4p + 1 ≥\displaystyle \ge 0
1≥4p  ⟹  4p≤1  ⟹  p≤141 ≥\displaystyle \ge 4p \implies 4p ≤\displaystyle \le 1 \implies p ≤\displaystyle \le 14\displaystyle \frac{1}{4}
Therefore, the roots are real if p≤1/4\displaystyle p ≤\displaystyle \le 1/4.

**Option d**

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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