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The roots of the quadratic equation x2−4x+k=0\displaystyle x^2 - 4x + k = 0 are coincident if

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

✅ Option a — k=4\displaystyle k = 4

All Options:

  • Ak=4\displaystyle k = 4
  • Bk=3\displaystyle k = 3
  • Ck=2\displaystyle k = 2
  • Dk=1\displaystyle k = 1

Detailed Solution & Explanation

Given the quadratic equation:
x2−4x+k=0x^2 - 4x + k = 0
Comparing with ax2+bx+c=0\displaystyle ax^2 + bx + c = 0, we have:
a=1,b=−4,c=k\displaystyle a = 1, b = -4, c = k
The roots are coincident (real and equal) if the discriminant (D\displaystyle D) is equal to zero:
D=b2−4ac=0D = b^2 - 4ac = 0
Substitute the coefficients:
(−4)2−4(1)(k)=0(-4)^2 - 4(1)(k) = 0
16−4k=016 - 4k = 0
4k=16  ⟹  k=44k = 16 \implies k = 4
Therefore, the roots are coincident when k=4\displaystyle k = 4, which corresponds to Option a.

*Note:* The value of the equal roots themselves is x=−b2a=−−42(1)=2\displaystyle x = -b2a\displaystyle \frac{b}{2a} = -−42(1)=2\displaystyle \frac{-4}{2(1)} = 2. Option c represents the value of the root, whereas the question asks for the value of the parameter k\displaystyle k, which is 4\displaystyle 4 (Option a).
**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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