EquationsMCQMTP Dec 23 Series IQuestion 1095 of 221
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If the roots of (K−4)x2−2Kx+(K+5)=0\displaystyle (K-4)x^2 - 2Kx + (K+5) = 0 are coincident. Then the value of K\displaystyle K?

Options

A14\displaystyle 14
B20\displaystyle 20
C18\displaystyle 18
D22\displaystyle 22
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Correct Answer

✅ Option b — 20\displaystyle 20

All Options:

  • A14\displaystyle 14
  • B20\displaystyle 20
  • C18\displaystyle 18
  • D22\displaystyle 22

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Detailed Solution & Explanation

To find the value of K\displaystyle K for which the roots of the quadratic equation (K−4)x2−2Kx+(K+5)=0\displaystyle (K-4)x^2 - 2Kx + (K+5) = 0 are coincident, we use the condition that the discriminant of the quadratic equation must be equal to zero (D=0\displaystyle D = 0).

The given quadratic equation is of the form Ax2+Bx+C=0\displaystyle Ax^2 + Bx + C = 0, where:
A=K−4\displaystyle A = K-4
B=−2K\displaystyle B = -2K
C=K+5\displaystyle C = K+5

The discriminant D\displaystyle D is given by:
D=B2−4ACD = B^2 - 4AC
Substituting the values of A\displaystyle A, B\displaystyle B, and C\displaystyle C:
D=(−2K)2−4(K−4)(K+5)D = (-2K)^2 - 4(K-4)(K+5)
D=4K2−4(K2+5K−4K−20)D = 4K^2 - 4(K^2 + 5K - 4K - 20)
D=4K2−4(K2+K−20)D = 4K^2 - 4(K^2 + K - 20)
D=4K2−4K2−4K+80D = 4K^2 - 4K^2 - 4K + 80
D=−4K+80D = -4K + 80

For the roots to be coincident (equal), we must have D=0\displaystyle D = 0:
−4K+80=0-4K + 80 = 0
4K=804K = 80
K=20K = 20

Thus, the value of K\displaystyle K is 20\displaystyle 20.

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

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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