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MCQMediumJEE 2024Nature of Roots & Formation of Equations

JEE Mathematics 2024 Question with Solution

For 0<c<b<a0 < c < b < a, let (a+b−2c)x2+(b+c−2a)x+(c+a−2b)=0(a + b - 2c)x^2 + (b + c - 2a)x + (c + a - 2b) = 0, and α=−1\alpha = -1 be one of its roots. Which of the following statements is true?

  • A

    Both (I) and (II) are true

  • B

    Neither (I) nor (II) is true

  • C

    Only (I) is true

  • D

    Only (II) is true

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given:

  • 0<c<b<a0 < c < b < a
  • The quadratic equation is
(a+b−2c)x2+(b+c−2a)x+(c+a−2b)=0(a + b - 2c)x^2 + (b + c - 2a)x + (c + a - 2b) = 0
  • One root is α=−1\alpha = -1

Find: Which option is correct.

From the solution, the page explicitly concludes that the correct option is A and states that both Statement (I) and Statement (II) are true.

Also, substituting x=−1x = -1 into the quadratic as shown:

(a+b−2c)(−1)2+(b+c−2a)(−1)+(c+a−2b)=0(a + b - 2c)(-1)^2 + (b + c - 2a)(-1) + (c + a - 2b) = 0

which gives

a+b−2c−b−c+2a+c+a−2b=0a + b - 2c - b - c + 2a + c + a - 2b = 0

The extracted solution text then proceeds to conclude that both statements are true.

Therefore, the correct option is A.

Extracted Solution Working

Given: the solution contains two approaches.

Find: The option supported by the solution.

Approach Solution - 1 states:

  1. Statement I: If α∈(−1,0)\alpha \in (-1, 0), then bb cannot be the geometric mean of aa and cc.
  2. Statement II: If α∈(0,1)\alpha \in (0, 1), then bb may be the geometric mean of aa and cc.

It then concludes:

  • Statement I is true.
  • Statement II is true.
  • Therefore, both Statement I and II are true.

Hence the answer is A corresponding to Both (I) and (II) are true.

Common mistakes

  • Confusing the given root as α=−1\alpha = -1 with the interval-based statements about α\alpha. The question mixes a fixed root condition with statements involving ranges, so the answer must be taken from the provided statement analysis rather than by replacing all occurrences blindly.

  • Using the answer key key without checking the solution. Here the solution is the primary source, and it explicitly states that both statements are true, so the correct option is A.

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