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MCQEasyJEE 2025Nature of Roots & Formation of Equations

JEE Mathematics 2025 Question with Solution

Let α1\alpha_1 and β1\beta_1 be the distinct roots of 2x2+(cos⁡θ)x−1=0, θ∈(0,2π)2x^2 + (\cos\theta)x - 1 = 0, \ \theta \in (0, 2\pi). If mm and MM are the minimum and the maximum values of α1+β1\alpha_1 + \beta_1, then 16(M+m)16(M + m) equals:

  • A

    2525

  • B

    2424

  • C

    1717

  • D

    2727

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: The quadratic equation is

2x2+(cos⁡θ)x−1=02x^2 + (\cos\theta)x - 1 = 0

with distinct roots α1\alpha_1 and β1\beta_1.

Find: The value of 16(M+m)16(M+m), where mm and MM are the minimum and maximum values mentioned in the question.

The solution explicitly states The Correct Option is D. However, the worked steps shown below it conclude with 2525, which is inconsistent with both the statement of the question and the declared correct option.

From Vieta's relations,

α1+β1=−cos⁡θ2,α1β1=−12\alpha_1 + \beta_1 = -\frac{\cos\theta}{2}, \qquad \alpha_1\beta_1 = -\frac{1}{2}

Since cos⁡θ∈[−1,1]\cos\theta \in [-1,1] for θ∈(0,2π)\theta \in (0,2\pi), we get

α1+β1=−cos⁡θ2∈[−12,12]\alpha_1 + \beta_1 = -\frac{\cos\theta}{2} \in \left[-\frac{1}{2}, \frac{1}{2}\right]

Hence,

m=−12,M=12m = -\frac{1}{2}, \qquad M = \frac{1}{2}

Therefore,

16(M+m)=16(12−12)=016(M+m) = 16\left(\frac{1}{2}-\frac{1}{2}\right)=0

So the algebra visible in the question leads to 00, while the solution declares the correct option as D and the worked text incorrectly ends at 2525. Following the instruction that the solution is the primary source, the recorded answer is D despite the discrepancy.

Common mistakes

  • Using the wrong Vieta relation for the sum of roots. For ax2+bx+c=0ax^2+bx+c=0, the sum is −ba-\frac{b}{a}, not −b2a-\frac{b}{2a} or any other variation. Here it must be α1+β1=−cos⁡θ2\alpha_1+\beta_1=-\frac{\cos\theta}{2}.

  • Ignoring the range of cos⁡θ\cos\theta. Since θ∈(0,2π)\theta \in (0,2\pi), one must use cos⁡θ∈[−1,1]\cos\theta \in [-1,1] to get the extrema of the sum. Without this, the minimum and maximum values cannot be found correctly.

  • Trusting inconsistent worked steps without checking against the original expression. The provided solution text mixes different expressions involving roots and reaches values unrelated to α1+β1\alpha_1+\beta_1. Always verify that the manipulated expression matches what the question asks.

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