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NVAMediumJEE 2023Nature of Roots & Formation of Equations

JEE Mathematics 2023 Question with Solution

Let α1,α2,…,α7\alpha_1, \alpha_2, \dots, \alpha_7 be the roots of the equation x7+3x5−13x3−15x=0x^7 + 3x^5 - 13x^3 - 15x = 0 and ∣α1∣≥∣α2∣≥⋯≥∣α7∣|\alpha_1| \geq |\alpha_2| \geq \dots \geq |\alpha_7|. Then α1α2−α3α4+α5α6\alpha_1 \alpha_2 - \alpha_3 \alpha_4 + \alpha_5 \alpha_6 is equal to:

Answer

Correct answer:3

Step-by-step solution

Standard Method

Given: The roots α1,α2,…,α7\alpha_1, \alpha_2, \dots, \alpha_7 belong to the equation x7+3x5−13x3−15x=0x^7 + 3x^5 - 13x^3 - 15x = 0.

Find: The value of α1α2−α3α4+α5α6\alpha_1 \alpha_2 - \alpha_3 \alpha_4 + \alpha_5 \alpha_6.

The solution states: The answer is 33.

Therefore, the required numerical value is 33.

Common mistakes

  • Assuming the expression can be evaluated directly from Vieta's formulas without first identifying or ordering the roots. The given expression depends on the ordering by modulus, so root structure matters. First analyze the polynomial and then apply the ordering condition.

  • Ignoring the factor xx in x7+3x5−13x3−15xx^7 + 3x^5 - 13x^3 - 15x. This misses one root immediately. Always factor out common terms before studying the remaining polynomial.

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