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JEE Advanced 2025 Mathematics question

MathematicsVector AlgebraJEE Advanced 2025

Consider the vectors

Missing argument for \vec\vec x = Missing argument for \hat\hat i + 2Missing argument for \hat\hat j + 3Missing argument for \hat\hat k, Missing argument for \vec\vec y = 2Missing argument for \hat\hat i + 3Missing argument for \hat\hat j + Missing argument for \hat\hat k, and Missing argument for \vec\vec z = 3Missing argument for \hat\hat i + Missing argument for \hat\hat j + 2Missing argument for \hat\hat k.

For two distinct positive real numbers α and β, define

Missing argument for \vec\vec X = αMissing argument for \vec\vec x + βMissing argument for \vec\vec y - Missing argument for \vec\vec z, Missing argument for \vec\vec Y = αMissing argument for \vec\vec y + βMissing argument for \vec\vec z - Missing argument for \vec\vec x, and Missing argument for \vec\vec Z = αMissing argument for \vec\vec z + βMissing argument for \vec\vec x - Missing argument for \vec\vec y.

If the vectors , , and lie in a plane, then the value of α + β − 3 is .

Answer: -2

Solution

, and \vec\vec\vec\begin{vmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{vmatrix}$ = -18

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Question ID VK-035477