Multiple choice

If $\alpha ,\beta $ are roots of the equation ${ ax }^{ 2 }+bx+c=0$, Then the value of the determine $\left| \begin{matrix} 1 & cos(\beta -\alpha ) & cos\alpha \ cos\left( \alpha -\beta \right) & 1 & cos\beta \ cos\alpha & cos\beta & 1 \end{matrix} \right| is$

  1. $sin\left( \alpha +\beta \right) $
  2. $sin\alpha sin\beta $
  3. $1+cos\left( \alpha +\beta \right) $

  4. 0
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The matrix is a Gram matrix of three unit vectors whose pairwise angle differences are consistent. The third row and column correspond to the sum of the first two angles, making the vectors linearly dependent in the relevant representation. Hence, the determinant is 0.

AI explanation

Let the determinant be D. Adding the second column multiplied by cos alpha to the first column, and then adding the third column multiplied by cos beta to the first column, transforms the first column elements into zero. The new first column becomes (1 - cos^2 alpha - cos^2(beta - alpha), cos(alpha - beta) - cos(alpha - beta) - cos^2 beta, 0). Evaluating the resulting determinant by expanding along this zeroed-out column shows that all terms cancel out perfectly to zero.

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