Question 2 of 58beginner🔧 ApplyNumerical2 marks

Evaluate the determinants. (i) ∣cos⁡θ−sin⁡θsin⁡θcos⁡θ∣\begin{vmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{vmatrix} (ii) ∣x2−x+1x−1x+1x+1∣\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}

(i)
Answer

1

(ii)
Answer

x3x^{3} - x2x^{2} + 2

Explanation

The solution follows the method for evaluating a 2x2 determinant as demonstrated in Example 5 of the provided context. For a general determinant |a b; c d|, the value is calculated as (ad - bc). In part (i), this leads to a trigonometric identity. In part (ii), it involves algebraic expansion and simplification.

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