Correct Answer - Option 3 : 0
Concept:
Properties of Determinants:
- Determinant evaluated across any row or column is same.
- If all the elements of a row or column are zeroes, then the value of the determinant is zero.
- The determinant of an Identity matrix is 1.
- If rows and columns are interchanged then the value of the determinant remains the same (value does not change).
- If any two-row or two-column of a determinant are interchanged the value of the determinant is multiplied by -1.
- If two rows or two columns of a determinant are identical the value of the determinant is zero.
Calculation:
Let Δ = \(\left| {\begin{array}{*{20}{c}} {{{\sec }^2}x}&{{{\tan }^2}x}&1\\ 2&1&1\\ {10}&{ 8}&2 \end{array}} \right|\)
Apply C1 → C1 - C2
Δ = \(\left| {\begin{array}{*{20}{c}} {{{\sec }^2}x -\tan^2 x}&{{{\tan }^2}x}&1\\ 2-1&1&1\\ {10-8}&{ 8}&2 \end{array}} \right|\)
Δ = \(\left| {\begin{array}{*{20}{c}} 1&{{{\tan }^2}x}&1\\ 1&1&1\\ {2}&{ 8}&2 \end{array}} \right|\) (∵ sec2 x - tan2 x = 1)
As we know, If two rows or two columns of a determinant are identical the value of the determinant is zero.
Here C1 and C3 are identical.
So, Δ = 0