Question

The value of \(\left| {\begin{array}{*{20}{c}} {{{\sec }^2}x}&{{{\tan }^2}x}&1\\ 2&1&1\\ {10}&{ 8}&2 \end{array}} \right|\)
1. -2sec²x − 6tan²x 
2. 2sec²x − 6tan²x + 2
3. 0
4. None of these

Answer 1

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