Question

The value of the determinant \(\begin{vmatrix}1&5&1\\\rm\log_ee&5&\sqrt\pi\\\rm\log_{10}10&5&\pi\end{vmatrix}\) is:
1. log_π π
2. π log_e 5
3. 0
4. log_e π
5. None of these.

Answer 1

Correct Answer - Option 3 : 0

Concept:

Properties of determinants:

• A linear combination of the rows/columns does not affect the value of the determinant.
• If two rows of a given matrix are interchanged, then the value of the determinant gets multiplied by -1.
• If a row of a given matrix is multiplied by a scalar k, then the value of the determinant is also multiplied by k.

Calculation:

Let A = \(\begin{vmatrix}1&5&1\\\rm\log_ee&5&\sqrt\pi\\\rm\log_{10}10&5&\pi\end{vmatrix}\).

We know that log_a a = 1.

∴ A = \(\begin{vmatrix}1&5&1\\1&5&\sqrt\pi\\1&5&\pi\end{vmatrix}\)

Using C₂ → C₂ - 5C₁, we get:

A = \(\begin{vmatrix}1&0&1\\1&0&\sqrt\pi\\1&0&\pi\end{vmatrix}\)

Expanding along C₂, we get:

A = 0.