Question

If \( \begin{vmatrix} 9 & 25 & 16 \\ 16 & 36 & 25 \\ 25 & 49 & 36 \end{vmatrix} = K \), then \( K, K+1 \) are the roots of the equation

• A. \( x^2 - 13x + 42 = 0 \)
• B. \( x^2 - 15x + 56 = 0 \)
• C. \( x^2 - 19x + 90 = 0 \)
• D. \( x^2 - 17x + 72 = 0 \)

Hint:

When evaluating determinants, look for recognizable patterns such as perfect squares or arithmetic progressions. Suitable row and column operations can simplify the determinant and make the computation much easier.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

We first evaluate the determinant \( K \). After finding its value, we form a quadratic equation whose roots are \( K \) and \( K+1 \).
Step 2: Key Formula or Approach:

If the roots of a quadratic equation are \( \alpha \) and \( \beta \), the equation can be written as \[ x^2 - (\alpha+\beta)x + \alpha\beta = 0 . \]
Step 3: Detailed Explanation:

Consider the determinant \[ D = \begin{vmatrix} 9 & 25 & 16 \\ 16 & 36 & 25 \\ 25 & 49 & 36 \end{vmatrix}. \] Observe that the entries are perfect squares: \[ 9=3^2,\; 25=5^2,\; 16=4^2,\quad 16=4^2,\; 36=6^2,\; 25=5^2,\quad 25=5^2,\; 49=7^2,\; 36=6^2 . \] Apply the column operation \( C_2 \rightarrow C_2 - C_1 \): \[ C_2 = (25-9,\; 36-16,\; 49-25) = (16,\; 20,\; 24). \] Thus, \[ D = \begin{vmatrix} 9 & 16 & 16 \\ 16 & 20 & 25 \\ 25 & 24 & 36 \end{vmatrix}. \] Factor \(4\) from the second column: \[ D = 4 \begin{vmatrix} 9 & 4 & 16 \\ 16 & 5 & 25 \\ 25 & 6 & 36 \end{vmatrix}. \] Next apply \( C_3 \rightarrow C_3 - C_1 \): \[ C_3 = (16-9,\; 25-16,\; 36-25) = (7,\; 9,\; 11). \] Hence, \[ D = 4 \begin{vmatrix} 9 & 4 & 7 \\ 16 & 5 & 9 \\ 25 & 6 & 11 \end{vmatrix}. \] Now perform row operations \( R_2 \rightarrow R_2 - R_1 \) and \( R_3 \rightarrow R_3 - R_2 \): \[ D = 4 \begin{vmatrix} 9 & 4 & 7 \\ 7 & 1 & 2 \\ 9 & 1 & 2 \end{vmatrix}. \] Then apply \( R_3 \rightarrow R_3 - R_2 \): \[ D = 4 \begin{vmatrix} 9 & 4 & 7 \\ 7 & 1 & 2 \\ 2 & 0 & 0 \end{vmatrix}. \] Expanding along the third row, \[ D = 4 \left[ 2(4 \cdot 2 - 7 \cdot 1) \right]. \] \[ D = 4 \left[ 2(8 - 7) \right] = 4 \times 2 = 8. \] Thus, \[ K = 8. \] The required quadratic equation has roots \( 8 \) and \( 9 \). Sum of roots: \[ S = 8 + 9 = 17 \] Product of roots: \[ P = 8 \times 9 = 72 \] Therefore, the quadratic equation is \[ x^2 - 17x + 72 = 0. \]
Step 4: Final Answer:

The required equation is \( x^2 - 17x + 72 = 0 \).