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: 

First, we must calculate the value of the determinant K. Then, we construct a quadratic equation with roots K and K+1.

Step 2: Key Formula or Approach:

The quadratic equation with roots α, β is:
x² - (α + β)x + αβ = 0

Step 3: Detailed Explanation:

Let D be the determinant:

 

9	25	16
16	36	25
25	49	36

Apply column operation C₂ → C₂ - C₁:

 

9	16	16
16	20	25
25	24	36

Now apply C₃ → C₃ - C₁:

 

9	16	7
16	20	9
25	24	11

Factor 4 from column 2:

D = 4 ×

9	4	7
16	5	9
25	6	11

Apply row operations:
R₂ → R₂ - R₁
R₃ → R₃ - R₂

 

9	4	7
7	1	2
2	0	0

Expand along third row:

D = 4 × [ 2 × (4×2 - 7×1) ]
D = 4 × [ 2 × (8 - 7) ]
D = 4 × 2 = 8

So, K = 8.

Roots are: 8 and 9
Sum = 17
Product = 72

Equation:
x² - 17x + 72 = 0

Step 4: Final Answer:

The equation is x² - 17x + 72 = 0.