Question

If one of the roots of \( \begin{vmatrix} 3 & 5 & x \\ 7 & x & 7 \\ x & 5 & 3 \end{vmatrix} = 0 \), then the other roots are

• A. 3, 7
• B. 4, 7
• C. 3, 9
• D. 3, 4

Hint:

Use property $C_1 \rightarrow C_1 + C_3$ or observation to find roots faster.

Answer 1

The Correct Option is A

Step 1: Expand Determinant
$3(3x - 35) - 5(21 - 7x) + x(35 - x^2) = 0$. $9x - 105 - 105 + 35x + 35x - x^3 = 0$.
Step 2: Simplify Equation
$-x^3 + 79x - 210 = 0 \Rightarrow x^3 - 79x + 210 = 0$.
Step 3: Factorization
Given one root is $-10$, $(x + 10)$ is a factor. $x^3 - 79x + 210 = (x + 10)(x^2 - 10x + 21) = 0$.
Step 4: Solve Quadratic
$x^2 - 10x + 21 = (x - 3)(x - 7) = 0$. The other roots are 3 and 7.
Final Answer: (a)