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Matrices & Determinants: Vmatrix Vmatrix Equal

JEE Maths question with a full step-by-step solution.

Question
If f(x)=∣1xx+12xx(x−1)(x+1)x3x(x−1)x(x−1)(x−2)(x+1)x(x−1)∣f(x) = \begin{vmatrix} 1 & x & x+1 \\ 2x & x(x-1) & (x+1)x \\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1) \end{vmatrix}, then f(100)f(100) is equal to:
A00correct
B1010
C5050
DNone
Solution
Step 1: Apply row operations R2→R2−xR1R_2 \to R_2 - xR_1 and R3→R3−(x−1)R2R_3 \to R_3 - (x-1)R_2:
f(x)=∣1xx+1x−x0x(x−1)−x(x−1)0∣f(x) = \begin{vmatrix} 1 & x & x+1 \\ x & -x & 0 \\ x(x-1) & -x(x-1) & 0 \end{vmatrix}
 Step 2: Factor from rows 2 and 3 Factor xx from R2R_2 and x(x−1)x(x-1) from R3R_3:
=x2(x−1)∣1xx+11−101−10∣= x^2(x-1)\begin{vmatrix} 1 & x & x+1 \\ 1 & -1 & 0 \\ 1 & -1 & 0 \end{vmatrix}
 Step 3: Identify identical rows Rows 2 and 3 are equal, so the determinant is zero. Hence f(x)=0f(x) = 0 for all xx, and f(100)=0f(100) = 0. Answer: (1)
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