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Matrices & Determinants: Value Determinant Vmatrix Vmatrix

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

Question
If x,y,zx, y, z are in A.P., then the value of the determinant
∣456x567y678zxyz0∣\begin{vmatrix} 4 & 5 & 6 & x \\ 5 & 6 & 7 & y \\ 6 & 7 & 8 & z \\ x & y & z & 0 \end{vmatrix}
 is:
A22
B−2-2
C00correct
DNone of these
Solution
Step 1: Apply the row operation R1→R1−2R2+R3R_1 \to R_1 - 2R_2 + R_3 Since 4−2(5)+6=04 - 2(5) + 6 = 0, 5−2(6)+7=05 - 2(6) + 7 = 0, 6−2(7)+8=06 - 2(7) + 8 = 0, and x−2y+z=0x - 2y + z = 0 (as x,y,zx, y, z are in A.P.), the entire first row becomes zero:
∣0000567y678zxyz0∣=0\begin{vmatrix} 0 & 0 & 0 & 0 \\ 5 & 6 & 7 & y \\ 6 & 7 & 8 & z \\ x & y & z & 0 \end{vmatrix} = 0
 Answer: (3)
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