2
Question
If Δ=⎛⎜
⎜
⎜
⎜⎝345x456y567zxyz0∣∣
∣
∣
∣∣ = 0, then
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Solution
The correct option is A x, y, z are in AP
Δ=⎛⎜
⎜
⎜
⎜⎝345x456y567zxyz0∣∣
∣
∣
∣∣ = 0
Applying C1→C1+C3−2C2, We get−⎛⎜
⎜
⎜
⎜⎝045x056y067zx+z−2yyz0∣∣
∣
∣
∣∣= (x+z−2y)⎛⎜⎝45x56y67z∣∣
∣
∣∣Now applying R1→R1+R3−2R2,We get(x+z−2y)⎛⎜⎝00x+z−2y56y67z∣∣
∣
∣∣= (x+z−2y)(x+z−2y)(5×7−6×6)= −(x+z−2y)2Since Δ = 0Therefore, (x+z−2y)2 = 0x+z = 2y
Δ=⎛⎜ ⎜ ⎜ ⎜⎝345x456y567zxyz0∣∣ ∣ ∣ ∣∣ = 0
Applying C1→C1+C3−2C2, We get−⎛⎜ ⎜ ⎜ ⎜⎝045x056y067zx+z−2yyz0∣∣ ∣ ∣ ∣∣= (x+z−2y)⎛⎜⎝45x56y67z∣∣ ∣ ∣∣Now applying R1→R1+R3−2R2,We get(x+z−2y)⎛⎜⎝00x+z−2y56y67z∣∣ ∣ ∣∣= (x+z−2y)(x+z−2y)(5×7−6×6)= −(x+z−2y)2Since Δ = 0Therefore, (x+z−2y)2 = 0x+z = 2y