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The value of the determinant `abs((1,"x","x"^3),(1,"y","y"^3),(1,"z","z"^3))` is ____________.

#### Options

2(x – y)(y – z)(z – x)

(x – y)(y – z)(z – x)

None of these

(x – y)(y – z)(z – x)(x + y + z)

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#### Solution

The value of the determinant `abs((1,"x","x"^3),(1,"y","y"^3),(1,"z","z"^3))` is **(x – y)(y – z)(z – x)(x + y + z)**.

**Explanation:**

`abs((1,"x","x"^3),(1,"y","y"^3),(1,"z","z"^3))`

Apply, R_{1}→ R_{1} - R_{2}, R_{2} → R_{2} - R_{3}

`abs((0, "x - y", "x"^3 - "y"^3),(0, "y - z", "y"^3 - "z"^3),(1,"z","z"^3))`

`=> ("x - y")("y - z") abs((0,1,"x"^2 + "y"^2 + "xy"),(0,1,"y"^2 + "z"^2 + "yz"),(1,"z","z"^3))`

= (x - y)(y - z)(y^{2} + z^{2} + yz - x^{2} - y^{2} - xy)

= (x - y)(y - z)(z - x)(x + y + z)

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