1.

सारणिकों के गुणधर्मों का प्रयोग करके निम्नलिखित को सिद्ध कीजिएः `|(x,x^(2),yz),(y,y^(2),zx),(z,z^(2),xy)|=(x-y)(y-z)(z-x)(xy+yz+zx)`

Answer» L.H.S `=|(x,x^(2),yz),(y,y^(2),zx),(z,z^(2),xy)|`
`=|(x-y,x^(2)-y^(2),yz-zx),(y-z,y^(2)-z^(2),zx-xy),(z,z^(2),xy)|`
`(R_(1)toR_(1)-R_(2),R_(2)toR_(2)-R_(3))`
`=|(x-y,(x-y)(x+y),-z(x-y)),(y-z,(y-z)(y+z),-x(y-z)),(z,z^(2),xy)|`
`=(x-y)(y-z)|(1,x+y,-z),(1,y+z,-x),(z,z^(2),xy)|`
`=(x-y)(y-z)|(1,x+y,-z),(0,z-x,z-x),(0,-yz,xy+zx)|`
`(R_(2)toR_(2)-R_(1),R_(3)toR_(3)-zR_(2))`
`=(x-y)(y-z)(z-x)|(1,x+y,-z),(0,1,1),(0,-yz,x+yz)|`
`=(x=y)(y-z)(z-x).1|(1,1),(-yz,xy+zx)|`
(`C_(1)`से विस्तार करने पर)
`=(x-y)(y-z)(z-x)(xy+zx+yz)`
`=(x=y)(y-z)(z-x)(xy+yz+zx)`


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