Prove: (i) (x-y-z)^2-(x^2+y^2+z^2)=2(yz-zx-xy) (ii) (3p-5q)^2-(3p+5q)^2+60pq=0 (iii) (5a-2)(5a-8)=25(a-1)^2-9

(i) (x-y-z)^2-(x^2+y^2+z^2)=2(yz-zx-xy) (ii) (3p-5q)^2-(3p+5q)^2+60pq=0 (iii) (5a-2)(5a-8)=25(a-1)^2-9

(i) (x-y-z)2 -(x2+y2+z2)

={(x+(-y)+(-z)}2 -(x2+y2+z2)

= x2+(-y)2+(-z)2 +2(x)(-y)+2(-y)(-z)+2(-z)(x) -x2-y2-z2

=x2+y2+z2-2xy+2yz-2zx-x2-y2-z2

= 2yz-2zx-2xy

=2(yz-zx-xy)

∴ (x-y-z)2 -(x2+y2+z2) = 2(yz-zx-xy) [Proved]

(ii) (3p-5q)2-(3p+5q)2+60pq

= {(3p)2-2.3p.5q+(5q)2}-{(3p)2+2.3p.5q+(5q)2}+60pq

=9p2 – 30pq+25q2 – (9p2+30pq+25q2)+60pq

=9p2-30pq+25q2-9p2-30pq-25q2+60pq

= -60pq+60pq

=0

∴ (3p-5q)2-(3p+5q)2+60pq =0 [Proved]

(iii) (5a-2)(5a-8)

= 5a(5a-8)-2(5a-8)

=25a2-40a-10a+16

=25a2-50a+16

= 25a2-50a+25-9

=25(a2-2a+1)-9

=25(a-1)2-9

∴ (5a-2)(5a-8) = 25(a-1)2-9 [Proved]

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