the value of the determinant |{:((a(1...

the value of the determinant
`|{:((a_(1)-b_(1))^(2),,(a_(1)-b_(2))^(2),,(a_(1)-b_(3))^(2),,(a_(1)-b_(4))^(2)),((a_(2)-b_(1))^(2),,(a_(2)-b_(2))^(2) ,,(a_(2)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(3)-b_(1))^(2),,(a_(3)-b_(2))^(2),,(a_(3)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(4)-b_(1))^(2),,(a_(4)-b_(2))^(2),,(a_(4)-b_(3))^(2),,(a_(4)-b_(4))^(2)):}|` is

dependent on `a_(i)` i=1,2,3,4

dependent on `b_(i)` i=1,2,3,4

dependent on `a_(i)b_(i)` i=1,2,3,4

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det[[2a_(1)b_(1),a_(1)b_(2)+a_(2)b_(1),a_(1)b_(3)+a_(3)b_(1)a_(1)b_(2)+a_(2)b_(1),2a_(2)b_(2),a_(2)b_(3)+a_(3)b_(2)a_(1)b_(3)+a_(3)b_(1),a_(3)b_(2)+a_(2)b_(3),2a_(3)b_(3)]]=

Show that if x_(1),x_(2),x_(3) ne 0 |{:(x_(1) +a_(1)b_(1),,a_(1)b_(2),,a_(1)b_(3)),(a_(2)b_(1),,x_(2)+a_(2)b_(2),,a_(2)b_(3)),(a_(3)b_(1),,a_(3)b_(2),,x_(3)+a_(3)b_(3)):}| =x_(1)x_(2)x_(3) (1+(a_(1)b_(1))/(x_(1))+(a_(2)b_(2))/(x_(2))+(a_(3)b_(3))/(x_(3)))

prove that (1)/((a-a_(1))^(2)),(1)/(a-a_(1)),(1)/(a_(1))(1)/((a-a_(2))^(2)),(1)/(a-a_(2)),(1)/(a_(2))(1)/((a-a_(3))^(2)),(1)/(a-a_(3)),(1)/(a_(3))]|=(-a^(2)(a_(1)-a_(2))(a_(2)-a_(3))(a_(3)-a_(1)))/(a_(1)a_(2)a_(3)(a-a_(1))^(2)(a-a_(2))^(2)(a-a_(3))^(2))

Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let /_\=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then

The determinant |(b_(1)+c_(1),c_(1)+a_(1),a_(1)+b_(1)),(b_(2)+c_(2),c_(2)+a_(2),a_(2)+b_(2)),(b_(3)+c_(3),c_(3)+a_(3),a_(3)+b_(3))|

The value of the determinant Delta = |(1 + a_(1) b_(1),1 + a_(1) b_(2),1 + a_(1) b_(3)),(1 + a_(2) b_(1),1 + a_(2) b_(2),1 + a_(2) b_(3)),(1 + a_(3) b_(1) ,1 + a_(3) b_(2),1 + a_(3) b_(3))| , is

If f(x)=(a_(1)x+b_(1))^(2)+(a_(2)x+b_(2))^(2)+...+(a_(n)x+b_(n))^(2), then prove that (a_(1)b_(1)+a_(2)b_(2)+...+a_(n)b_(n))^(2)<=(a_(1)^(2)+a_(2)^(2)+...+a_(n)^(2))(b_(1)^(2)+b_(2)^(2)+...+b_(n)^(2))

(a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=(2a_(2))/(a_(2)+a_(3))

Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let Delta=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then prove that

DISHA PUBLICATION-DETERMINANTS-EXERCISE -2 CONCEPT APPLICATOR

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