1.

A homogeneous polynomial of the second degree in n variables i.e., the expression phi=sum_(i=1)^(n)sum_(j=1)^(n)a_(ij)x_(i)x_(j) where a_(ij)=a_(ji) is called a quadratic form in n variables x_(1),x_(2)….x_(n) if A=[a_(ij)]_(nxn) is a symmetric matrix and x=[{:(x_(1)),(x_(2)),(x_(n)):}] then X^(T)AX=[X_(1)X_(2)X_(3) . . . .X_(n)][{:(a_(11),a_(12)....a_(1n)),(a_(21),a_(22)....a_(2n)),(a_(n1),a_(n2)....a_(n n)):}][{:(x_(1)),(x_(2)),(x_(n)):}] =sum_(i=1)^(n)sum_(j=1)^(n)a_(ij)x_(i)x_(j)=phi Matrix A is called matrix of quadratic form phi. Q. If number of distinct terms in a quadratic form is 10 then number of variables in quadratic form is

Answer»

4
3
5
can not FOUND uniquely

Solution :If number of variable is `n` then there are `n` terms of the type `x_(i)^(2)` and `.^(n)C_(2)` terms of the type `x_(ij)`.
Hence total number of distinct terms
`=n+.^(n)C_(2)=10`
`impliesn+(n(n-1))/(2)=10impliesn(n+1)=20impliesn=4`


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