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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
A. 4
B. 3
C. 5
D. can not found uniquely

1 Answer

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Best answer
Correct Answer - B
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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