Let `|[x^2+3x,x-1,x+3],[x+1,-2x,x-4],[x-3,x+4, 3x]|=a x^4+b x^3+c x^2+e` be an identity in `x ,` where `a , b , c , d , e` are independent of `xdot` Then the value of `e` is (a) 4 (b) 0 (c) 1 (d) none of these

Options

A. 3
B. 2
C. 4
D. none of these

Answer

A

Explanation

Let `Delta = |(1 +x,x,x^(2)),(x,1 +x,x^(2)),(x^(2),x,1 +x)|`
Applying `C_(1) rarr C_(1) + C_(2) + C_(3)`, we get
`Delta = (1 +x)^(2) |(1,x,x^(2)),(1,1 +x,x^(2)),(1,x,1 +x)|`
Clearly, `lamda = ((d Delta)/(dx))_(x = 0)`
We have,
`(d Delta)/(dx) = 2 (1 + x) |(1,x,x^(2)),(1,1 +x,x^(2)),(1,x,1 +x)| + (1 +x)^(2) |(1,1,x^(2)),(1,1,x^(2)),(1,1,1 +x)| + (1 +x)^(2) |(1,x,2x),(1,1 +x,2x),(1,x,1)|`
`:. ((d Delta)/(dx))_(x =0) = 2 |(1,0,0),(1,1,0),(1,0,1)| + 0 + |(1,0,0),(1,1,0),(1,0,1)| = 2 + 1 = 3`.
Hence, `lamda = 3`