1.

Let `|(1 +x,x,x^(2)),(x,1 +x,x^(2)),(x^(2),x,1 +x)| = ax^(5) + bx^(4) + cx^(3) + dx^(2) + lamdax +mu` be an identity in x, where a, b, c, `lamda, mu` are independent of x. Then, the value of `lamda` isA. 3B. 2C. 4D. none of these

Answer» Correct Answer - A
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`


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