1.

`" If " Delta (x)=|underset(9" "x" "-7)underset(6" "4x" "3)(1" "x^(2)" "x^(2))|` then find the value of `overset(1)underset(0)(int)Delta (x) dx` without expanding `Delta (x)`.

Answer» Taking x common from `C_(2)` then multiplying each element of `R_(1)` with x we get
`:. Delta (x)= |{:( x,,x^(2),,x^(3)),(6,,4,,3),(9,,1,,-7):}|`
`|{:(overset(1)underset(0)(int) xdx,,overset(1)underset(0)(int)x^(2)dx,,overset(1)underset(0)(int)x^(3)dx),(6,,4,,3),(9,,1,,-7):}|`
`|{:([[x^(2))/(2)]_(0)^(2),,[[x^(3))/(3)]_(0)^(1),,[[x^(4))/(4)]_(0)^(1)),(6,,4,,3),(9,,1,,-7):}|`
`= |{:(9,,1,,-7),(6,,4,,3),((1)/(2),,(1)/(3),,(1)/(4)):}|`
`=(1)/(12) |{:(6,,4,,3),(6,,4,,3),(9,,1,,-7):}|` (Multiplying each element of `R_(1)` with 12)
`=0" "(R_(1) " and "R_(2) " are identical")`


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