Consider the system of equations
x-2y+3z=-1
-x+y-2z=k
x-3y+4z=1
Statement -1 The system of equation has no solutions for `k ne 3`.
statement -2 The determinant `|{:(1,3,-1),(-1,-2,k),(1,4,1):}| ne0, "for"" " kne3.`

Options

A. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1
C. Statement 1 is true, Statement 2 is False
D. Statement 1 is False, Statement 2 is true

Answer

A

Explanation

We have,
`D = |(1,-2,3),(-1,1,-2),(1,-3,4)|`
`rArr D = |(1,-2,3),(0,-1,1),(0,-1,1)| = 0 " " [("Applying " R_(2) rarr R_(2) - R_(1)),(R_(3) rarr R_(3) - R_(1))]`
and `D_(2) = |(1,-1,3),(-1,k,-2),(1,1,4)| = - |(1,3,-1),(-1,-2,k),(1,4,1)| " " ["Applying " C_(2) hArr C_(3)]`
`rArr D_(2) = - |(1,3,-1),(0,1,k -1),(0,1,2)| = k -3 " " [("Applying " R_(2) rarr R_(2) - R_(1)),(R_(3) rarr R_(3) - R_(1))]`
Clearly, `D_(2) != 0 " for " k != 3` and the determinant is not zero for `k != 3`.
Hence, both the statement are true and statement 2 is a correct explanation for statement 1