Determinants

Let A be a 3 3 real matrix such that A pmatrix1\\0\\1 pmatrix=2 pmatrix1\\0\\1 pmatrix, A pmatrix-1\\0\\1 pmatrix=4 pmatrix-1\\0\\1 pmatrix, A pmatrix0\\1\\0 pmatrix=2 pmatrix0\\1\\0 pmatrix. Then, the system (A-3I) pmatrixx\ \ pmatrix= pmatrix1\\2\\3 pmatrix has :

1. exactly two solutions
2. infinitely many solutions
3. unique solution
4. no solution

Consider the system of linear equations x+y+z=4 , x+2y+2 z=10 , x+3y+4 2z= 2+15 where , . Which one of the following statements is NOT correct ?

1. The system has unique solution if 12 and 1, 15
2. The system has infinite number of solutions if = 12 and =15
3. The system is consistent if 12
4. The system is inconsistent if = 12 and 1

Consider the system of linear equations x+y+z=5, x+2y+ 2 z=9, x+3y+ z= , where , R. Then, which of the following statements is NOT correct?

1. System is consistent if 1 and =13
2. System is inconsistent if =1 and 13
3. System has unique solution if 1 and 13
4. System has infinite number of solutions if =1 and =13

Let for any three distinct consecutive terms a, b, c of an A.P, the lines ax+by+c=0 be concurrent at the point P and Q( , ) be a point such that the system of equations x+y+z=6, 2x+5y+ z= and x+2y+3z=4, has infinitely many solutions. Then (PQ)2 is equal to .