Determinants · Question 31

If the system of linear equations 3x + y + z = 3 2x + y - z = -3 x + 2y + z = 4 has infinitely many solutions, then the value of 22 - 9 is :

1. 31
2. 37
3. 43
4. 49

Answer

(1) 31

Solution

For infinitely many solutions: = 0 and x = y = z = 0.

= vmatrix 3 & 1 & \ 2 & & -1 \ 1 & 2 & 1 vmatrix = 3( + 2) - 1(2 + 1) + (4 - )

= 3 + 3 + (4 - ) = 0 ...(i)

z = vmatrix 3 & 1 & 3 \ 2 & & -3 \ 1 & 2 & 4 vmatrix = 3(4 + 6) - 1(8 + 3) + 3(4 - )

z = 9 + 19 = 0 = - 199

From (i): - 193 + 3 + (4 + 199 ) = 0

559 = 103 = 611

Check: x = vmatrix 3 & 1 & \ -3 & & -1 \ 4 & 2 & 1 vmatrix = 3 + 5 - (6 + 4 ) = - 43 + 43 = 0 (similarly y = 0), so the system is consistent.

22 - 9 = 12 + 19 = 31

Answer: (1)