Among the statements : I: If vmatrix 1 & & \ & 1 & \ & & 1 vmatrix = vmatrix 0 & & \ & 0 & \ & & 0 vmatrix, then 2 + 2 + 2 = 32, and II: If vmatrix x2 + x & x + 1 & x - 2 \ 2x2 + 3x - 1 & 3x & 3x - 3 \ x2 + 2x + 3 & 2x - 1 & 2x - 1 vmatrix = px + q, then p2 = 196q2.
- both are true
- both are false
- only I is true
- only II is true
Answer
(2) both are false
Solution
Statement I: let a = , b = , c = . Expanding,
LHS = 1(1 - c2) - a(a - bc) + b(ac - b) = 1 - a2 - b2 - c2 + 2abc
RHS = 0 - a(0 - bc) + b(ac - 0) = 2abc
Equating them, 1 - a2 - b2 - c2 + 2abc = 2abc
2 + 2 + 2 = 1 32, so Statement I is false.
Statement II: the determinant equals px + q for every x.
Putting x = 0: q = vmatrix 0 & 1 & -2 \ -1 & 0 & -3 \ 3 & -1 & -1 vmatrix = 0 - 1(1 + 9) - 2(1 - 0) = -12
Putting x = 1: p + q = vmatrix 2 & 2 & -1 \ 4 & 3 & 0 \ 6 & 1 & 1 vmatrix = 2(3) - 2(4) - 1(4 - 18) = 12
Hence p = 24, q = -12, so p2 = 576 while 196q2 = 196 144 = 28224; thus p2 196q2 and Statement II is false.
Both statements are false.
Answer: (2)