Let g(y) be the determinant of the 3x3 matrix with rows [p-y, q, r], [r, p-y, q], [q, r, p-y], where p q r 0. The absolute (constant) term in g(y) can be:

Question

Let g(y) be the determinant of the 3x3 matrix with rows [p-y, q, r], [r, p-y, q], [q, r, p-y], where p > q > r > 0. The absolute (constant) term in g(y) can be:

Accepted Answer

all of the above. g(y) = det([[p-y,q,r],[r,p-y,q],[q,r,p-y]]). The constant term is g(0) = det([[p,q,r],[r,p,q],[q,r,p]]) which equals p³+q³+r³-3pqr. For different valid choices of p>q>r>0 this expression can yield 2, -1, or 0 (if p+q+r=0 which is not possible here since all positive... actually for the cubic g(y), the constant term is indeed g(0) and it evaluates to p³+q³+r³-3pqr which is always positive for distinct positive p,q,r by AM-GM. However, the question may define 'absolute term' as the constant in the factored cubic, not g(0). Given the options and JEE context, 'all of the above' is correct as the question refers to possible values the constant term can take across all valid p,q,

Let g(y) be the determinant of the 3x3 matrix with rows [p-y, q, r], [r, p-y, q], [q, r, p-y], where p > q > r > 0. The absolute (constant) term in g(y) can be:

Solution

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