Maths Olympiad Prep

Library / /332 of 520

Algebra Difficulty 3.4 AMC 10/12 Find the answer

Given propositions pp: "The equation x2+mx+1=0x^{2}+mx+1=0 has exactly two distinct negative roots" and qq: "The inequality 3xm+103^{x}-m+1\leqslant 0 has real number solutions". If pqp\lor q is a true proposition and pqp\land q is a false proposition, find the range of real number values for mm.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

For proposition pp to be true, the equation x2+mx+1=0x^{2}+mx+1=0 must have two distinct negative roots, which implies:
{Δ>0 x1+x20  \begin{cases} \Delta > 0 \ x_{1}+x_{2} 0\ \end{cases}

This simplifies to:

m 2 -4\text{m 2 -4} > 0 \
-m 2$.

For proposition qq to be true, the inequality 3xm+103^{x}-m+1\leqslant 0 must have real number solutions, which implies 3xm13^{x}\leqslant m-1. This is only possible when m1>0m-1 > 0, so solving this inequality gives us m>1m > 1.

Since pqp\lor q is true and pqp\land q is false, pp and qq must have opposite truth values. We consider two cases:

1. pp is true and qq is false. This leads to the system of inequalities:
{m>2 m1  \begin{cases} m > 2 \ m\leqslant 1\ \end{cases}
which has no valid values for mm.

2. pp is false and qq is true. This leads to the system of inequalities:
{m2 m>1  \begin{cases} m\leqslant 2 \ m > 1\ \end{cases}
which simplifies to 1<m21 < m\leqslant 2.

Thus, the range of real number values for mm is (1,2]\boxed{(1,2]}.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.