Question

16．已知p：ax²-x+$\frac{1}{16}$a＞0對(duì)于任意x恒成立；q：a≥1，如果命題“p∨q為真，p∧q為假”，求實(shí)數(shù)a的取值范圍．

Answer 1

分析 p：ax²-x+$\frac{1}{16}$a＞0對(duì)于任意x恒成立，則$\left\{\begin{array}{l}{a＞0}\\{△=1-4a×\frac{a}{16}＜0}\end{array}\right.$，解得a范圍．由命題“p∨q為真，p∧q為假”，可得p與q一真一假．

解答 解：p：ax²-x+$\frac{1}{16}$a＞0對(duì)于任意x恒成立，則$\left\{\begin{array}{l}{a＞0}\\{△=1-4a×\frac{a}{16}＜0}\end{array}\right.$，解得a＞2．
∵命題“p∨q為真，p∧q為假”，∴p與q一真一假．
若p真q假，則$\left\{\begin{array}{l}{a＞2}\\{a＜1}\end{array}\right.$，解得a∈∅，
若p假q真時(shí)，$\left\{\begin{array}{l}{a≤2}\\{a≥1}\end{array}\right.$，解得1≤a≤2．
綜上，實(shí)數(shù)a的取值范圍是1≤a≤2．

點(diǎn)評(píng) 本題考查了不等式的解法與性質(zhì)、簡(jiǎn)易邏輯的判定方法，考查了推理能力與計(jì)算能力，屬于中檔題．