diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index a8a16978..2dc42776 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -252388,7 +252388,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) 错误, 例如$a=2$, $b=-2$; (2) 正确, 根据乘法性质; (3) 正确, 若$a=0$, 则$a=0$或$b=0$成立, 若$a\\ne 0$, 则在$ab=0$两边同时乘$\\dfrac{1}{a}$, 得$b=0$, 故$a=0$或$b=0$也成立; (4) 正确, 令$\\dfrac{a}{c}=\\dfrac{b}{d}=k$, 则$a=ck$, $b=dk$, 因此$\\dfrac{a+b}{c+d}=\\dfrac{ck+dk}{c+d}=k=\\dfrac{a}{c}$.", "solution": "", "duration": -1, "usages": [], @@ -252412,7 +252412,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "当$a=1$时, 解集为$\\mathbf{R}$; 当$a\\ne 1$时, 解集为$\\{a+1\\}$.", "solution": "", "duration": -1, "usages": [], @@ -252464,7 +252464,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "当$a=2$时, 解集为$\\{1\\}$; 当$a>2$时, 解集为$\\varnothing$; 当$a< 2$时, 解集为$\\{\\dfrac{2+\\sqrt{4-2a}}{a},\\dfrac{2-\\sqrt{4-2a}}{a}\\}$.", "solution": "", "duration": -1, "usages": [], @@ -252488,7 +252488,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $3$; (2) $\\dfrac{4}{3}$; (3) $7$; (4) $-17$", "solution": "", "duration": -1, "usages": [], @@ -252538,7 +252538,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [], @@ -252562,7 +252562,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) 正确, 因$c^2>0$, 由不等式的乘法性质可知结论正确;\\\\\n(2) 错误, 如$a=-2$, $b=-3$, $c=3$;\\\\\n(3) 正确, $a-b=(\\sqrt{a}-\\sqrt{b})(\\sqrt{a}+\\sqrt{b})>0$, 因为$\\sqrt{a}+\\sqrt{b}>0$, 故$\\sqrt{a}-\\sqrt{b}>0$, 因此$\\sqrt{a}>\\sqrt{b}$.", "solution": "", "duration": -1, "usages": [], @@ -267297,7 +267297,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "当$k\\ne -2$时, 解集为$\\{(\\dfrac{4}{k+2},\\dfrac{k-6}{k+2}\\}$; 当$k=-2$时, 解集为$\\varnothing$.", "solution": "", "duration": -1, "usages": [], @@ -267323,7 +267323,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$\\{a+2,a-2\\}$", "solution": "", "duration": -1, "usages": [], @@ -267347,7 +267347,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$a=1$, $c=4$", "solution": "", "duration": -1, "usages": [], @@ -267395,7 +267395,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $x^2-3x-3=0$; (2) $x^2+4x-17=0$; (3) $x^2-3x-\\dfrac{1}{3}=0$; (4) $x^2-15x+9=0$", "solution": "", "duration": -1, "usages": [], @@ -267472,7 +267472,7 @@ "第一单元" ], "genre": "选择题", - "ans": "", + "ans": "B", "solution": "", "duration": -1, "usages": [], @@ -267527,7 +267527,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [], @@ -267722,7 +267722,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "当$m=-1$时, 解集为$\\{-\\dfrac{5}{3}\\}$; 当$m>\\dfrac{1}{8}$时, 解集为$\\varnothing$; 当$m=\\dfrac{1}{8}$时, 解集为$\\{-\\dfrac{1}{3}\\}$; 当$m<\\dfrac{1}{9}$且$m\\ne -1$时, 解集为$\\{\\dfrac{-3m-\\sqrt{1-8m}}{1+m},\\dfrac{-3m+\\sqrt{1-8m}}{1+m}\\}$", "solution": "", "duration": -1, "usages": [], @@ -267746,7 +267746,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$a=2$, $b=1$, $c=-2$", "solution": "", "duration": -1, "usages": [], @@ -267797,7 +267797,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $x^2+2x+\\dfrac{3}{4}=0$; (2) $x^2-\\dfrac{17}{2}x+\\dfrac{33}{2}=0$; (3) $x^2+\\dfrac{13}{6}x+1=0$; (4) $x^2-\\dfrac{97}{16}x+\\dfrac{81}{16}=0$.", "solution": "", "duration": -1, "usages": [], @@ -267822,7 +267822,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$1$或$-\\dfrac{1}{2}$", "solution": "", "duration": -1, "usages": [], @@ -267846,7 +267846,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [], @@ -311449,7 +311449,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) 正确, 根据加法性质; (2) 正确, 根据乘法性质; (3) 正确, $\\dfrac{1}{a}-\\dfrac{1}{b}=\\dfrac{b-a}{ab}=0$; (4) 正确, 反复使用乘法性质; (5) 错误, 如$a=2$, $b=-1$, $c=0$; (6) 正确, 因为实数的平方非负, 因此$a-b=0$, $b-c=0$, 从而$a=b=c$.", "solution": "", "duration": -1, "usages": [], @@ -311471,7 +311471,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) 当$a\\ne 0$时, 解集为$\\{\\dfrac{1}{a}\\}$; 当$a=0$时, 解集为$\\varnothing$;\\\\\n(2) 当$a\\ne 1$时, 解集为$\\{-1\\}$; 当$a=1$时, 解集为$\\mathbf{R}$;\\\\\n(3) 当$a=0$且$b=0$时, 解集为$\\mathbf{R}$; 当$a=0$且$b\\ne 0$时, 解集为$\\varnothing$; 当$a\\ne 0$时, 解集为$\\{\\dfrac{b}{a}\\}$.", "solution": "", "duration": -1, "usages": [], @@ -311677,7 +311677,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $3$; (2) $\\sqrt{13}$; (3) $-10$", "solution": "", "duration": -1, "usages": [], @@ -311699,7 +311699,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) $x^2-3x-3=0$; (2) $x^2-x-\\dfrac{1}{3}=0$", "solution": "", "duration": -1, "usages": [], @@ -311723,7 +311723,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$x=\\dfrac{1}{3}$或$-1$", "solution": "", "duration": -1, "usages": [], @@ -311745,7 +311745,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$x=90$或$-99$", "solution": "", "duration": -1, "usages": [], @@ -311767,7 +311767,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [], @@ -311789,7 +311789,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [], @@ -311815,7 +311815,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [], @@ -311841,7 +311841,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "(1) 证明略; (2) 证明略", "solution": "", "duration": -1, "usages": [], @@ -311863,7 +311863,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [], @@ -493753,7 +493753,7 @@ "第一单元" ], "genre": "4em", - "ans": "", + "ans": "(1) 当$k=-3$或$k=1$时, 解集为$\\varnothing$; 当$k\\ne -3$且$k\\ne 1$时, 解集为$\\{\\dfrac{1}{k^2+2k-3}\\}$;\\\\\n(2) 当$k=1$时, 解集为$\\mathbf{R}$; 当$k=-1$时, 解集为$\\varnothing$; 当$k\\ne \\pm 1$时, 解集为$\\{\\dfrac{k+3}{k+1}\\}$.", "solution": "", "duration": -1, "usages": [], @@ -493780,7 +493780,7 @@ "第一单元" ], "genre": "4em", - "ans": "", + "ans": "(1) 解集为$\\{(1,1),(3,1),(2,2)\\}$;\\\\\n(2) 解集为$\\{(-1,-3),(5,-3),(-1,5),(5,5),(-2,-2),(6,-2),(-2,4),(3,4)\\}$.", "solution": "", "duration": -1, "usages": [], @@ -493885,7 +493885,7 @@ "不等式" ], "genre": "解答题", - "ans": "", + "ans": "最小值为$-10$; 当且仅当$x=-2$, $y=4$时取到最小值", "solution": "", "duration": -1, "usages": [], @@ -493908,7 +493908,7 @@ "不等式" ], "genre": "选择题", - "ans": "", + "ans": "B", "solution": "", "duration": -1, "usages": [], @@ -493931,7 +493931,7 @@ "不等式" ], "genre": "解答题", - "ans": "", + "ans": "(1) $\\alpha-\\beta$的最大值为$0$, 最小值为$-\\pi$;\\\\\n(2) $2\\alpha-\\beta$的最大值为$2\\pi$, 最小值为$-\\pi$.", "solution": "", "duration": -1, "usages": [], @@ -493954,7 +493954,7 @@ "不等式" ], "genre": "解答题", - "ans": "", + "ans": "$b-a>b>-a>a-b$", "solution": "", "duration": -1, "usages": [], @@ -508029,7 +508029,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "(1) 假; (2) 真; (3) 真; (4) 假; (5) 假; (6) 真.", "solution": "", "duration": -1, "usages": [], @@ -508051,7 +508051,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\{\\dfrac{3}{2}\\}$", "solution": "", "duration": -1, "usages": [], @@ -508073,7 +508073,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "当$a=3$且$b=-1$时, 解集为$\\mathbf{R}$; 当$a=3$且$b\\ne -1$时, 解集为$\\varnothing$; 当$a\\ne 3$时, 解集为$\\{\\dfrac{b+1}{a-2}\\}$", "solution": "", "duration": -1, "usages": [], @@ -508095,7 +508095,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "$\\{(8,-4)\\}$", "solution": "", "duration": -1, "usages": [], @@ -508117,7 +508117,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "$a=\\dfrac{5}{3}$, $b=-\\dfrac{10}{9}$", "solution": "", "duration": -1, "usages": [], @@ -508139,7 +508139,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "$a=\\dfrac{13}{2}$, $b=-4$", "solution": "", "duration": -1, "usages": [], @@ -508161,7 +508161,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "当$k=\\pm 1$时, 解集为$\\varnothing$; 当$k\\ne \\pm 1$时, 解集为$\\{(\\dfrac{3k-1}{2k^2-2},\\dfrac{k-3}{k^2-1})\\}$.", "solution": "", "duration": -1, "usages": [], @@ -508186,7 +508186,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$a=-3,-2,0,4,12$", "solution": "", "duration": -1, "usages": [], @@ -508209,7 +508209,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "当$a>-\\dfrac{1}{3}$且$a\\ne 0$时, 解集为$\\{\\dfrac{-1-\\sqrt{1+3a}}{a},\\dfrac{-1+\\sqrt{1+3a}}{a}\\}$; 当$a=0$时, 解集为$\\{\\dfrac{3}{2}\\}$; 当$a=-\\dfrac{1}{3}$时, 解集为$\\{3\\}$; 当$a<-\\dfrac{1}{3}$时, 解集为$\\varnothing$.", "solution": "", "duration": -1, "usages": [], @@ -508231,7 +508231,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "证明略", "solution": "", "duration": -1, "usages": [], @@ -508253,7 +508253,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "$(1,+\\infty)$", "solution": "", "duration": -1, "usages": [], @@ -508277,7 +508277,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "$(-\\dfrac{1}{8},1)$", "solution": "", "duration": -1, "usages": [], @@ -508301,7 +508301,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "$c=-1$", "solution": "", "duration": -1, "usages": [], @@ -508323,7 +508323,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$a=\\dfrac{5\\pm \\sqrt{17}}{2}$", "solution": "", "duration": -1, "usages": [], @@ -508345,7 +508345,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "$19$", "solution": "", "duration": -1, "usages": [], @@ -508367,7 +508367,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$(-\\dfrac{3}{2},-\\sqrt{2})\\cup (\\sqrt{2},+\\infty)$", "solution": "", "duration": -1, "usages": [], @@ -508390,7 +508390,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$k_{\\max}=-1$", "solution": "", "duration": -1, "usages": [], @@ -508412,7 +508412,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "$a=-\\dfrac{1}{3}$或$1$", "solution": "", "duration": -1, "usages": [], @@ -508434,7 +508434,7 @@ "第一单元" ], "genre": "填空题", - "ans": "", + "ans": "(1) $\\times$; (2) \\checkmark; (3) $\\times$; (4) \\checkmark", "solution": "", "duration": -1, "usages": [], @@ -508923,7 +508923,7 @@ "第一单元" ], "genre": "解答题", - "ans": "", + "ans": "当$c>-1$时, $m$的取值范围为$(-\\infty,\\dfrac{1}{c+1}]$; 当$c=-1$时, $m$的取值范围为$\\mathbf{R}$; 当$c<-1$时, $m$的取值范围为$[\\dfrac{1}{c+1},+\\infty)$", "solution": "", "duration": -1, "usages": [