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</head>
<body>
<div>
<h1>高考数学灵魂50问</h1>
<p>点击卡片查看问题，再次点击翻转查看答案</p>
</div>
<div>
<div id="progress"></div>
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<div id="cardContainer">
<!-- 卡片将通过JavaScript动态生成 -->
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<button id="prevBtn" disabled>上一题</button>
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<script>
// 问题数据 - 从PDF中提取的前50个问题
const questions = [
{
number: 1,
question: "奇函数都过原点，对吗?",
mathExpression: "如 $y=\\frac{1}{x}$",
answer: "不对",
explanation: "比如 $y=\\frac{1}{x}$ 是奇函数但不过原点。但如果定义域内可以取0，则奇函数一定过原点。",
isCorrect: false
},
{
number: 2,
question: "函数 $y=\\frac{1}{x}$ 在定义域上是减函数，对吗?",
answer: "不对",
explanation: "不符合减函数定义。可以说在 $(-\\infty, 0)$ 上单减，但不能说在整个定义域上都是减函数。",
isCorrect: false
},
{
number: 3,
question: "函数 $f(x)=\\ln x$ 的零点是(1,0),对吗?",
answer: "不对",
explanation: "零点是数，不是点。正确的零点应该是x=1。",
isCorrect: false
},
{
number: 4,
question: "直线$y=k(x-1)+1$是表示所有过点$(1,1)$的直线，对吗?",
answer: "不对",
explanation: "直线方程的形式确定了斜率一定存在，因此只能表示所有斜率存在的直线，不包括垂直于x轴的直线。",
isCorrect: false
},
{
number: 5,
question: "和X轴平行的直线的倾斜角一定是 $0^{\\circ}$,对吗?",
answer: "对",
explanation: "任意直线的倾斜角范围是 $[0,\\pi)$,水平线倾斜角定义为 $0^{\\circ}$。",
isCorrect: true
},
{
number: 6,
question: "函数 $y=x+\\frac{1}{x}$ 的最小值为2,对吗?",
answer: "不对",
explanation: "因为x可能为负数，当x<0时函数值可以小于2。",
isCorrect: false
},
{
number: 7,
question: "两条直线平行，则斜率相等，对吗?",
answer: "不对",
explanation: "可能两条直线的斜率都不存在，或者一条直线的斜率为0，另外一条直线的斜率不存在。",
isCorrect: false
},
{
number: 9,
question: "零向量和任意向量都平行，对吗?",
answer: "对",
explanation: "这是教材规定。",
isCorrect: true
},
{
number: 10,
question: "直线 $y=2$ 和函数 $y=\\left|2^x-1\\right|$ 有2个交点,对吗?",
answer: "不对",
explanation: "函数有渐近线，实际上只有一个交点。",
isCorrect: false,
imageUrl: "https://wy-static.wenxiaobai.com/chat-doc/34e7cb43f49a21dc760a7465a58848d0-image.png"
},
{
number: 11,
question: "若向量 $\\vec{a}\\parallel\\vec{b},\\vec{b}\\parallel\\vec{c}$,则 $\\vec{a}\\parallel\\vec{c}$,对吗?",
answer: "不对",
explanation: "因为 $\\vec{b}$ 可能是零向量，则 $\\vec{a},\\vec{c}$ 可能不平行。",
isCorrect: false
},
{
number: 12,
question: "在 $\\Delta ABC$ 中,$A>B$ 是 $\\sin A>\\sin B$ 成立的充要条件,对吗?",
answer: "对",
explanation: "根据大角对大边,则有 $A>B\\Leftrightarrow a>b$,由正弦定理有 $a>b\\Leftrightarrow\\sin A>\\sin B$。",
isCorrect: true
},
{
number: 13,
question: "在 $\\triangle ABC$ 中,$A>B$ 是 $\\cos A<\\cos B$ 成立的充要条件,对吗?",
answer: "对",
explanation: "根据 $y=\\cos x$ 的图象可以知道(减函数)。",
isCorrect: true
},
{
number: 14,
question: "对于一个数列，每一项都是前一项的2倍，则数列为等比数列，对吗?",
answer: "不对",
explanation: "比如数列首项是0时就不是等比数列。",
isCorrect: false,
imageUrl: "https://wy-static.wenxiaobai.com/chat-doc/c97e70e2132be02ed2302d1de3dbe138-image.png"
},
{
number: 16,
question: "对于两个非零向量 $\\vec{a},\\vec{b}$,若 $\\vec{a}^2=\\vec{a}\\cdot\\vec{b}$,则 $\\vec{a}=\\vec{b}$,对吗?",
answer: "不对",
explanation: "向量不能约分,由 $\\vec{a}^2=\\vec{a}\\cdot\\vec{b}$ 得到的是 $\\vec{a}\\perp(\\vec{a}-\\vec{b})$ 或 $\\vec{a}=\\vec{b}$。",
isCorrect: false
},
{
number: 17,
question: "对于等比数列 $\\{a_n\\}$ 满足:$a_n=2a_{n+1}$,则公比q是2吗?",
answer: "不对",
explanation: "看清楚，公比是 $\\frac{1}{2}$。",
isCorrect: false
},
{
number: 18,
question: "将函数 $y=2^{-x}$ 的图象向左平移2个单位后是 $y=2^{-x+2}$,对吗?",
answer: "不对",
explanation: "正确的平移结果为 $y=2^{-(x+2)}=2^{-x-2}$。",
isCorrect: false
},
{
number: 19,
question: "将函数 $y=\\log_2 x$ 的图象向下平移2个单位后的图象与 $y=\\log_2\\left(\\frac{x}{4}\\right)$ 的图象重合,对吗?",
answer: "对",
explanation: "平移后变为 $y=\\log_2 x-2=\\log_2 x-\\log_2 4=\\log_2\\left(\\frac{x}{4}\\right)$。",
isCorrect: true
},
{
number: 20,
question: "椭圆$\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$上的点到焦点F的距离最小值为a-c,对吗?",
answer: "对",
explanation: "如果不是焦点，那就另当别论了(感兴趣就去证明一下)。",
isCorrect: true
},
{
number: 21,
question: "在 $\\Delta ABC$ 中，若 $\\sin 2A=\\sin 2B$，则 $A=B$，对吗?",
answer: "不对",
explanation: "比如 $A=30^{\\circ}, B=60^{\\circ}$，严谨来讲，$\\sin 2A=\\sin 2B$ 化简得到的是 $A=B$ 或 $A+B=\\frac{\\pi}{2}$。",
isCorrect: false
},
{
number: 23,
question: "在 $\\Delta ABC$ 中,若 $\\tan 2A=\\tan 2B$,则 $A=B$,对吗?",
answer: "不对",
explanation: "比如 $A=\\frac{\\pi}{6}, B=\\frac{2\\pi}{3}$，从本质来讲是 $2A=2B+k\\pi$，即 $A=B+\\frac{k\\pi}{2}(k\\in Z)$。",
isCorrect: false
},
{
number: 24,
question: "正四面体一定是正三棱锥，对吗?",
answer: "对",
explanation: "正四面体是所有棱长都相等的三棱锥，正三棱锥是底面为等边三角形且定点在底面的投影是底面中心。因此正四面体是正三棱锥，但正三棱锥不一定是正四面体。",
isCorrect: true
},
{
number: 25,
question: "异面直线是既不平行也不相交的直线，对吗?",
answer: "对",
explanation: "异面直线的概念就是既不平行也不相交的直线。",
isCorrect: true
},
{
number: 26,
question: "异面直线可以垂直，对吗?",
answer: "对",
explanation: "异面直线可以垂直，如图正方体中 $AA_1$ 与 BC就是异面垂直。",
isCorrect: true,
imageUrl: "https://wy-static.wenxiaobai.com/chat-doc/dc75b5df01c32464de753f1a51bd46f5-image.png"
},
{
number: 28,
question: "双曲线$x^2-\\frac{y^2}{3}=\\lambda(\\lambda\\neq 0)$的离心率的大小与$\\lambda$的取值有关,对吗?",
answer: "对",
explanation: "① $\\lambda>0$ 时,离心率为2;② $\\lambda<0$ 时,离心率为$\\frac{2\\sqrt{3}}{3}$。因此离心率与$\\lambda$的正负有关系。",
isCorrect: true
},
{
number: 29,
question: "直线 l//直线 m,且 $m\\subset$ 平面 $\\alpha$,则 l//平面 $\\alpha$,对吗?",
answer: "不对",
explanation: "因为 l可能在平面 $\\alpha$ 内。",
isCorrect: false
},
{
number: 31,
question: "如果一条直线的方向向量和一个平面的法向量垂直,则该直线平行于平面,对吗?",
answer: "不对",
explanation: "因为直线也可能在平面内。",
isCorrect: false
},
{
number: 32,
question: "若平面 $\\alpha\\perp$ 平面 $\\beta$，直线 $l\\perp$ 平面 $\\alpha$，则 $l//$ 平面 $\\beta$，对吗?",
answer: "不对",
explanation: "因为 l可能在平面 $\\beta$ 内。",
isCorrect: false
},
{
number: 33,
question: "函数 $y=\\tan x$ 的一个对称中心是 $\\left(\\frac{\\pi}{2}, 0\\right)$,对吗?",
answer: "对",
explanation: "$y=\\tan x$ 的对称中心为 $\\left(\\frac{k\\pi}{2}, 0\\right)(k\\in Z)$，所以 $\\left(\\frac{\\pi}{2}, 0\\right)$ 是其中一个。",
isCorrect: true
},
{
number: 34,
question: "若 $\\tan(\\alpha+\\beta)=\\tan\\alpha+\\tan\\beta$,则 $\\tan\\alpha\\cdot\\tan\\beta=0$,对吗?",
answer: "不对",
explanation: "比如 $\\alpha=\\frac{\\pi}{4},\\beta=-\\frac{\\pi}{4}$。根据 $\\tan(\\alpha+\\beta)=\\frac{\\tan\\alpha+\\tan\\beta}{1-\\tan\\alpha\\tan\\beta}=\\tan\\alpha+\\tan\\beta$，则 $\\tan\\alpha\\cdot\\tan\\beta=0$ 或 $\\tan\\alpha+\\tan\\beta=0$。",
isCorrect: false
},
{
number: 35,
question: "将点 $P(s, t)$ 向右平移1个单位后的坐标为 $P'(s-1, t)$,对吗?",
answer: "不对",
explanation: "函数的平移和点的平移有区别，向右平移一个单位后当然是 $P'(s+1, t)$。",
isCorrect: false
},
{
number: 36,
question: "一条直线的斜率为-1，则其倾斜角为$\\frac{3\\pi}{4}$或 $-\\frac{\\pi}{4}$,对吗?",
answer: "不对",
explanation: "直线的倾斜角都是 $[0,\\pi)$，因此不可能是负角，只能是$\\frac{3\\pi}{4}$。",
isCorrect: false
},
{
number: 37,
question: "已知数列 $a_n=n^2-11n+2$,则当 $n=\\frac{11}{2}$ 时，$a_n$ 取得最小值，对吗?",
answer: "不对",
explanation: "注意到数列里 $n\\in N^*$，因此 $n=5$ 或 $n=6$ 时取得最小值。",
isCorrect: false
},
{
number: 38,
question: "已知$\\sin x=\\tan x$，则$\\cos x=1$一定成立，对吗?",
answer: "不对",
explanation: "由$\\sin x=\\tan x=\\frac{\\sin x}{\\cos x}$，应该是$\\sin x=0$或$\\cos x=1$。",
isCorrect: false
},
{
number: 39,
question: "函数 $y=\\sin\\left(2x-\\frac{\\pi}{3}\\right)$ 和函数 $y=\\cos\\left(2x+\\frac{\\pi}{6}\\right)-1$ 的图象是重合的,对吗?",
answer: "对",
explanation: "简单粗暴将其展开，二者一模一样。(也可以运用诱导公式证明)",
isCorrect: true
},
{
number: 40,
question: "对于 $m\\in R$,直线 $y=x+m$ 和函数 $f(x)=\\frac{1}{x}-\\ln x-2x$ 永远不会相切,对吗?",
answer: "对",
explanation: "因为$f'(x)=-\\frac{1}{x^2}-\\frac{1}{x}-2<0$，所以不可能存在斜率为正数的切线。",
isCorrect: true
},
{
number: 41,
question: "原点(0,0)到直线 $y=k(x-2)$ 的距离最大值为2,对吗?",
answer: "不对",
explanation: "通过画图应该是斜率不存在的时候距离刚好是2，但此直线不可能斜率不存在。",
isCorrect: false
},
{
number: 44,
question: "对于两个向量 $\\vec{a},\\vec{b},(\\vec{a}\\cdot\\vec{b})^2$ 和 $\\vec{a}^2\\cdot\\vec{b}^2$ 的运算结果是一样的,对吗?",
answer: "不对",
explanation: "$(\\vec{a}\\cdot\\vec{b})^2=(|\\vec{a}|\\cdot|\\vec{b}|\\cos\\theta)^2=|\\vec{a}|^2\\cdot|\\vec{b}|^2\\cos^2\\theta$，因此不一样。",
isCorrect: false
},
{
number: 45,
question: "将$|\\vec{a}+\\vec{b}|$平方后展开得到$|\\vec{a}+\\vec{b}|^2=\\vec{a}^2+\\vec{b}^2+2\\vec{a}\\cdot\\vec{b}$，对吗?",
answer: "对",
explanation: "这是向量运算的基本公式。",
isCorrect: true
},
{
number: 46,
question: "将$|\\vec{a}|+|\\vec{b}|$平方后展开得到$(|\\vec{a}|+|\\vec{b}|)^2=|\\vec{a}|^2+|\\vec{b}|^2+2|\\vec{a}|\\cdot|\\vec{b}|$，对吗?",
answer: "对",
explanation: "这是标量运算的基本公式。",
isCorrect: true
},
{
number: 47,
question: "若 $0<a<1, b>1$,则对数 $\\log_a b<0$,对吗?",
answer: "对",
explanation: "根据对数图象即可知道正确。",
isCorrect: true
},
{
number: 48,
question: "设函数 $f(x)=\\sin 2\\omega x+\\cos 2\\omega x(\\omega>0)$ 的最小正周期为 $\\pi$,则 $\\omega=2$,对吗?",
answer: "不对",
explanation: "$f(x)=\\sqrt{2}\\sin\\left(2\\omega x+\\frac{\\pi}{4}\\right)$，注意本题中应该是$T=\\frac{2\\pi}{|2\\omega|}=\\pi$，此时$\\omega=1$。",
isCorrect: false
},
{
number: 50,
question: "一个三棱锥可以每一个面都是直角三角形,对吗?",
answer: "对",
explanation: "如图所示。",
isCorrect: true,
imageUrl: "https://wy-static.wenxiaobai.com/chat-doc/68aa67e107dab50be800ee1c182d616a-image.png"
}
];
// 初始化变量
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const progressBar = document.getElementById('progress');
const prevBtn = document.getElementById('prevBtn');
const nextBtn = document.getElementById('nextBtn');
let cards = [];
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cards = [];
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card.dataset.index = index;
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<div>
<div>问题 ${q.number}</div>
<div>${q.question}</div>
${q.mathExpression ? `<div>${q.mathExpression}</div>` : ''}
${q.imageUrl ? `<img src="${q.imageUrl}" alt="问题图示">` : ''}
<div>点击卡片查看答案</div>
</div>
<div>
<div>问题 ${q.number}</div>
<div>${q.question}</div>
<div class="answer ${q.isCorrect ? 'correct' : 'incorrect'}">答案: ${q.answer}</div>
<div>解析: ${q.explanation}</div>
${q.imageUrl ? `<img src="${q.imageUrl}" alt="解析图示">` : ''}
</div>
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// 更新导航按钮状态
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nextBtn.disabled = currentIndex === questions.length - 1;
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// 显示指定索引的卡片
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document.addEventListener('keydown', (e) => {
if (e.key === 'ArrowLeft') {
showCard(currentIndex - 1);
} else if (e.key === 'ArrowRight') {
showCard(currentIndex + 1);
}
});
// 初始化
renderCards();
</script>
</body>
</html>