收录高三寒假作业24新题

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wangweiye7840 2024-01-25 11:01:57 +08:00
parent b03c04a367
commit 2fbd328f12
2 changed files with 235 additions and 7 deletions

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20240125-105739
023799:023805,003102,023806:023807
20240125-110148
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"023808": {
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"content": "若 $180^{\\circ}<\\alpha<360^{\\circ}$, 则 $\\cos \\dfrac{\\alpha}{2}$ 的值等于\\bracket{20}.\n\\fourch{$-\\sqrt{\\dfrac{1-\\cos \\alpha}{2}}$}{$\\sqrt{\\dfrac{1-\\cos \\alpha}{2}}$}{$-\\sqrt{\\dfrac{1+\\cos \\alpha}{2}}$}{$\\sqrt{\\dfrac{1+\\cos \\alpha}{2}}$}",
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"content": "若 $\\dfrac{\\sin \\alpha-\\cos \\alpha}{\\sin \\alpha+\\cos \\alpha}=\\dfrac{1}{2}$, 则 $\\tan 2 \\alpha=$\\blank{50}.",
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"content": "若 $\\alpha$、$\\beta \\in(0, \\dfrac{\\pi}{2})$, 且 $\\cos \\alpha=\\dfrac{5}{13}$, $\\sin (\\alpha+\\beta)=\\dfrac{3}{5}$, 则 $\\cos \\beta=$\\blank{50}.",
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"content": "若 $\\sin (\\dfrac{\\pi}{6}-\\alpha)=\\dfrac{1}{3}$, 则 $\\cos (\\dfrac{2 \\pi}{3}+2 \\alpha)=$\\blank{50}.",
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"content": "若 $\\tan (\\alpha+\\dfrac{\\pi}{6})=\\dfrac{3}{2}$, 则 $\\cos (2 \\alpha-\\dfrac{\\pi}{6})=$\\blank{50}.",
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"023813": {
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"content": "已知 $\\alpha+\\beta=\\dfrac{\\pi}{4}$, 则 $(1+\\tan \\alpha)(1+\\tan \\beta)$ 的值为\\blank{50}.",
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"023814": {
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"content": "已知 $0<\\alpha<\\dfrac{\\pi}{2}$, $\\cos \\alpha=\\dfrac{4}{5}$. 若 $0<\\beta<\\dfrac{\\pi}{2}$ 且 $\\cos (\\alpha+\\beta)=-\\dfrac{1}{2}$, 则 $\\sin \\beta$ 的值为\\blank{50}.",
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"content": "若 $\\sin \\alpha+\\sin \\beta=\\dfrac{1}{4}$, $\\cos \\alpha+\\cos \\beta=\\dfrac{1}{3}$, 则 $\\tan (\\alpha+\\beta)$ 的值为\\blank{50}.",
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"023816": {
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"content": "已知 $\\alpha$ 与 $\\beta$ 都是锐角, 且 $\\sin (\\alpha-\\beta)=\\dfrac{1}{3}$, $\\cos (\\alpha+\\beta)=\\dfrac{\\sqrt{3}}{2}$.\\\\\n(1) 求 $\\sin 2 \\alpha$ 的值;\\\\\n(2) 求证: $\\sin \\alpha \\cos \\beta=5 \\cos \\alpha \\sin \\beta$.",
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"content": "如图, 已知面积为 $\\dfrac{\\pi}{6}$ 的扇形 $AOB$, 半径为 $1$, $C$ 是弧 $AB$ 上任意一点, 作矩形 $CDEF$ 内接于该扇形.\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\t{35}\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (60:2) node [above] {$B$} coordinate (B);\n\\draw (\\t:2) node [above right] {$C$} coordinate (C);\n\\draw (C) ++ ({-2/sin(120)*sin(60-\\t)},0) node [above left] {$D$} coordinate (D);\n\\draw ($(O)!(D)!(A)$) node [below] {$E$} coordinate (E);\n\\draw ($(O)!(C)!(A)$) node [below] {$F$} coordinate (F);\n\\draw (E)--(D)--(C)--(F)(A)--(O)--(B) arc (60:0:2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求扇形圆心角 $\\angle AOB$ 的大小;\\\\\n(2) 点 $C$ 在什么位置时, 矩形 $CDEF$ 的面积最大?并说明理由.",
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"content": "若$x,y,z$都是实数, 则:(填写``\\textcircled{1} 充分非必要、\\textcircled{2} 必要非充分、\\textcircled{3} 充要、\\textcircled{4} 既非充分又非必要''之一)\\\\\n(1) ``$xy=0$''是``$x=0$''的\\blank{50}条件;\\\\\n(2) ``$x\\cdot y=y\\cdot z$''是``$x=z$''的\\blank{50}条件;\\\\\n(3) ``$\\dfrac xy=\\dfrac yz$''是``$xz=y^2$''的\\blank{50}条件;\\\\\n(4) ``$|x |>| y|$''是``$x>y>0$''的\\blank{50}条件;\\\\\n(5) ``$x^2>4$''是``$x>2$'' 的\\blank{50}条件;\\\\\n(6) ``$x=-3$''是``$x^2+x-6=0$'' 的\\blank{50}条件;\\\\\n(7) ``$|x+y|<2$''是``$|x|<1$且$|y|<1$'' 的\\blank{50}条件;\\\\\n(8) ``$|x|<3$''是``$x^2<9$'' 的\\blank{50}条件;\\\\\n(9) ``$x^2+y^2>0$''是``$x\\ne 0$'' 的\\blank{50}条件;\\\\\n(10) ``$\\dfrac{x^2+x+1}{3x+2}<0$''是``$3x+2<0$'' 的\\blank{50}条件;\\\\\n(11) ``$0<x<3$''是``$|x-1|<2$'' 的\\blank{50}条件.",
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