录入25届测验2补充题目
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"023093": {
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"id": "023093",
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"content": "设 $\\angle A$ 与 $\\angle B$ 的两边分别平行, 若 $\\angle A=\\dfrac{\\pi}{4}$, 则 $\\angle B$ 的大小为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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"023094": {
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"id": "023094",
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"content": "若一个圆锥的轴截面是边长为 $2 \\mathrm{cm}$ 的正三角形, 那么这个圆锥的表面积为 $\\mathrm{cm}^2$.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"duration": -1,
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"space": "4em",
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},
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"023095": {
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"id": "023095",
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"content": "已知正三棱柱的侧面积为 $12 \\sqrt{2}$, 高为 $2$ , 则它的体积为\\blank{50}.",
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"objs": [],
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"genre": "填空题",
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"ans": "",
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"023096": {
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"id": "023096",
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"content": "正四面体的棱长为 $2$ , 则对棱所在的异面直线距离为\\blank{50}.",
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"objs": [],
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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"023097": {
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"id": "023097",
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"content": "已知正棱锥 $P-O$, 点 $O$ 为底面正多边形的中心, 点 $P$ 为该正棱锥的顶点, 现给出下列结论: \\textcircled{1} $PO \\perp$ 底面正多边形; \\textcircled{2} 正棱锥所有棱长都相等; \\textcircled{3} 侧面是全等的等腰三角形.\n其中所有正确结论的序号是\\blank{50}.",
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"genre": "填空题",
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"023098": {
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"id": "023098",
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"content": "已知 $ABCD$ 是边长为 $a$ 的正方形, 点 $P$ 在平面 $ABCD$ 外, 侧棱 $PA=a$, $PB=PD=\\sqrt{2}a$,则该几何体 $P-ABCD$ 的 5 个面中, 互相垂直的面有对.",
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"space": "4em",
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"023099": {
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"id": "023099",
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"content": "《九章算术》是我国古代内容极为丰富的数学名著, 书中有如下问题:``今有委米依垣内角, 下周八尺, 高五尺. 问: 为米几何?''其意思为:``在屋内墙角处堆放米(如图, 米堆为一个圆锥的四分之一), 米堆底部的弧长为 $8$ 尺, 米堆的高为 $5$ 尺,则米堆堆放的米为\\blank{50}斛''. (已知 $1$ 斛米的体积约为 $1.62$ 立方尺, 计算结果精确到整数)\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw (0,0,{16/pi}) coordinate (S) -- (0,5,0) coordinate (P);\n\\draw ({16/pi},0,0) coordinate (T) -- (0,5,0);\n\\draw (S) --++ (0,0,1) (T) --++ (1,0,0) (P) --++ (0,1,0);\n\\draw [dashed] (0,0,0) -- (T) (0,0,0) -- (S) (0,0,0) -- (P);\n\\draw [domain = 0:90, samples = 100] plot ({16/pi*cos(\\x)},0,{16/pi*sin(\\x)});\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"remark": "",
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"unrelated": []
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},
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"023100": {
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"id": "023100",
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"content": "已知一个矩形的周长为 $16$, 则矩形绕它的一条边旋转一周形成的圆柱的侧面积最大值为\\blank{50}.",
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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},
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"023101": {
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"id": "023101",
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"content": "已知某圆台上、下底面的半径均为整数且它们的面积之比为 $1: 4$, 母线长为 $2$. 若该圆台的体积是 $\\dfrac{7 \\sqrt{3}}{3}\\pi$, 则它的轴截面面积为\\blank{50}.",
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"objs": [],
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"genre": "填空题",
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"duration": -1,
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"023102": {
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"id": "023102",
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"content": "已知一个圆锥 $P-O$ ($P$ 为圆锥的顶点, $O$ 为底面圆心) 的母线长为 $20$, 底面半径为 $12$, 有一条平行于底面的直线 $l$ 和圆锥的高 $PO$ 的距离为 $3 \\sqrt{3}$, 和底面距离为 $8$. 则直线 $l$ 被圆锥所截的线段的长为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"duration": -1,
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"023103": {
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"id": "023103",
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"content": "如图, 三棱锥 $P-ABC$ 中, $BC=1$, $AC=2$, $PC=3$, $PA=AB$, $PA \\perp AC$, $PB \\perp BC$. 点 $Q$ 在棱 $PB$ 上且 $BQ=1$,则直线 $CQ$ 与平面 $ABC$ 所成角的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0,0) node [left] {$A$} coordinate (A);\n\\draw (-2,2,1) node [above] {$P$} coordinate (P);\n\\draw (0,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw ($(B)!{1/sqrt(8)}!(P)$) node [left] {$Q$} coordinate (Q);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B)(C)--(Q);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"genre": "填空题",
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"ans": "",
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"023104": {
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"id": "023104",
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"content": "用一个平面截正方体, 如果截面是三角形, 则截面三角形的形状不可能是\\bracket{20}.\n\\fourch{直角三角形}{等腰三角形}{锐角三角形}{等边三角形}",
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"objs": [],
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"genre": "选择题",
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"ans": "",
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"duration": -1,
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"remark": "",
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},
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"023105": {
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"id": "023105",
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"content": "如图, 在透明材料制成的长方体容器 $ABCD-A_1 B_1 C_1 D_1$内灌注一些水, 固定容器底面一边 $BC$ 于桌面上, 再将容器倾斜根据倾斜度的不同, 有下列命题: \\textcircled{1} 水的部分始终呈棱柱形; \\textcircled{2} 水面四边形 $EFGH$ 的面积不会改变; \\textcircled{3} 棱 $A_1 D_1$ 始终与水面 $EFGH$ 平行; \\textcircled{4} 当容器倾斜如图所示时, $BE\\cdot BF$是定值, 其中所有正确命题的序号是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, x = {(-15:1cm)}, y = {(75:1cm)}]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(A)!0.6!(B)$) node [below] {$E$} coordinate (E);\n\\draw ($(D)!0.6!(C)$) node [above] {$H$} coordinate (H);\n\\draw ($(B)!0.2!(B_1)$) node [right] {$F$} coordinate (F);\n\\draw ($(C)!0.2!(C_1)$) node [right] {$G$} coordinate (G);\n\\fill [pattern = north west lines] (E)--(F)--(G)--(H)--cycle;\n\\draw (E)--(F)--(G);\n\\draw [dashed] (E)--(H)--(G);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{4}}",
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"objs": [],
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"tags": [],
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"genre": "选择题",
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"ans": "",
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"duration": -1,
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},
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"023106": {
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"id": "023106",
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"content": "如图, 圆柱的一条母线为 $PB$, 底面圆上内接三角形 $ABC$, 且 $AB=AC=1$, $PB=4$,且点 $M$ 为线段 $PA$ 的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0) node [left] {$B$} coordinate (B);\n\\draw (1,0) node [right] {$C$} coordinate (C);\n\\draw (B) ++ (0,2.5) node [left] {$P$} coordinate (P);\n\\draw (-60:1 and 0.25) node [below] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(P)$) node [left] {$M$} coordinate (M);\n\\draw (B) arc (180:360:1 and 0.25);\n\\draw (B)--(P) (C)--++(0,2.5);\n\\draw (P) arc (180:-180:1 and 0.25);\n\\draw [dashed] (B) arc (180:0:1 and 0.25);\n\\draw [dashed] (A)--(P)(P)--(C)(B)--(A)--(C)(M)--(B)(B)--(C);\n\\end{tikzpicture}\n\\end{center}\n(1) 若 $\\angle BAC=90^{\\circ}$, 求证: 平面 $PAC \\perp$ 平面 $PAB$;\\\\\n(2) 若 $\\angle BAC=60^{\\circ}$, 求异面直线 $BM$ 与 $PC$ 所成的角的大小. (结果用反三角表示);\\\\\n(3) 若 $\\angle BAC=150^{\\circ}$, 求二面角 $B-PA-C$ 的大小.",
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"objs": [],
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"genre": "解答题",
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"ans": "",
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},
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"023107": {
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"id": "023107",
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"content": "已知正方体 $ABCD-A_1B_1C_1D_1$ 的棱长为 $1$ , 点 $E$ 是线段 $AC$ 上的动点(包括端点).\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw [dashed] (A)--(C);\n\\draw (A_1)--(B)--(C_1)--cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 证明: $D_1E \\parallel $ 平面 $A_1BC_1$;\\\\\n(2) 求三棱锥 $E-A_1BC_1$ 的体积;\\\\\n(3) 若直线 $EA_1, EB, EC_1$ 与平面 $A_1BC_1$ 所成的角分别为 $\\alpha, \\beta, \\gamma$, 求 $\\dfrac{1}{\\sin ^2 \\alpha}+\\dfrac{1}{\\sin ^2 \\beta}+\\dfrac{1}{\\sin ^2 \\gamma}$ 的取值范围.",
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"objs": [],
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"030001": {
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"id": "030001",
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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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Reference in New Issue