OCR Output Comparison | Datalab (Chandra & Surya)

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$\therefore \left[(2b)^2 + \left(\frac{3\sqrt{2}}{b}\right)^2\right] + \left[(2b-b)^2 + \left(\frac{6\sqrt{2}}{b} - \frac{3\sqrt{2}}{b}\right)^2\right] = b^2 + \left(\frac{6\sqrt{2}}{b}\right)^2,$

$\therefore b = \sqrt{3},$

$\therefore B(\sqrt{3}, 2\sqrt{6}), D(2\sqrt{3}, \sqrt{6}),$

$\therefore$ 直线 $OB$ 的解析式为: $y = 2\sqrt{2}x$ , $\therefore$ 直线 $DF$ 的解析式为: $y = 2\sqrt{2}x - 3\sqrt{6}$ , 当 $y = 0$ 时, $2\sqrt{2}x - 3\sqrt{6} = 0$ ,

$\therefore x = \frac{3\sqrt{3}}{2},$

$\therefore F\left(\frac{3\sqrt{3}}{2}, 0\right),$

$\therefore OE = \sqrt{3}, OF = \frac{3\sqrt{3}}{2},$

$\therefore EF = OF - OE = \frac{\sqrt{3}}{2},$

$\therefore \frac{EF}{OE} = \frac{1}{2},$

故答案为: $\frac{1}{2}, \left(\frac{3\sqrt{3}}{2}, 0\right)$ .

得分
阅卷人

三、解答题: 本题共 8 小题, 每小题 5 分, 请考生在 22、23 题中选择一题, 并在答题纸上涂黑, 不涂黑、多涂或多答均按第一题评分.

17. (1) 计算: $(x+1)(x-1) + x(2-x)$ . (2) 解不等式组: $\begin{cases} 4x-3 > 9 \\ 2+x \ge 0 \end{cases}$

[答案]: 【小问 1 详解】解: 原式 $= x^2 - 1 + 2x - x^2 = 2x - 1$ ; 【小问 2 详解】解: $\begin{cases} 4x-3 > 9(1) \\ 2+x \ge 0(2) \end{cases}$ , 解不等式 (1), 得 $x > 3$ , 解不等式 (2), 得 $x \ge -2$ , 所以原不等式组的解是 $x > 3$ . [解析]:

18. 图 1, 图 2 都是由边长为 1 的小等边三角形构成的网格, 每个小等边三角形的顶点称为格点, 线段的端点均在格点上, 分别按要求画出图形.

Figure 1 shows a grid pattern formed by small equilateral triangles, resembling a large hexagon. The vertices of the small triangles are grid points. Two points, A and B, are marked on the grid. A and B are separated by two small triangle side lengths horizontally along the grid lines.

Figure 1: A hexagonal grid structure composed of small equilateral triangles. Points A and B are marked on the boundary.

图1

Figure 2 shows a grid pattern formed by small equilateral triangles, resembling a large hexagon. The vertices of the small triangles are grid points. Two points, A and B, are marked on the grid, identical to their positions in Figure 1.

Figure 2: A hexagonal grid structure composed of small equilateral triangles. Points A and B are marked on the boundary.

图2

(1) 在图 1 中画出等腰三角形 $ABC$ , 且点 $C$ 在格点上. (画出一个即可) (2) 在图 2 中画出以 $AB$ 为边的菱形 $ABDE$ , 且点 $D, E$ 均在格点上.

[答案]: (1) 答案不唯一

A possible solution for problem 18(1) showing Figure 1 with an isosceles triangle ABC drawn. Points A and B are fixed. Point C is a grid point located such that AC and BC are equal in length (each spanning two small triangle sides along the grid lines). C is positioned above the segment AB.

Figure 1 showing an isosceles triangle ABC drawn on the grid, with C being a lattice point.

Another possible solution for problem 18(1) showing Figure 1 with an isosceles triangle ABC drawn. Points A and B are fixed. Point C is a grid point located such that AC and BC are equal in length (each spanning two small triangle sides along the grid lines). C is positioned below the segment AB.

Figure 1 showing an isosceles triangle ABC drawn on the grid, with C being a lattice point.

(2)

A solution for problem 18(2) showing Figure 2 with a rhombus ABDE drawn. A and B are fixed. D and E are grid points such that ABDE forms a rhombus with AB as a side. E is located one unit left and one unit down from A, and D is located one unit right and one unit down from B (relative to the grid structure).

Figure 2 showing a rhombus ABDE drawn on the grid, with D and E being lattice points.

[解析]:

19. 如图, 正比例函数 $y = -\frac{2}{3}x$ 的图像与反比例函数 $y = \frac{k}{x}(k \neq 0)$ 的图像都经过点 $A(a, 2)$ (1) 求点 $A$ 的坐标和反比例函数表达式. (2) 若点 $P(m, n)$ 在该反比例函数图像上, 且它到 $y$ 轴距离小于 3, 请根据图像直接写出 $n$ 的取值范围.

数学试题第 7 页 (共 14 页)

数学试题第 8 页 (共 14 页)