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124731ae-cf00-4677-83ec-fe7f452a2e36 | Given the flowchart of an algorithm as shown, then $$f(-3)+f(2)=$$______． | 0 | math | |
199d4e01-0ec1-4b66-a494-eb44c87d86e6 | Rewriting the scale of a line segment as a numerical scale is ______. | 1:2000000 | math | |
6a6d31d5-9c2a-4b1e-b287-30e355e33abb | As shown in the figure, if a student numbered 1, who is $$x\ cm$$ tall, stands next to a student numbered 2, who is $$y\ cm$$ tall, and we use an inequality to represent their height relationship, this can be expressed as $$x$$___$$y$$ (fill in with "$$>$$" or "$$<$$"). | $$< $$ | math | |
0e4c3187-436b-401b-9854-bcebcb7be050 | As shown in the figure, points $$A$$ and $$B$$ are on circle $$\odot O$$. Line $$AC$$ is a tangent to $$\odot O$$, and $$OC \bot OB$$. Line segment $$AB$$ intersects $$OC$$ at point $$D$$. Given that $$AC = 2$$ and $$AO = \sqrt{5}$$, the length of $$OD$$ is ___. | $$1$$ | math | |
2544de9f-c521-4dae-aa1c-3ee9fc0c0764 | To prevent influenza, a school uses a medicinal fumigation method to disinfect classrooms. It is known that during the release of the drug, the amount of drug $$y$$ (milligrams) per cubic meter of air in the room is directly proportional to time $$t$$ (hours); after the drug release is complete, the functional relationship between $$y$$ and $$t$$ is $$y=\left (\dfrac{1}{16} \right ) ^{t-a}$$ ($$a$$ is a constant), as shown in the figure. Based on the information provided in the figure, the function representing the amount of drug $$y$$ (milligrams) per cubic meter of air from the start of the drug release to time $$t$$ (hours) is ___. | $$y=\left\lbrace\begin{align}& 10t,0\leqslant t\leqslant \dfrac{1}{10}\cr&\left ( \dfrac{1}{16}\right )^{t-\frac{1}{10}},t>\dfrac{1}{10} \end{align}\right .$$ | math | |
82df2f87-1e5c-41bd-ae0f-694e80e3b11c | As shown in the figure, $$A$$, $$B$$, and $$C$$ are three points on the circle $$\odot O$$, and $$\angle OAB = 55\unit{^{\circ}}$$. What is the measure of $$\angle ACB$$ in degrees? | $$35$$ | math | |
c3100585-1ecc-4c79-bf2d-1cf24e3a4824 | To make a lidless paper box with a regular pentagon base (as shown in Figure 2) from a regular pentagon paper piece (as shown in Figure 1), a quadrilateral needs to be cut from each vertex, such as the quadrilateral $$ABCD$$ shown in Figure 1. What is the measure of $$ \angle BAD$$? | $$\number{72}^{\circ}$$ | math | |
533cd92a-f78f-4390-aa3a-8a067eeffc53 | As shown in the figure, the side length of two identical rhombuses (all four sides are equal) is 1 cm. An ant starts from point A and moves along the edges of the rhombuses in the order ABCDEFCGA. After traveling 2012 cm, the ant stops. At which point does the ant stop? | $$E$$ | math | |
978de471-323d-4103-8a94-c1b7e4fdf58c | Execute the program flowchart as shown in the figure, the output value of $n$ is. | $4$ | math | |
4aa54aae-cea2-4118-a260-43e0ee9bd244 | A square with a side length of 2 is cut to leave the part enclosed by the solid lines as shown in the figure. This remaining part is folded into a regular square pyramid. When the volume of this pyramid is maximized, the length of the base edge is. | $\frac{4}{5}$ | math | |
1c60085d-dc66-481d-a5b2-b612193a9150 | As shown in the figure, the side length of the regular hexagon $ABCDEF$ is $1$, and let $\overrightarrow{AB}=\overrightarrow{a}$. If any two points are chosen from $A$, $B$, $C$, $D$, $E$, and $F$ as the start and end points of vector $\overrightarrow{b}$, then the maximum value of $\overrightarrow{a}\cdot \overrightarrow{b}$ is. | 2 | math | |
fe0a53b0-21ac-4638-808a-5a57fbdbe08c | In △ABC, AB = 8, CA = 6, BC = CD = 4, BD is the angle bisector of ∠ABC, and BD intersects AC at point E. Find the length of CE. | 2 | math | |
830c17d2-b327-4070-a3fc-58b68b0e2d2b | As shown in the figure, quadrilateral $ABCD$ is a rhombus, and circle ⊙O passes through points $A, C, D$, intersecting $BC$ at point $E$. Connecting $AC$ and $AE$, if $\angle EAC=15^\circ$, then $\angle B=$°. | 70 | math | |
90adab49-63ec-43e9-85b7-d8fa0cae9f6c | As shown in the figure, a series of patterns are formed using matches. Following this pattern, when 10 matches are placed on each side (i.e., n=10), the total number of matches needed is . | 165 | math | |
25709a7c-6468-4598-9d58-3352f0a9bcdb | In triangle ABC, AB = AC, and D is a point on AB such that AD = CD. If ∠ACD = 40°, then ∠B = °. | 70 | math | |
186a8607-1852-4fd9-a93e-2d1aed7fa3fb | The solution set of the inequality 3x - 2a ≤ -2 with respect to x is shown in the figure. What is the value of a? | $-\frac{1}{2}$ | math | |
b52bfac8-6875-4b20-b89f-1885603a67c5 | In the figure, in △ABC, ∠C=90°, AC=3, if cosA=$\frac{3}{5}$, then the length of BC is. | 4 | math | |
0b589743-2a15-4182-bde9-92783b54a25a | As shown in the figure, in parallelogram $ABCD$, $AB \perp BC$ at point $E$. With point $B$ as the center, and the rotation angle equal to $\angle ABC$, $\triangle B{A}'{E}'$ is rotated clockwise to get $\triangle B{A}'{E}'$. Line $DA'$ is then connected. If $\angle ADC = 60^\circ$ and $\angle ADA' = 50^\circ$, then $\angle DA' E' =$． | $160^\circ$ | math | |
a52b0557-9031-4f78-9b9c-e9c5f2246405 | As shown in the figure, D and E are points on the sides AB and AC of △ABC, respectively, such that $\frac{\text{AD}}{\text{AB}}$ = $\frac{\text{AE}}{\text{AC}}$. Given AE = 2, EC = 6, and AB = 12, find the length of AD. | 3 | math | |
024d09de-4871-4c31-87f3-fb479caa1c21 | Following the operation steps shown in the figure, if the output value of $y$ is $22$, then the input value of $x$ is. | 9 | math | |
de33abe9-2f7f-49c1-aead-f2b5f51d2aba | The following is a diagram of YaYa's 'Object Immersion in Water' experiment. She first puts 2 large balls and 1 small ball into the water, at which point the water level in the rectangular container is 5 cm. Then, she adds 6 more small balls, and the water level in the rectangular container rises to 10 cm. What is the volume of each small ball in cubic centimeters? (Unit: cm) | 30 | math | |
e632c2fe-8b3c-4fa4-bc7a-2b027fd8c4ca | As shown in the figure, $$P$$ is a point on $$\odot O$$ and $$\angle APB=50\degree$$. Point $$C$$ is the midpoint of arc $$AB$$. Then $$\angle BOC=$$______ degrees. | $$50$$ | math | |
06e2f205-f0be-4156-9bf6-938631ad989c | As shown in the figure, six rays $$OA$$, $$OB$$, $$OC$$, $$OD$$, $$OE$$, and $$OF$$ are drawn with endpoint $$O$$. Starting from a point on ray $$OA$$, points are marked and connected in a counterclockwise direction on each ray. If the points marked on each ray are sequentially labeled as $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$\cdots$$, then the 2013th point marked will be on ray ______. | $$OC$$ | math | |
bbe4584f-9c2a-4bd0-809a-95e8ccf191f0 | Read the program flowchart on the right, then the output data $$S$$ is ______. | $$31$$ | math | |
e0dcd8b6-f96e-4d90-b5aa-19f3b8106e14 | The ages (in years) of some middle school students participating in a summer camp were statistically analyzed, and the results are shown in the table. The mode of these students' ages is ______. | 17 years | math | |
3b76a0ad-7a66-48ad-b769-e8f7be354557 | The diagram shows a part of a Chinese chessboard. If the 'General' is located at point $(1, -2)$ and the 'Advisor' is located at point $(3, -2)$, then the 'Cannon' is located at point ___. | $(-2,1)$ | math | |
0c130769-36ed-4d3f-b787-9d29b4013e0e | As shown in the figure, the graphs of the functions y = 3x and y = kx + 6 intersect at point A(a, 3). The solution set for the inequality 3x ≤ kx + 6 is. | x ≤ 1 | math | |
1cd16b61-72c3-46ce-b88a-840dff2f65de | In a cube $$ABCD-A'B'C'D'$$ with edge length $$a$$, points $$M$$ and $$N$$ are the midpoints of $$CD$$ and $$AD$$, respectively. The positional relationship between $$MN$$ and $$A'C'$$ is ___. | $$MN \parallel A'C'$$ | math | |
8f82bd14-055f-41db-b040-33ee978c7dce | In the figure, $$A$$, $$B$$, and $$C$$ are three points on the circle $$\odot O$$, $$\angle OBA=50^{\circ}$$, $$\angle OBC=60^{\circ}$$, then $$\angle OAC=$$ ___. | $$20^{\circ}$$ | math | |
0c4b9037-615d-4e62-a569-8145ffde2295 | As shown in the figure, $$A_{1}$$, $$A_{2}$$ are the left and right vertices of the major axis of the ellipse $$\dfrac{x^{2}}{9}+\dfrac{y^{2}}{5}=1$$, and $$O$$ is the origin. If $$S$$, $$Q$$, and $$T$$ are three points on the ellipse different from $$A_{1}$$ and $$A_{2}$$, and the lines $$QA_{1}$$, $$QA_{2}$$, $$OS$$, and $$OT$$ form a parallelogram, then $$|OS|^{2}+|OT|^{2}=$$ ___. | $$14$$ | math | |
966cce20-b58a-4b51-a3dd-dd08f85a9b1e | As shown in the figure, the line $$y=x-2$$ intersects the $$y$$-axis at point $$C$$, and the $$x$$-axis at point $$B$$. It intersects the graph of the inverse proportion function $$y=\dfrac{k}{x}$$ in the first quadrant at point $$A$$. Connecting $$OA$$, if $$S_{\triangle AOB}:S_{\triangle BOC}=1:2$$, then the value of $$k$$ is ___. | $$3$$ | math | |
6296b9b2-3046-4ca6-bcc9-9dbb342169f8 | As shown in the figure, in trapezoid $$ABCD$$, $$AD \parallel BC$$, $$AB=5$$, $$AC=9$$, $$\angle BCA=30^{\circ}$$, $$\angle ADB=45^{\circ}$$, then $$BD=$$ ___. | $$\dfrac{9\sqrt{2}}{2}$$ | math | |
bf6494ba-72ee-455c-b378-d3f582861307 | As shown in the figure, the side length of rhombus $ABCD$ is $4cm$, point $P$ is the midpoint of side $BC$, and $AP \bot BC$. What is the area of rhombus $ABCD$ in $cm^2$? (Express the result in radical form) | $8\sqrt{3}$ | math | |
2037e943-5114-4e1e-a564-122d4172b8d6 | As shown in the figure, the area of rhombus ABCD is 8, with side AD on the x-axis and the midpoint E of side BC on the y-axis. The graph of the inverse proportion function y = $\frac{k}{x}$ passes through vertex B. What is the value of k? | 4 | math | |
b3e1d516-4b4e-40a6-b821-a99a8d998407 | As shown in the figure, in $\Delta ABC$, $\angle ABC=90{}^\circ$, BD is the median of AC, CE is perpendicular to BD at point E, and a line through point A parallel to BD intersects the extension of CE at point F. On the extension of AF, a segment FG is taken such that FG=BD, and BG and DF are connected. If AF=8 and CF=6, then the perimeter of quadrilateral BDFG is. | 20 | math | |
aaf1a682-5a05-4279-98ec-72e32ddf2c04 | In the parallelogram $ABCD$, $DE$ bisects $\angle ADC$ and intersects $BC$ at point $E$, $AF \perp DE$, with the foot of the perpendicular at $F$. If $\angle DAF = 40{}^\circ$, then $\angle B$ equals. | $100{}^\circ$ | math | |
90dbb443-e501-4f48-bd8e-b292a4c5474e | As shown in the figure, a rectangular paper ABCD is folded along line EF, such that point C lands on the midpoint H of side AD, and point B lands on point G. Given AB=9, BC=6, find the length of CF. | 5 | math | |
8634740e-1b0b-4f8d-9f8b-9b619d31f8ae | As shown in the figure, in $\vartriangle ABC$, points D and E are on AB and BC, respectively, and $DE\parallel AC$. AE and CD intersect at point F. If the ratio of the areas ${{S}_{\vartriangle BDE}}:{{S}_{\vartriangle DEC}}=1:3$, then the ratio ${{S}_{\vartriangle DEF}}:{{S}_{\vartriangle AFC}}$ is . | 1:16 | math | |
db6952a3-17a9-4c19-bb65-ceb4b6b9bf97 | As shown in the figure, in trapezoid ABCD, $AD//BC$, and AD：$BC=1$：3. The diagonals AC and BD intersect at point O. Then, ${{S}_{\vartriangle AOD}}$：${{S}_{\vartriangle BOC}}$：${{S}_{\vartriangle AOB}}=$. | 1：9：3 | math | |
02c1808c-2a22-4f30-959e-f66a100e5b93 | As shown in the figure, point O is on the line DB. Given that ∠1 = 15° and ∠AOC = 90°, find the measure of ∠2. | 105° | math | |
878c09ab-2978-4a2e-9895-bd158239ceea | As shown in the figure, AB=AC, BD=DC, ∠BAC=36°, then the measure of ∠BAD is °. | 18 | math | |
556b1180-cda5-4e0a-ace8-ca7369d4f9dc | Arrange the numbers in Pascal's Triangle from top to bottom and from left to right to form a sequence: 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, ... Denote this sequence as $\{a_n\}$. If the sum of the first n terms of the sequence $\{a_n\}$ is $S_n$, then $S_{69}=$. | 2114 | math | |
7d83b548-c618-4593-a92e-269502a94b75 | New Year's Day is coming, and everyone is preparing for the party. The cultural committee member made a hat using a semicircle with a radius of $8\text{cm}$. After everyone tried it on, it was found that Li Hua fits it perfectly. Therefore, Li Hua's head circumference is approximately $\text{cm}$ (result should include $\text{ }\!\!\pi\!\!\text{ }$). | $8\text{ }\!\!\pi\!\!\text{ }$ | math | |
82727647-04f8-4602-88b0-73563f63c92b | Let $$X$$ be a discrete random variable with the following probability distribution: Then the value of $$q$$ is ___. | $$1-\dfrac{\sqrt{2}}{2}$$ | math | |
1260eb1e-7141-4007-ae63-1a6e1358c8e0 | As shown in the figure, points $$A$$, $$C$$, $$F$$, and $$B$$ lie on the same straight line, $$CD$$ bisects $$\angle ECB$$, and $$FG \parallel CD$$. If $$\angle ECA$$ is $$\alpha$$ degrees, then $$\angle GFB$$ is ___ degrees (express your answer in terms of $$\alpha$$). | $$90^{ \circ }-\dfrac{ \alpha }{2}$$ | math | |
c969f383-5474-4234-8a77-9c733e1f294e | As shown in Figure 1, there is a small cup inside a container filled with water. The small cup already contains some water. Water is now being poured into the small cup at a constant rate. After the small cup is filled, the pouring continues. The relationship between the height of the water $$y$$ (cm) in the small cup and the time $$x$$ (s) of pouring is represented by the graph in Figure 2. How many seconds at least are needed to fill the small cup? | $$5$$ | math | |
147be507-b531-4871-9405-abf77fb35d27 | The diagram represents a numerical transformation machine. If the input value of $$x$$ is $$6$$ and the value of $$y$$ is $$-4$$, then the output result is ___. | $$7$$ | math | |
c6f664d5-7205-4b6f-ae2b-29be00f8a837 | As shown in the figure, it is known that $$PA$$ is perpendicular to the plane of circle $$O$$. $$AB$$ is the diameter of circle $$O$$, and $$C$$ is a point on the circumference of the circle. How many pairs of planes are perpendicular to each other in the figure? | $$3$$ | math | |
9354e3ca-97a0-4c2f-bc53-90405a6b1222 | Arrange tables and chairs as shown in the figure: If tables are arranged in the same way, and n tables are placed, the number of chairs that should be placed is ______. | 4n+2 | math | |
fbc2a95b-9834-46f9-809d-d9a682feab88 | As shown in the figure, in two concentric circles, the chord $$AB$$ of the larger circle intersects the smaller circle at points $$C$$ and $$D$$. Given that $$AB = 2CD$$ and the distance from $$AB$$ to the center $$OM = \dfrac{1}{2}CD$$, the ratio of the radii of the larger circle to the smaller circle is ___. | $$\sqrt{5}:\sqrt{2}$$ | math | |
1ef1e420-5490-449f-88f5-934c27053b46 | In the 'Everyone Can Play an Instrument' performance competition for primary school students at a certain school, the scores of 10 students from Class 1, Grade 4 are shown in the chart below. What is the median score of these 10 students? | 90 | math | |
6fc1a7f7-1d28-4868-a56d-4273bb16292c | As shown in the figure, a drone measures the angle of elevation to the top of a building $$B$$ from point $$A$$ to be $$30\degree$$, and the angle of depression to the bottom of the building $$C$$ to be $$60\degree$$. At this moment, the horizontal distance $$AD$$ between the drone and the building is $$90$$ meters. The height of the building $$BC$$ is approximately ______ meters. (Round to the nearest $$1$$ meter, reference data $$\sqrt3\approx1.73$$) | $$208$$ | math | |
67de2964-37a6-4b11-b701-9823234025b5 | Yangyang has an empty bottle. The upper part of the bottle is gourd-shaped, and the lower part is cylindrical. The diameter of the base is $$8\text{cm}$$, to measure its volume, he filled the bottle with water and conducted the experiment as shown in the figure (unit: $$\text{cm}$$). The volume of this bottle is ______ $$\text{c}{{\text{m}}^{3}}$$. ($$\pi$$ is taken as $$3.14$$) | 753.6 | math | |
675e1d1c-2415-434f-b2a4-b9b49564643a | As shown in the figure, in quadrilateral $$ABCD$$, $$DC \parallel AB$$, $$BC=1$$, $$AB=AC=AD=2$$. Then $$BD=$$ ___. | $$\sqrt{15}$$ | math | |
e12cebec-3764-4df7-80cc-699d799f8339 | As shown in the figure, there is an animation program where the square $$ABCD$$ on the screen is a black area (including the square's boundary), with points $$A\left ( 1,1\right ) $$, $$B\left ( 2,1\right ) $$, $$C\left ( 2,2\right ) $$, and $$D\left ( 1,2\right ) $$. A signal gun fires a signal along the line $$y=2x+b$$. When the signal hits the black area, the area turns white. The range of values for $$b$$ that can turn the black area white is ___. | $$-3\leqslant b\leqslant 0$$ | math | |
9cf309b2-0694-43ae-ba4a-3314f7fd5387 | As shown in Figure 1, a sealed container in the shape of a regular quadrilateral prism has a solid decorative block in the shape of a regular quadrilateral pyramid embedded at the bottom. When the container holds $$a\ \unit{L}$$ of water, the water surface exactly passes through the apex $$P$$ of the pyramid. If the container is inverted, the water surface also exactly passes through point $$P$$ (as shown in Figure 2). There are the following four statements: (1) The height of the regular quadrilateral pyramid is half the height of the regular quadrilateral prism; (2) When the container is placed horizontally on its side, the water surface also exactly passes through point $$P$$; (3) Regardless of how the container is placed, when the water surface is still, it will always exactly pass through point $$P$$; (4) If $$a\ \unit{L}$$ of water is added to the container, the container will be exactly full. The correct statements are ___. | (2)(4) | math | |
72961f52-5b38-47a4-8857-ee3876d37fba | As shown in the figure, in rectangle $$ABCD$$, $$DE \perp AC$$ at point $$E$$, $$AB=12$$, $$AC=20$$, then $$\cos \angle ADE=$$ ___. | $$\dfrac{3}{5}$$ | math | |
d17917d9-430f-42bb-9361-3130cc750348 | As shown in the figure, grass is planted around a rectangular plot of land with a length of 8 meters and a width of 6 meters, with the same width on all sides. Flowers are planted in the middle, making the area of the grass half of the area of the rectangular plot. What is the width of the grass? | 1m | math | |
9bc46b83-f045-4227-addf-93e747a491c8 | Arrange consecutive positive integers according to the following pattern: If the positive integer $$565$$ is located in the $$a$$th row and the $$b$$th column, then $$a+b=$$ ___. | $$147$$ | math | |
af6528b6-ab59-4278-b25f-ef6c66e289ef | Calculate according to the program shown in the figure. If the input value of $$x$$ is $$3$$, then the output value is ___. | $$-3$$ | math | |
11b25328-dad7-4c96-b0e9-a10bc972321c | The graph of an inverse proportion function in the third quadrant is shown in the figure. A is any point on the graph, AM is perpendicular to the x-axis at point M, and O is the origin. If the area of △AOM is 3, then the expression of this inverse proportion function is ______. | y = 6/x | math | |
bd00cecd-f874-493f-9f5c-a0ffeb598dea | A slow train and a fast train depart from locations A and B simultaneously, traveling towards each other at constant speeds. After meeting en route, both trains continue to travel to location A at their original speeds. The graph shows the relationship between the distance $$s\ \unit{(km)}$$ between the two trains and the travel time $$t\ \unit{(h)}$$ of the slow train. When the fast train arrives at location A, the slow train is ___$$\unit{km}$$ away from location A. | $$60$$ | math | |
bc0788f5-9f76-4efc-8bc8-36a14f41bd6f | The position of the real number $$a$$ on the number line is shown in the figure. Then, the simplified form of $$\sqrt{(a-4)^2}+\sqrt{(a-11)^2}$$ is ___. | $$7$$ | math | |
60e00e59-12d2-4e27-afd9-7e8f421b6cd4 | A factory conducted a survey on the defective parts produced by a group over 10 days. The number of defective parts produced by this group each day is shown in the table: What is the standard deviation of the number of defective parts produced by this group each day over these 10 days? | $$\sqrt{2}$$ | math | |
21691094-1ba4-4a20-a658-fffa3761f7bb | To understand the situation of students participating in club activities during the May Day holiday this year, 100 students were randomly selected for statistical analysis, and the frequency distribution histogram shown in the figure was created. It is known that the school has a total of 1,000 students. Based on this, estimate the number of students who participated in club activities for 8 to 10 hours during the May Day holiday. | 280 | math | |
dd9d814c-0042-42f8-9da4-bd46c9a48ab0 | Non-negative real numbers $$x$$, $$y$$. satisfy $$\begin{cases} 2x+y-4 \leqslant 0, x+y-3 \leqslant 0, \end{cases}$$ then the maximum value of $$x+3y$$ is ___. | $$9$$ | math | |
62235d6b-7ca2-4aa5-a355-a54fd899c0cc | To understand the growth of a windbreak forest, the base circumference (unit: cm) of 100 randomly selected trees was measured. Based on the collected data, a frequency distribution histogram was drawn (as shown below). In these 100 trees, the number of trees with a base circumference greater than 110 cm is ______. | 30 | math | |
5f714729-4f80-4057-b7fa-957598d8e522 | The graph shows the function of the total production $$C$$ of a certain product over time $$t$$ (years) for a factory over eight years. Among the following four statements: (1) The production growth rate increased in the first three years; (2) The production growth rate decreased in the first three years; (3) Production of this product stopped after the third year; (4) The total production remained constant after the third year. The correct statement(s) is/are ___ (fill in the sequence number). | (2)(4) | math | |
b6fb20ad-0c92-41c5-af3b-c1b1910122b6 | Arrange positive odd numbers in the following form, where $$a_{ij}$$ represents the $$j$$th number in the $$i$$th row $$(i \in \mathbf{N}^{*}, j \in \mathbf{N}^{*})$$, for example, $$a_{32}=9$$. If $$a_{ij}=2009$$, then $$i+j=$$ ___. | 60 | math | |
c903c06e-d3c1-4e19-8f84-be229d535824 | Below are the function values of the continuous function $$f\left ( x\right ) $$ at some points in the interval $$\left \lbrack 1,2\right \rbrack $$: From this, it can be determined that an approximate solution to the equation $$f\left (x\right )=0$$ is ___. (Accuracy: $$0.1$$) | $$1.438$$ | math | |
a8f23014-a44c-49e5-b534-d51882a252b0 | As shown in the figure, an equilateral triangle $$ADE$$ is constructed outside the square $$ABCD$$. The degree measure of $$∠BED$$ is ___. | $$45^{\circ}$$ | math | |
2d462373-5d30-42a2-9736-387a72ba86c1 | As shown in the figure, there is an ancient tower $$AB$$ on the opposite bank of the river. Xiao Min measures the angle of elevation to the top of the tower $$A$$ from point $$C$$ as $$\alpha$$. Moving forward $$s\ \unit{m}$$ to point $$D$$, she measures the angle of elevation to $$A$$ as $$\beta$$. The height of the tower is ___$$\unit{m}$$. | $$\dfrac{s\cdot \tan \alpha \cdot \tan \beta}{ \tan \beta -\tan \alpha}$$ | math | |
58314254-6c2b-4224-9c4b-97894337f967 | Given the lines $$y_{1}=mx+n$$ and $$y_{2}=ax+b$$ in the Cartesian coordinate system as shown in the figure, the solution set of the inequality $$mx+n\leqslant ax+b$$ is ___. | $$x\leqslant -2$$ | math | |
b9eb7f6a-8936-498b-ac10-c18f020d2bf3 | The equation = 0 has an extraneous root, then m=______. | m=9 | math | |
ad4b332d-b70b-4083-a358-6e345ac59522 | As shown in the figure, in the right prism $$ABC-A_{1}B_{1}C_{1}$$, the base is a right triangle. $$\angle ACB=90^{\circ}$$, $$AC=6$$, $$BC=CC_{1}=\sqrt{2}$$, and $$P$$ is a moving point on $$BC_{1}$$. The minimum value of $$CP+PA_{1}$$ is ___. | $$5\sqrt{2}$$ | math | |
8c6a2746-7b5a-4a43-a28a-b88db44b58cc | As shown in the figure, in $\vartriangle ABC$, $\angle B = \angle C = 60{}^\circ$, point $D$ is on side $AB$, $DE \perp AB$, and intersects side $AC$ at point $E$. If $AD = 1$ and $BC = 6$, then $CE =$． | 4 | math | |
1d4b63a5-8041-4105-9ecc-98d5a030f42c | As shown in the figure, in $\vartriangle ABC$, $\angle B = 40^\circ$, $\angle C = 45^\circ$. The perpendicular bisector of $AB$ intersects $BC$ at point $D$, and the perpendicular bisector of $AC$ intersects $BC$ at point $E$. Then $\angle DAE = $ degrees. | 10 | math | |
fa45fbdd-2448-44b5-abcd-a22458b249a2 | The area of the region bounded by the curve $y=\cos 2x$ (as shown in the shaded part of the figure below) is. | $\frac{5}{4}$ | math | |
89a941f7-24a6-4b68-8631-84809023887f | In the quadrilateral region $ABCD$ shown in the figure, $AB=BC=1$, $CD=3$, $\angle ABC=\angle BCD=120{}^\circ$. Now, a landscape gardener plans to add a scenic area $ADM$ outside the region with $AD$ as one of its sides. When $\angle AMD=45{}^\circ$, the maximum area of the scenic area is. | $3(\sqrt{2}+1)$ | math | |
8ce898c8-3af5-46f8-9690-cd050c4f7a2d | We refer to the number of points in the two arrangement forms shown in the figure as 'triangular numbers' (such as 1, 3, 6, 10...) and 'square numbers' (such as 1, 4, 9, 16...). Among numbers less than 200, let the largest 'triangular number' be m, and the largest 'square number' be n. What is the value of m + n? | 386 | math | |
7619ca9f-206f-4374-aa64-31f8294d7994 | As shown in the figure, CD is the diameter of circle O, and AB is a chord. If CD is perpendicular to AB, and ∠BCD = 25°, then ∠AOD = °. | 50 | math | |
9eaecde4-a149-4ca5-af8d-969d9361451b | In △ABC, AB=AC, BC=8, AD⊥BC at D, then BD=． | 4 | math | |
c4ae9627-1994-4d7e-96bc-2961b66cbb82 | As shown in the figure, perform the calculation according to the program. If the input number is $-2$, then the output number is. | -50 | math | |
b5fecd90-63b7-4655-ae62-71554d95601a | “●” “□” “△” represent three different objects. As shown in the figure, scales 1 and 2 are balanced. If we want scale 3 to also be balanced, how many “□” should be placed on the right side of scale 3? | 5 | math | |
079f0825-2863-4daa-93a0-a1d4c471f3d3 | As shown in the figure, $$\angle E= \angle F=90^{ \circ }$$, $$\angle B= \angle C$$, $$AE=AF$$. Among the conclusions: 1. $$EM =FN$$; 2. $$CD=DN$$; 3. $$\angle EAN= \angle EAM$$; 4. $$\triangle ACN\cong \triangle ABM$$, the number of correct conclusions is ___ . | $$3$$ | math | |
1c9ddac0-c159-49b8-81e4-3f140138abaf | According to the pseudocode shown in the figure, the value of the output $$S$$ is ___. | $$13$$ | math | |
6c2e567f-9b56-4291-aab8-5ed9bbdb538a | As shown in the figure, based on the following shapes and the number of vertices in each corresponding shape, find the pattern, then the number of vertices in the $$n$$th shape is ___. | $$n^{2}+5n+6$$ | math | |
02c8d33f-47e3-45a2-8b57-15cc253ed0f4 | Someone found that watching too much TV can make people more indifferent. The table below shows the results of a survey conducted by an organization on this phenomenon: Therefore, there is ___ confidence that watching too much TV is related to people becoming indifferent. | $$99.9\%$$ | math | |
88174f5d-3e3b-4977-826c-db6c62a69fc8 | Use the bisection method to find a zero of the function $$f ( x ) = 3^{x} - x - 4$$. The reference data is as follows: According to this data, an approximate solution (with an error not exceeding $$0.005$$) to the equation $$3^{x} - x - 4 = 0$$ is ___. | $$\number{1.55935}$$ | math | |
17139316-db56-419b-94ab-b0346280c1b7 | As shown in the figure, $$DE \parallel AB$$, $$BC \perp CD$$ at $$C$$. If $$\angle D=42^{ \circ }$$, then $$\angle B=$$ ___. | $$48^{ \circ }$$ | math | |
186dca38-8b5a-4451-8729-20ac440e5827 | As shown in the diagram, it is a schematic of a numerical conversion machine. If the input x is -3 and y is 2, then the output result is. | -1 | math | |
cf52f681-319b-4f87-b3cb-3be90194c2f4 | As shown in the figure, given that a, b, and c are the lengths of the three sides of the right triangle ABC, with ∠C = 90°, the linear function of the form y = $\frac{a}{c}x + \frac{b}{c}$ is called a 'Pythagorean linear function'. If point P (1, $\frac{3\sqrt{5}}{5}$) lies on the graph of the 'Pythagorean linear function', and the area of the right triangle ABC is 5, then the value of c is. | 5 | math | |
27efbc2d-4a34-4ffa-80de-192e4b85e411 | As shown in the figure, given AB⊥BD, ED⊥BD, AC⊥CE, points B, D, and C are the feet of the perpendiculars, point C is the midpoint of segment BD. If ED＝1, BD＝4, then AB＝. | 4 | math | |
2955430d-f741-4969-ab37-21674b3f5999 | The three views of a geometric solid represented by a stone are shown in the figure. If the stone is cut, polished, and processed into a sphere, then the maximum volume of the sphere that can be obtained is . | $\frac{32}{3}\text{ }\!\!\pi\!\!\text{ }$ | math | |
0028db5d-f2c1-4516-a839-0f69596f8f88 | As shown in the figure, the radius of the hemisphere O is R, and its inscribed rectangular prism $ABCD-{{A}_{1}}{{B}_{1}}{{C}_{1}}{{D}_{1}}$ has one of its faces ABCD on the base of the hemisphere O. The maximum value of the sum of all the edges of the rectangular prism is. | $12R$ | math | |
af80546e-66f5-4ab9-93e9-0faebf2733f3 | In the figure, rectangle ABCD has E as the midpoint of BC. When triangle ABE is folded along line AE, point B lands on point F. Connecting FC, if ∠DAF = 18°, then ∠DCF = ____ degrees. | 36 | math | |
94902b44-484a-4c97-8ece-18e0aed76fb7 | In the figure, △ABC has points D, E, and F as the midpoints of each side. If a grain of rice is randomly thrown inside △ABC, what is the probability that the grain of rice lands in the shaded area? | $\frac{1}{4}$ | math | |
5e1c2b87-e211-4680-9966-f7a8f3c8373b | As shown in the figure, AB is the diameter of circle O, $\overset\frown{BC}=\overset\frown{CD}=\overset\frown{DE}$, ∠COD=48°, then the measure of ∠AOE is. | 36° | math | |
561bcfd5-6ac2-4650-bdd8-50592d1d0e64 | Execute the program flowchart as shown. If the input $t \in \left[ -1,3 \right]$, then the range of the output $s$ is. | 0,1 | math | |
cb3ed696-ceda-4d4f-b20a-0f04e719f1d2 | In △ABC, F is the intersection point of the altitudes AD and BE, and AD=BD, AC=8cm. Find the length of BF. | 8cm | math |
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