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<image>Given the quadratic function $$y=ax^2+bx+c$$, the graph is shown below, and $$OA=OC$$. Write the equations or inequalities involving $$a$$, $$b$$, and $$c$$ based on the characteristics of the parabola: 1. $$\frac{4ac-b^2}{4a}=-1$$ 2. $$ac+b+1=0$$ 3. $$abc>0$$ 4. $$a-b+c>0$$. The number of correct ones is ( ).
A. $$4$$
B. $$3$$
C. $$2$$
D. $$1$$
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A
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<image>In the right triangle ABC, ∠ACB=90°, CD is the altitude, ∠A=30°, CD=$2\sqrt{3}$cm. What is the length of AB?
A. 4cm
B. 6cm
C. 8cm
D. 10cm
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C
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<image>As shown in the figure, $$⊙O$$ is the circumcircle of $$\triangle ABC$$, $$∠B=60^{\circ}$$, and the radius of $$⊙O$$ is $$4$$. Then the length of $$AC$$ is ( )
A. $$4\sqrt{3}$$
B. $$6\sqrt{3}$$
C. $$2\sqrt{3}$$
D. $$8$$
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A
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<image>A store records a total of $$N$$ data points $$a_1$$, $$a_2$$, $$\cdots$$, $$a_N$$ for its income and expenses in a month, where income is recorded as positive numbers and expenses as negative numbers. The store uses the flowchart shown to calculate the total monthly income $$S$$ and the net profit $$V$$. Therefore, what should be filled in the blank decision box and processing box in the figure are ( )
A. $$A>0$$, $$V=S-T$$
B. $$A < 0$$, $$V=S-T$$
C. $$A>0$$, $$V=S+T$$
D. $$A < 0$$, $$V=S+T$$
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C
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<image>The figure shows a part of the graph of the quadratic function y=ax$^{2}$+bx+c, which passes through point A (3, 0), and the axis of symmetry of the quadratic function graph is x=1. The following conclusions are given: 1. b$^{2}$ > 4ac; 2. ac > 0; 3. When x > 1, y decreases as x increases; 4. 3a + c > 0; 5. For any real number m, a + b ≥ am$^{2}$ + bm. Which of the conclusions are correct?
A. 1, 2, 3.
B. 1, 4, 5.
C. 3, 4, 5.
D. 1, 3, 5.
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D
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<image>As shown in the figure, the number of rectangles (including squares) that do not contain the shaded area is ( )
A. $15$
B. $ 24$
C. $ 25$
D. $ 26$
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C
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<image>As shown in the subtraction table below, the equation in the red circle is ( ).
A. $$11-5$$
B. $$12-5$$
C. $$11-4$$
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A
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<image>As shown in the figure, the graph of the function $f(x)$ consists of two line segments $AB$ and $BC$, where the coordinates of points $A$, $B$, and $C$ are $(0,1)$, $(2,2)$, and $(3,0)$, respectively. The value of $f(f(f(3)))$ is ( )
A. $0$
B. $1$
C. $2$
D. $\frac{3}{2}$
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D
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<image>In $$\triangle ABC$$, points $$D$$ and $$E$$ are on $$AB$$ and $$AC$$ respectively. If $$AD=2$$ and $$BD=3$$, which of the following conditions can determine that $$DE//BC$$?
A. $$\frac{DE}{BC}=\frac{2}{3}$$
B. $$\frac{DE}{BC}=\frac{2}{5}$$
C. $$\frac{AE}{AC}=\frac{2}{3}$$
D. $$\frac{AE}{AC}=\frac{2}{5}$$
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D
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<image>The 'Nine Chapters on the Mathematical Art' is an extremely rich ancient Chinese mathematical classic. It contains the following problem: 'Now there is rice piled up against the inner corner of a wall, with a base circumference of 10 feet and a height of 6 feet. Question: What is the volume of the pile and how much rice is there?' It means: 'Rice is piled up in the corner of a room (as shown in the figure, the rice pile is a quarter of a cone), the arc length at the base of the pile is 10 feet, and the height of the pile is 6 feet. What is the volume of the pile and how much rice is there?' It is known that 1 hu of rice has a volume of approximately 1.62 cubic feet, and the value of pi is approximately 3. Estimate the amount of rice to be ( ).
A. 17 hu
B. 25 hu
C. 41 hu
D. 58 hu
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C
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<image>The graphs of the functions $y=6-x$ and $y=\dfrac{4}{x}\left(x > 0\right)$ intersect at points $A$ and $B$. Let the coordinates of point $A$ be $(x_{1}, y_{1})$. The area and perimeter of a rectangle with sides of lengths $x_{1}$ and $y_{1}$ are respectively ( )
A. $4$, $12$
B. $4$, $6$
C. $8$, $12$
D. $8$, $6$
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A
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<image>Carefully observe the table below, the two corresponding quantities ( )
A. are directly proportional
B. are inversely proportional
C. are not proportional
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B
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<image>As shown in the figure, in $\vartriangle \text{ABC}$, points $\text{D}$ and $\text{E}$ are on $\text{AB}$ and $\text{AC}$, respectively. There are four conditions: $(1)\frac{\text{AD}}{\text{AB}}=\frac{\text{AE}}{\text{AC}}$; $(2)\frac{\text{DB}}{\text{AB}}=\frac{\text{EC}}{\text{AC}}$; $(3)\frac{\text{AD}}{\text{DB}}=\frac{\text{AE}}{\text{EC}}$; $(4)\frac{\text{AD}}{\text{DB}}=\frac{\text{DE}}{\text{BC}}$. Among these, the conditions that can definitely determine $\text{DE} \parallel \text{BC}$ are ( )
A. $1$ condition
B. $2$ conditions
C. $3$ conditions
D. $4$ conditions
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C
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<image>Given the graph of the function $y=A\sin (\omega x+\varphi )(A > 0,\omega > 0,|\varphi |\le \pi )$ as shown in the figure, then ( )
A. $\omega =2$, $\varphi =\pi $
B. $\omega =2$, $\varphi =\frac{\pi }{2}$
C. $\omega =\frac{1}{2}$, $\varphi =\frac{\pi }{4}$
D. $\omega =\frac{1}{2}$, $\varphi =-\frac{3\pi }{4}$
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D
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<image>As shown in the figure, in the spatial quadrilateral $$ABCD$$, $$M$$ and $$G$$ are the midpoints of $$BC$$ and $$CD$$, respectively. Then $$\overrightarrow {AB}+\frac12\overrightarrow {BC}+\frac12\overrightarrow {BD}$$ equals ( ).
A. $$\overrightarrow {AD}$$
B. $$\overrightarrow {GA}$$
C. $$\overrightarrow {AG}$$
D. $$\overrightarrow {MG}$$
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C
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<image>As shown in the figure, which of the following conditions cannot be used to prove that $\triangle BOE$ ≌ $\triangle COD$?
A. $AB=AC$, $BE=CD$
B. $AB=AC$, $OB=OC$
C. $BE=CD$, $BD=CE$
D. $BE=CD$, $OB=OC$
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D
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<image>As shown in the figure, point $E$ is on the side $BC$ of square $ABCD$. When $\triangle ABE$ is folded along the line $AE$, point $B$ lands at point $P$ inside the square. Extend $EP$ to intersect $CD$ at point $F$, and connect $AF$. If point $E$ moves along $BC$, which of the following conclusions is correct?
A. The perimeter of $\triangle AEF$ remains constant
B. The area of $\triangle AEF$ remains constant
C. The perimeter of $\triangle CEF$ remains constant
D. The area of $\triangle CEF$ remains constant
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C
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<image>The graph of the function $y=f(x)$ $(x \in \text{R})$ is shown in the figure. Then, the interval of monotonic decrease for the function $g(x)=f(-\ln x)$ is ( )
A. $\left( 0,\frac{1}{\sqrt{e}} \right]$
B. $\left[ \frac{1}{\sqrt{e}},1 \right]$
C. $[1,+\infty)$
D. $\left( 0,\frac{1}{\sqrt{e}} \right]$ and $[1,+\infty)$
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B
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<image>The 'Nine Chapters on the Mathematical Art' is the first systematic mathematical treatise in ancient China, representing the pinnacle of Eastern mathematics. Its algorithm system continues to drive the development and application of computers to this day. The book records: 'There is a cylindrical piece of wood buried in a wall, the size of which is unknown. When sawed, the saw cut is 1 inch deep (ED=1 inch), and the length of the saw cut is 1 foot (AB=1 foot=10 inches). What is the diameter of the cylindrical piece of wood?' As shown in the figure, please calculate based on your knowledge: The diameter AC of the cylindrical piece of wood is ( ).
A. 13 inches
B. 20 inches
C. 26 inches
D. 28 inches
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C
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<image>According to the program shown in the figure, if the initial value of $$n$$ is $$1$$, then the final output result is ( ).
A. $$7$$
B. $$10$$
C. $$77$$
D. $$1541$$
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B
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<image>In the mid-3rd century, during the Wei and Jin periods, the mathematician Liu Hui pioneered the 'circle-cutting method,' which involves inscribing regular polygons in a circle to approximate the value of $$\pi$$. Liu Hui described his 'circle-cutting method' as follows: 'The finer the cuts, the less the loss; cut again and again until it cannot be cut further, then the polygon will coincide with the circle, and there will be no loss.' When the number of sides of the inscribed regular polygon in a circle is increased infinitely, the area of the polygon can approach the area of the circle infinitely. Using the 'circle-cutting method,' Liu Hui obtained an approximate value of $$\pi$$ accurate to two decimal places, which is $$\number{3.14}$$. This is the famous 'Hui's value'. The figure below shows a flowchart designed based on the idea of Liu Hui's 'circle-cutting method.' What is the output value of $$n$$? (Reference data: $$\sin 15^{ \circ }=\number{0.259}$$, $$\sin 7.5^{ \circ }=\number{0.131}$$)
A. $$6$$
B. $$12$$
C. $$24$$
D. $$48$$
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C
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<image>In the figure, in $$\triangle ABC$$, the angle bisectors of $$\angle ABC$$ and $$\angle ACB$$ intersect at point $$F$$. A line through $$F$$ parallel to $$BC$$ intersects $$AB$$ and $$AC$$ at points $$D$$ and $$E$$, respectively. If $$AB=5$$ and $$AC=4$$, then the perimeter of $$\triangle ADE$$ is ( ).
A. $$9$$
B. $$10$$
C. $$12$$
D. $$14$$
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A
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<image>As shown in the figure, in parallelogram $ABCD$, the two diagonals $AC$ and $BD$ intersect at point $O$, $AF \perp BD$ at $F$, and $CE \perp BD$ at $E$. The number of pairs of congruent triangles in the figure is ( )
A. $5$ pairs
B. $6$ pairs
C. $7$ pairs
D. $8$ pairs
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C
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<image>In the figure, in $\triangle ABC$, $AD \perp BC$ at point $D$, and $BD = DC$. Point $E$ is on the extension of $BC$, and point $C$ lies on the perpendicular bisector of $AE$. The following conclusions are given: 1. $AB = AC = CE$; 2. $AB + BD = DE$; 3. $AD = \dfrac{1}{2}AE$; 4. $BD = DC = CE$. Which of the following conclusions are correct?
A. Only 1.
B. Only 1. and 2.
C. Only 1., 2., and 3.
D. Only 1. and 4.
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B
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<image>The graph of the function $f(x)=A\cos (\omega x+\varphi )\left( A > 0,\omega > 0,|\varphi | < \frac{\pi }{2} \right)$ is shown below. The monotonic increasing interval of $f(x)$ is ( )
A. $(16k\pi -6,16k\pi +2)(k\in \mathbb{Z})$
B. $(8k\pi -6,8k\pi +2)(k\in \mathbb{Z})$
C. $(8k-6,8k+2)(k\in \mathbb{Z})$
D. $(16k-6,16k+2)(k\in \mathbb{Z})$
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D
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<image>As shown in the figure, a 13 cm long chopstick is placed in a cylindrical cup with a base diameter of 6 cm and a height of 8 cm. The length of the chopstick protruding from the cup is at least ( ) cm.
A. 1
B. 2
C. 3
D. 4
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C
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<image>As shown in Figure 1, a square piece of paper with side length $$a$$ is cut into two small rectangles, resulting in a pattern $$S$$ as shown in Figure 2. Then, the two small rectangles are combined to form a new rectangle, as shown in Figure 3. The perimeter of the new rectangle can be expressed as ( ).
A. $$2a-3b$$
B. $$4a-8b$$
C. $$2a-4b$$
D. $$4a-10b$$
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B
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<image>(1) When spinning the right wheel, the pointer is most likely to stop in the ( ) area.
A. Red
B. Yellow
C. Green
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A
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<image>The line graph below shows the daily minimum temperatures for a certain period in a certain place. The median, mode, and mean of the minimum temperatures during this period are ( )
A. 4℃, 5℃, 4℃
B. 5℃, 5℃, 4.5℃
C. 4.5℃, 5℃, 4℃
D. 4.5℃, 5℃, 4.5℃
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C
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<image>As shown in the figure, there is a building $$OP$$. To measure its height, a baseline $$AB$$ of length $$\quantity{40}{m}$$ is chosen on the ground. If the angle of elevation to point $$P$$ from point $$A$$ is $$30^{\circ}$$, and from point $$B$$ is $$45^{\circ}$$, and $$\angle AOB=30^{\circ}$$, then the height of the building is ( )
A. $$\quantity{20}{m}$$
B. $$20\sqrt{2}\ \unit{m}$$
C. $$20\sqrt{3}\ \unit{m}$$
D. $$40\ \unit{m}$$
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D
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<image>As shown in the figure, in $$\triangle ABC$$, the perpendicular bisector of $$AB$$ intersects $$AB$$ at point $$D$$ and $$BC$$ at point $$E$$. If $$BC=6$$ and $$AC=5$$, then the perimeter of $$\triangle ACE$$ is ()
A. $$8$$
B. $$11$$
C. $$16$$
D. $$17$$
|
B
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<image>In the figure, in $$\triangle ABC$$, $$CD$$ is the altitude on side $$AB$$. If $$AB=1.5$$, $$BC=0.9$$, and $$AC=1.2$$, then the value of $$CD$$ is ( ).
A. $$0.72$$
B. $$2.0$$
C. $$1.125$$
D. Cannot be determined
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A
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<image>As shown in the figure, a rectangular paper sheet $ABCD$ is folded so that point $B$ coincides with point $D$, with the fold line being $EF$. Given that $AB=6cm$ and $BC=18cm$, the area of $Rt\triangle CDF$ is ( )
A. $27cm^{2}$
B. $ 24cm^{2}$
C. $ 22cm^{2}$
D. $ 20cm^{2}$
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B
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<image>The three views of a geometric solid are shown in the figure. The front view and the left view are both composed of a triangle and a semicircle, while the top view is composed of a circle and an inscribed triangle. What is the volume of the solid?
A. $$\frac{\sqrt{2}\pi}{3}+\frac{1}{6}$$
B. $$\frac{\sqrt{2}\pi}{6}+\frac{1}{2}$$
C. $$\frac{\sqrt{2}\pi}{6}+\frac{1}{6}$$
D. $$\frac{\sqrt{2}\pi}{3}+\frac{1}{2}$$
|
C
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<image>The graph of the quadratic function $$y=ax^2+bx+c(a≠0)$$ is shown in the figure. Which of the following statements are true: 1. $$b^2-4ac=0$$; 2. $$2a+b=0$$; 3. If $$(x_1,y_1)$$ and $$(x_2,y_2)$$ are points on the graph, when $$x_1 < x_2$$, then $$y_1 < y_2$$; 4. $$a-b+c<0$$. The correct choices are ( )
A. 2, 4.
B. 3, 4.
C. 2, 3, 4.
D. 1, 2, 4.
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A
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<image>The number of tourists received daily by two attractions, Attraction A and Attraction B, in our city during the first ten days of May this year is shown in the figure below. The relationship between the variances of the daily number of tourists received by Attraction A and Attraction B is ( ).
A. $$S_\text{A}^2 > S_\text{B}^2$$
B. $$S_\text{A}^2 < S_\text{B}^2$$
C. $$S_\text{A}^2 = S_\text{B}^2$$
D. Cannot be determined
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A
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<image>There are four cards, each with two sides, one side written with a number, the other side written with an English letter. It is stipulated that if one side of the card is the letter $$P$$, then the other side must be the number $$2$$. As shown in the figure, the visible sides of the four cards are $$P$$, $$Q$$, $$2$$, $$3$$. To check if any of these four cards violate the rule, which cards must be flipped? ( )
A. The first and third cards
B. The first and fourth cards
C. The second and fourth cards
D. The second and third cards
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B
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<image>If the graph of the quadratic function $y=ax^2+bx+a^2-2$ (where $a$ and $b$ are constants) is as shown in the figure, then the value of $a$ is ( ).
A. 1
B. $\sqrt{2}$
C. $-\sqrt{2}$
D. -2
|
C
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<image>In the figure, in $\vartriangle ABC$, $DE \parallel BC$, and $\frac{\text{AD}}{\text{AB}} = \frac{2}{3}$. What is the value of $\frac{S_{\vartriangle ADE}}{S_{quadrilateral DBCE}}$?
A. $\frac{4}{5}$
B. 1
C. $\frac{2}{3}$
D. $\frac{4}{9}$
|
A
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<image>Xiao Ming divides the wheel shown in the figure into $$n$$ (where $$n$$ is a positive integer) sectors, making the area of each sector equal. He then labels each sector with consecutive even numbers $$2$$, $$4$$, $$6$$, ..., $$2n$$ (each sector is labeled with one number, and the numbers in different sectors are distinct). The wheel is spun once, and when it stops, the probability that the number in the sector where the pointer lands is greater than $$8$$ is $$\frac{5}{6}$$. What is the value of $$n$$?
A. $$36$$
B. $$30$$
C. $$24$$
D. $$18$$
|
C
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<image>As shown in the figure, in the rectangle $$OABC$$, points $$A$$ and $$C$$ lie on the $$x$$-axis and $$y$$-axis, respectively, and point $$B$$ lies on the graph of the inverse proportion $$y=\dfrac{k}{x}$$ in the second quadrant. The area of the rectangle is $$6$$. What is the value of $$k$$?
A. $$3$$
B. $$6$$
C. $$-6$$
D. $$-3$$
|
C
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<image>Let the universal set $U$ be the set of natural numbers $N$, set $A=\left\{x\left|\right.x^{2} > 4, x\in N\right\}$, and $B=\left\{0,2,3\right\}$. The shaded part in the diagram represents the set ( )
A. $\left\{x\left|\right.x > 2, x\in N\right\}$
B. $ \left\{x\left|\right.x\leqslant 2, x\in N\right\}$
C. $ \left\{0,2\right\}$
D. $ \left\{1,2\right\}$
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C
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<image>After running the program below, the output value is ( )
A. 90
B. 29
C. 13
D. 54
|
D
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<image>As shown in the figure, in parallelogram ABCD, AB=6, AD=9, the angle bisector of ∠BAD intersects BC at point E and the extension of DC at point F. BG is perpendicular to AE at G, and BG= 4 \sqrt{2}. Then the perimeter of trapezoid AECD is ( )
A. 22
B. 23
C. 24
D. 25
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A
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<image>As shown in the figure, it is a frequency distribution histogram of a sample. Based on the data in the graph, the mode and median can be estimated as ( )
A. 12.5; 12.5
B. 13; 13
C. 13; 12.5
D. 12.5; 13
|
D
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<image>Given the spatial quadrilateral $$OABC$$, its diagonals are $$OB$$ and $$AC$$, and $$M$$ and $$N$$ are the midpoints of sides $$OA$$ and $$CB$$, respectively. Point $$G$$ lies on segment $$MN$$ such that $$MG=2GN$$. Express the vector $$\overrightarrow{OG}$$ in terms of vectors $$\overrightarrow{OA}$$, $$\overrightarrow{OB}$$, and $$\overrightarrow{OC}$$ ( ).
A. $$\overrightarrow{OG}=\frac{1}{6}\overrightarrow{OA}+\frac{1}{3}\overrightarrow{OB}+\frac{1}{3}\overrightarrow{OC}$$
B. $$\overrightarrow{OG}=\frac{1}{6}\overrightarrow{OA}+\frac{1}{3}\overrightarrow{OB}+\frac{2}{3}\overrightarrow{OC}$$
C. $$\overrightarrow{OG}=\overrightarrow{OA}+\frac{2}{3}\overrightarrow{OB}+\frac{2}{3}\overrightarrow{OC}$$
D. $$\overrightarrow{OG}=\frac{1}{2}\overrightarrow{OA}+\frac{2}{3}\overrightarrow{OB}+\frac{2}{3}\overrightarrow{OC}$$
|
A
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<image>The following is part of the graph of the quadratic function $$f(x) = x^2 - bx + a$$. What is the minimum value of the slope of the tangent line to the function $$g(x) = 2\ln x + f(x)$$ at the point $$(b, g(b))$$?
A. 1
B. $$\sqrt{3}$$
C. 2
D. $$2\sqrt{2}$$
|
D
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<image>As shown in the figure, the line $$y=-\dfrac{1}{2}x+2$$ intersects the $$x$$-axis at point $$B$$ and the $$y$$-axis at point $$A$$. Through the midpoint $$A_{1}$$ of segment $$AB$$, a perpendicular $$A_{1}B_{1} \bot x$$-axis is drawn at point $$B_{1}$$. Through the midpoint $$A_{2}$$ of segment $$A_{1}B$$, a perpendicular $$A_{2}B_{2} \bot x$$-axis is drawn at point $$B_{2}$$. Through the midpoint $$A_{3}$$ of segment $$A_{2}B$$, a perpendicular $$A_{3}B_{3} \bot x$$-axis is drawn at point $$B_{3}$$, and so on. What is the area of triangle $$A_{n}B_{n}B_{n-1}$$?
A. $$\dfrac{1}{2^{n-1}}$$
B. $$\dfrac{1}{2^{n}}$$
C. $$\dfrac{1}{4^{n-1}}$$
D. $$\dfrac{1}{4^{n}}$$
|
C
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<image>A set of triangular rulers is placed as shown in the figure, and the degree of $$\angle 1$$ is $$50^{\circ}$$ greater than that of $$\angle 2$$. If we set $$\angle 1=x^{\circ}$$ and $$\angle 2=y^{\circ}$$, then the system of equations can be written as ( ).
A. $$\begin{cases} x=y-50 \\ x+y=180 \end{cases}$$
B. $$\begin{cases} x=y+50 \\ x+y=180 \end{cases}$$
C. $$\begin{cases} x=y-50 \\ x+y=90 \end{cases}$$
D. $$\begin{cases} x=y+50 \\ x+y=90 \end{cases}$$
|
D
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<image>The positions of rational numbers a and b on the number line are shown in the figure. Which of the following statements is correct?
A. a + b = 0
B. b < a
C. ab > 0
D. |b| < |a|
|
D
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<image>Execute the program flowchart shown in the figure. If the output result is $$\frac{31}{32}$$, then the input $$a$$ is ( ).
A. $$3$$
B. $$4$$
C. $$5$$
D. $$6$$
|
C
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<image>Cai Lin surveyed the heights of 6 classmates in their group, as shown in the table below: Which of the following statements is incorrect?
A. Classmate No.4 is the tallest
B. Classmate No.1 is the shortest
C. Classmate No.3 is 4 cm shorter than Classmate No.4
D. Classmate No.4 definitely eats more than the other classmates
|
D
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<image>From the four cards printed with car brand logos shown in the figure, the probability of randomly picking a card with a logo that is a centrally symmetric figure is ( )
A. $\dfrac{1}{4}$
B. $\dfrac{1}{2}$
C. $\dfrac{3}{4}$
D. $1$
|
A
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<image>As shown in the figure, the mathematical formula you can derive from the area relationship is ( )
A. $a^{2}-b^{2}=(a+b)(a-b)$
B. $(a+b)^{2}=a^{2}+2ab+b^{2}$
C. $(a-b)^{2}=a^{2}-2ab+b^{2}$
D. $a(a+b)=a^{2}+ab$
|
C
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<image>As shown in the figure, in the regular quadrilateral prism $ABCD-{A}_{1}{B}_{1}{C}_{1}{D}_{1}$, $E$ and $F$ are the midpoints of ${AB}_{1}$ and ${BC}_{1}$, respectively. Which of the following statements is incorrect? ( )
A. $EF$ is perpendicular to ${BB}_{1}$
B. $EF$ is perpendicular to $BD$
C. $EF$ and $CD$ are skew lines
D. $EF$ and ${A}_{1}{C}_{1}$ are skew lines
|
D
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<image>As shown in the figure, $\angle ACB=90^\circ$, $\angle BDC=\angle ADC=90^\circ$. Which of the following statements is incorrect?
A. $\angle ACD$ is equal to $\angle B$
B. $\angle BCD$ and $\angle B$ are complementary
C. $\angle B$ and $\angle BCD$ are complementary
D. $\angle A$ and $\angle BCD$ are complementary
|
D
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<image>In triangle ABC, ∠ABC = 90°, ∠C = 52°, BE is the median of side AC, AD bisects ∠BAC and intersects side BC at point D, and BF is perpendicular to AD, with the foot of the perpendicular at F. What is the measure of ∠EBF?
A. 19°
B. 33°
C. 34°
D. 43°
|
B
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<image>As shown in the figure, during the Three Kingdoms period, the mathematician Zhao Shuang provided an ingenious proof of the Pythagorean theorem in the 'Zhou Bi Suan Jing' using a string diagram. The figure contains four congruent right-angled triangles and a small square (shaded). Suppose one of the internal angles of the right-angled triangle is 30°, if 1000 grains of rice (neglecting their size) are randomly thrown into the string diagram, then the number of grains that fall into the small square (shaded) is approximately ( )
A. 134
B. 866
C. 300
D. 500
|
A
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<image>As shown in the figure, in the right triangle $\triangle ABC$, $\angle C=90^{\circ}$, $AD$ is the angle bisector, and $DE \bot AB$ at $E$. Which of the following conclusions is incorrect?
A. $BD + DE = BC$
B. $DE$ bisects $\angle ADB$
C. $AD$ bisects $\angle EDC$
D. $AC + DE > AD$
|
B
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<image>In the Cartesian coordinate system $$xoy$$, the rough graph of the quadratic function $$y=ax^2+bx+c$$$$(a\not =0)$$ is shown in the figure below, then which of the following conclusions is correct?
A. $$a<0, b<0, c>0$$
B. $$-{b\over 2a}=1$$
C. $$a+b+c<0$$
D. The equation $$ax^2+bx+c=-1$$ with respect to $$x$$ has two distinct real roots
|
D
|
|
<image>To understand the situation of employees' 'love donations' to disaster areas, a company randomly selected the donation amounts of some employees and organized them into the histogram shown in the figure. Based on the information in the figure, which of the following conclusions is correct?
A. The median of the sample is 200 yuan
B. The sample size is 20
C. The range of the donation amounts by the company's employees is 450 yuan
D. The maximum donation amount by the company's employees is 500 yuan
|
B
|
|
<image>As shown in the figure, in △ABC, ∠B = ∠C, BF = CD, BD = CE, ∠FDE = α. Which of the following conclusions is correct?
A. 2α + ∠A = 180°
B. α + ∠A = 90°
C. 2α + ∠A = 90°
D. α + ∠A = 180°
|
A
|
|
<image>As shown in the figure, a highway is being constructed, and a tunnel needs to be built from point B to point C (B and C are on the same horizontal plane). To measure the distance between B and C, an engineer took a hot air balloon from point C and ascended vertically to point A, 200 meters above C. At point A, the angle of depression to point B is α. The distance between B and C is ()
A. $200\text{sin }\alpha$ meters
B. $200\text{tan }\alpha$ meters
C. $\frac{200}{\text{sin }\alpha}$ meters
D. $\frac{200}{\text{tan }\alpha}$ meters
|
D
|
|
<image>As shown in the figure, point $$O$$ is the origin of the number line. The points on the number line that represent opposite numbers are ( )
A. Point $$A$$ and Point $$C$$
B. Point $$C$$ and Point $$D$$
C. Point $$A$$ and Point $$D$$
D. Point $$B$$ and Point $$E$$
|
B
|
|
<image>As shown in the figure, $$AB \parallel CD$$, $$AE$$ bisects $$∠CAB$$, and intersects $$CD$$ at point $$D$$. If $$∠C=100^{\circ}$$, then the measure of $$∠CDA$$ is ( )
A. $$110^{\circ}$$
B. $$55^{\circ}$$
C. $$40^{\circ}$$
D. $$35^{\circ}$$
|
C
|
|
<image>As shown in the figure, in the equilateral triangle ABC, BD = CE, and AD intersects BE at point P. The measure of ∠APE is ( )
A. 45°
B. 60°
C. 55°
D. 75°
|
B
|
|
<image>Given that $y=f(x)$ is an increasing function on $R$, and some of its corresponding values are shown in the table, then the range of $x$ that satisfies $|f(x)| < 3$ is ( )
A. $(0,2)$
B. $(-1,2)$
C. $(-\infty, -1)$
D. $(2, +\infty)$
|
B
|
|
<image>When $$m=7$$ and $$n=3$$, execute the program flowchart as shown in the figure, and the output value of $$S$$ is ( ).
A. $$7$$
B. $$42$$
C. $$210$$
D. $$840$$
|
C
|
|
<image>As shown in the figure, the paper of a regular book is obtained by multiple halvings of the original paper. Rectangle $$ABCD$$ is halved along $$EF$$, and then rectangle $$EFCD$$ is halved along $$MN$$, and so on. If all the resulting rectangles are similar, then $$\dfrac{AD}{AB}$$ equals ( )
A. $$0.618$$
B. $$\dfrac{\sqrt{2}}{2}$$
C. $$\sqrt{2}$$
D. $$2$$
|
C
|
|
<image>The flowchart of a program is shown in the figure. The output values of (x, y) are sequentially recorded as: (x_1, y_1), (x_2, y_2), ... (x_n, y_n), ... If one of the output arrays during the program's execution is (x, -10), then the value of x in the array is ( ).
A. 64
B. 32
C. 16
D. 8
|
B
|
|
<image>As shown in the figure, in the regular tetrahedron $P-ABC$, $D$, $E$, and $F$ are the midpoints of $AB$, $BC$, and $CA$ respectively. Which of the following statements is false?
A. $BC$ ∥ plane $PDF$
B. $DF \bot$ plane $PAE$
C. Plane $PDF \bot$ plane $PAE$
D. Plane $PDE \bot$ plane $ABC$
|
D
|
|
<image>As shown in the figure, to measure the height of a tower $$AB$$ whose base cannot be reached, two students, Jia and Yi, measured from points $$C$$ and $$D$$, respectively. It is known that points $$B$$, $$C$$, and $$D$$ lie on the same straight line. The angles of elevation to the top of the tower $$A$$ from points $$C$$ and $$D$$ are $$60^{\circ}$$ and $$30^{\circ}$$, respectively, and $$AB \bot BD$$, $$CD = 12$$ meters. The height of the tower $$AB$$ is ( )
A. $$12\sqrt{3}$$ meters
B. $$6\sqrt{3}$$ meters
C. $$12$$ meters
D. $$6$$ meters
|
B
|
|
<image>As shown in the figure, to estimate the width of a river, Xiao Ming adopted the following method: he selected a point $$A$$ on the opposite bank of the river, and took points $$D$$ and $$B$$ on the near bank such that $$A$$, $$D$$, and $$B$$ are collinear and perpendicular to the riverbank. He measured $$BD=10\ \unit{m}$$, and then took point $$C$$ on a line perpendicular to $$AB$$, measuring $$BC=30 \ \unit{m}$$. If $$DE=20 \ \unit{m}$$, then the width of the river $$AD$$ is ( )
A. $$20\ \unit{m}$$
B. $$\dfrac{20}{3}\ \unit{m}$$
C. $$10\ \unit{m}$$
D. $$30\ \unit{m}$$
|
B
|
|
<image>Let $$f(x)$$ be a periodic function defined on $$\bf{R}$$ with a period of $$3$$. The graph of the function in the interval $$(-2,1]$$ is shown below, then $$f(2013)+f(2014)=$$（）.
A. 3
B. 2
C. 1
D. 0
|
C
|
|
<image>As shown in the figure, given $OA=OB$ and $OC=OD$, the number of pairs of congruent triangles in the figure is ( )
A. $1$ pair
B. $2$ pairs
C. $3$ pairs
D. $4$ pairs
|
C
|
|
<image>As shown in the figure, in △ABC, DE∥BC. If $\frac{AD}{AB}=\frac{1}{3}$, then $\frac{AE}{AC}=$ ( )
A. $\frac{1}{2}$
B. $\frac{1}{3}$
C. $\frac{2}{3}$
D. $\frac{1}{4}$
|
B
|
|
<image>As shown in the figure, which of the following conditions can determine that the lines $$l_1 \parallel l_2$$?
A. $$\angle 1 = \angle 2$$
B. $$\angle 1 = \angle 5$$
C. $$\angle 1 + \angle 3 = 180°$$
D. $$\angle 3 = \angle 5$$
|
C
|
|
<image>Two people, A and B, start from a village and travel along a road to a county town to handle affairs. A drives a car, and B rides a bicycle, both moving at constant speeds. A arrives at the county town first, spends half an hour handling affairs, and then returns along the same route at a different constant speed until meeting B. B's speed on the bicycle is $20km/h$. The graph shows the relationship between the distance $y\left(km\right)$ between the two and B's travel time $x\left(\min \right)$. Which of the following conclusions are correct? （1）The distance between the village and the county town is $15km$; （2）A's speed driving to the county town is $80km/h$; （3）The equation of line segment $AB$ is $y=-\dfrac{1}{3}x+20$; （4）A's speed driving back is $40km/h$.
A. $\left(2\right)\left(3\right)\left(4\right)$
B. $ \left(1\right)\left(3\right)\left(4\right)$
C. $ \left(1\right)\left(2\right)\left(4\right)$
D. $ \left(1\right)\left(2\right)\left(3\right)$
|
A
|
|
<image>As shown in the figure, an advertisement board, composed of a rectangle and a triangle, is tightly attached to a wall. The area of the overlapping part (shaded) is $$4\rm m^2$$, and the area occupied by the advertisement board is $$30\rm m^2$$ (ignoring thickness). The area of the remaining part of the rectangle, excluding the overlapping part, is $$2\rm m^2$$ more than the area of the remaining part of the triangle. Let the area of the rectangle be $$x\rm m^2$$ and the area of the triangle be $$y\rm m^2$$. According to the problem, the system of linear equations can be written as ( ).
A. $$\left\{\eqalign{&x+y-4=30\cr&(x-4)-(y-4)=2\cr }\right. $$
B. $$\left\{\eqalign{&x+y=26\cr&(x-4)-(y-4)=2\cr }\right. $$
C. $$\left\{\eqalign{&x+y-4=30\cr&(y-4)-(x-4)=2\cr }\right. $$
D. $$\left\{\eqalign{&x-y+4=30\cr&x-y=2\cr }\right. $$
|
A
|
|
<image>The perspective drawing of isosceles triangle $ABC$ is ( )
A. 1.2.
B. 2.3.
C. 2.4.
D. 3.4.
|
D
|
|
<image>As shown in the figure, in △ABC, DE∥BC, AB＝9, BD＝3, AE＝4, then the length of EC is ( )
A. 1
B. 2
C. 3
D. 4
|
B
|
|
<image>As shown in the figure, this circle is divided into 4 equal parts by two diameters, each 5 cm long. The perimeter of the shaded part is ( )
A. 1.57 cm
B. 7.71 cm
C. 8.925 cm
D. 17.85 cm
|
C
|
|
<image>As shown in the figure, in $$\triangle ABC$$, $$\overrightarrow{AD}=2\overrightarrow{DB}$$, $$\overrightarrow{BC}=2\overrightarrow{BE}$$, $$AE$$ intersects $$CD$$ at point $$F$$. A line $$QP$$ is drawn through point $$F$$, intersecting $$AB$$ and $$AC$$ at points $$Q$$ and $$P$$, respectively. If $$\overrightarrow{AQ}=\lambda \overrightarrow{AB}$$ and $$\overrightarrow{AP}=\mu \overrightarrow{AC}$$, then the minimum value of $$\lambda +\mu$$ is ( )
A. $$\dfrac{8}{5}$$
B. $$\dfrac{9}{5}$$
C. $$2$$
D. $$\dfrac{11}{5}$$
|
A
|
|
<image>A unit needs to purchase a batch of screws with a diameter of 10 mm. They randomly selected 20 screws from each of four factories (Factory A, Factory B, Factory C, and Factory D) to measure. The table below records the mean and variance of the diameters of these screws. Based on the data in the table, the factory to choose for purchasing is ( ).
A. A
B. B
C. C
D. D
|
C
|
|
<image>The frequency histograms of two sets of data, A and B, are shown below. The set with the larger variance is ( )
A. A
B. B
C. The same
D. Cannot be determined
|
A
|
|
<image>The program flowchart is shown in the figure. If the input $$x=-1$$, then the output $$y$$ value is ( )
A. $$-1$$
B. $$0$$
C. Uncertain
D. $$1$$
|
D
|
|
<image>As shown in the figure, it is known that the parabola is $$y_{1}=-2x^{2}+2$$, and the line is $$y_{2}=2x+2$$. When $$x$$ takes any value, the corresponding function values are $$y_{1}$$ and $$y_{2}$$ respectively. If $$y_{1} \neq y_{2}$$, take the smaller value of $$y_{1}$$ and $$y_{2}$$ and denote it as $$M$$; if $$y_{1}=y_{2}$$, denote $$M=y_{1}=y_{2}$$. For example: when $$x=1$$, $$y_{1}=0$$, $$y_{2}=4$$, $$y_{1} < y_{2}$$, at this time $$M=0$$. The following judgments: 1. When $$x > 0$$, $$y_{1} > y_{2}$$; 2. When $$x < 0$$, the larger the $$x$$ value, the smaller the $$M$$ value; 3. There does not exist an $$x$$ value that makes $$M$$ greater than $$2$$; 4. The $$x$$ value that makes $$M=1$$ is $$-\dfrac{1}{2}$$ or $$\dfrac{\sqrt{2}}{2}$$. The number of correct conclusions is ( )
A. $$1$$
B. $$2$$
C. $$3$$
D. $$4$$
|
B
|
|
<image>As shown in the figure, in rhombus ABCD, diagonals AC and BD intersect at O, and E is the midpoint of BC. Given AD = 6, the length of OE is ( )
A. 6
B. 4
C. 3
D. 2
|
C
|
|
<image>As shown in the figure, in $$\triangle ABC$$, $$∠B=90^{\circ}$$, $$BC=2AB$$, then $$cosA$$ equals ( )
A. $$\dfrac{\sqrt{5}}{2}$$
B. $$\dfrac{1}{2}$$
C. $$\dfrac{2\sqrt{5}}{5}$$
D. $$\dfrac{\sqrt{5}}{5}$$
|
D
|
|
<image>The flowchart of a certain program is shown in the figure. Given the following four functions, which function can be output?
A. $f=\ln (1-x)-\ln (1+x)$
B. $f=\frac{2^x+1}{2^x-1}$
C. $f=2^x+2^{-x}$
D. $f=x^2\ln (1+x^2)$
|
A
|
|
<image>The function corresponding to the image shown in the figure might be ( )
A. $$y=2^x$$
B. The inverse function of $$y=2^x$$
C. $$y=2^{-x}$$
D. The inverse function of $$y=2^{-x}$$
|
D
|
|
<image>A geometric figure composed of 6 identical small cubes is shown in the figure. Which of the following statements is correct?
A. The area of the left view is the largest
B. The area of the front view is the largest
C. The area of the top view is the largest
D. The areas of the three views are equal
|
B
|
|
<image>If the output value of S equals 16, then the content that should be filled in the decision box in the program flowchart is ( )
A. $i > 3$
B. $i > 4$
C. $i > 5$
D. $i > 6$
|
C
|
|
<image>As shown in the figure, the side length of each square grid on the grid paper is 1 unit length. The thick lines in the figure represent the three views of a certain geometric solid. The surface area of the solid is ( )
A. $$16$$
B. $$8\sqrt{5}$$
C. $$32$$
D. $$16\sqrt{5}$$
|
B
|
|
<image>As shown in the figure, in the Cartesian coordinate system, the coordinates of point $$A_1$$ are $$(-1,0)$$, and an isosceles right triangle $$\text{Rt}\triangle O{{A}_{1}}{{A}_{2}}$$ is constructed with $$O{{A}_{1}}$$ as one of the legs. Then, another isosceles right triangle $$\text{Rt}\triangle O{{A}_{2}}{{A}_{3}}$$ is constructed with $$O{{A}_{2}}$$ as one of the legs, followed by another isosceles right triangle $$\text{Rt}\triangle O{{A}_{3}}{{A}_{4}}$$ with $$O{{A}_{3}}$$ as one of the legs, and so on. Following this pattern, what is the x-coordinate of point $${{A}_{2020}}$$?
A. $$-{{2}^{1009}}$$
B. $${{2}^{1009}}$$
C. $$-{{2}^{1010}}$$
D. $${{2}^{1010}}$$
|
B
|
|
<image>As shown in the figure, the area of $$\triangle ABC$$ is $$12$$, and $$D$$, $$E$$, $$F$$, and $$G$$ are the midpoints of $$BC$$, $$AD$$, $$BE$$, and $$CE$$, respectively. Then the area of $$\triangle AFG$$ is ( )
A. $$4.5$$
B. $$5$$
C. $$5.5$$
D. $$6$$
|
A
|
|
<image>The three views of a geometric solid are shown in the figure. The front view and the side view are two congruent isosceles right triangles with a leg length of $$1$$. The surface area of the circumscribed sphere of the solid is ( )
A. $$12\pi $$
B. $$3\pi $$
C. $$4\sqrt{3}\pi $$
D. $$12\sqrt{2}\pi $$
|
B
|
|
<image>As shown in the figure, in rectangle $ABCD$, $AB=6cm$, $BC=12cm$. Point $P$ starts from $A$ and moves towards point $B$ along $AB$ at a speed of $1cm/s$, while point $Q$ starts from point $B$ and moves towards point $C$ along $BC$ at a speed of $2cm/s$ (both $P$ and $Q$ stop moving once they reach points $B$ and $C$, respectively). If the area of pentagon $CAPQCD$ at the $t$-th second of the motion is denoted as $S cm^{2}$, then the function relationship between $S$ and $t$ is ( )
A. $S=t^{2}-6t+72$
B. $ S=t^{2}+6t+72$
C. $ S=t^{2}-6t-72$
D. $ S=t^{2}+6t-72$
|
A
|
|
<image>As shown in the figure, point $$A$$ is any point on the graph of the inverse proportion function $$y=\frac{2}{x}(x>0)$$, $$AB\text{//}x$$-axis intersects the graph of the inverse proportion function $$y=\frac{-3}{x}$$ at point $$B$$, and a parallelogram $$ABCD$$ is constructed with $$AB$$ as one side, where $$C$$ and $$D$$ are on the $$x$$-axis. Then, $${{S}_{parallelogramABCD}}$$ is ( ).
A. $$2$$
B. $$3$$
C. $$4$$
D. $$5$$
|
D
|
|
<image>Given the data between $$x$$ and $$y$$ as shown in the table, the regression line passes through the point ( ).
A. $$\left ( 0,0\right ) $$
B. $$\left ( 2,1.8\right ) $$
C. $$\left ( 3,2.5\right ) $$
D. $$\left ( 4,3.2\right ) $$
|
C
|
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