id string | image_1 string | image_2 string | image_3 string | image_4 string | image_5 string | question string | options list | answer string | knowledge-source string | course string |
|---|---|---|---|---|---|---|---|---|---|---|
52011046_4 | null | null | null | null | null | Set\( M = \{ f(x), f'(x), f''(x), \cdots \} \), then the following can be expressions of\( f(x) \) are () | [
"\\(\\sin x\\)",
"\\(e^x\\)",
"\\(\\ln x\\)",
"\\(x^2 + 2x + 3\\)"
] | C | 集合与常用逻辑用语 | math |
52011046_17 | null | null | null | null | null | Given the sets \( A = \left\{ x \mid x = \frac{k}{2} + \frac{1}{4}, k \in \mathbb{Z} \right\} \), \( B = \left\{ y \mid y = \frac{l}{4} + \frac{1}{2}, l \in \mathbb{Z} \right\} \), then ( ) | [
"\\( A \\cap B = \\emptyset \\)",
"\\( A = B \\)",
"\\( A \\not\\subseteq B \\)",
"\\( B \\not\\subseteq A \\)"
] | C | 集合与常用逻辑用语 | math |
52011046_45 | iVBORw0KGgoAAAANSUhEUgAAAXcAAAFXCAYAAABZbA7IAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAEnQAABJ0Ad5mH3gAAGAKSURBVHhe7Z0HmFRVtrb194a54+gY0FGMIAaCCgZUlCAgSVCSoICAJEEFARUJioKogCiSlKAIiAgIkiQIChIEEyKiGEDEgNk7zui9c+em/d93Oatmd1EN3U3VqRPW9zz7aaiuqj5n77W/s/aKBzmDwWAwxA5G7gaDwRBDGLkbDAZDDGHkbjAYDDGEkbvBYDDEEEbuBoPBEEMY... | null | null | null | null | <image_1>If the Venn graph representing the relationship between sets\( M \) and\( N \) is as shown in the graph, then\( M, \, N \) may be () | [
"\\( M = \\{0, 2, 4, 6\\}, \\, N = \\{4\\} \\)",
"\\( M = \\{x \\mid x^2 < 1\\}, \\, N = \\{x \\mid x > - 1\\} \\)",
"\\( M = \\{x \\mid y = \\log x\\}, \\, N = \\left\\{y \\mid y = e^x + \\frac{1}{e^x}\\right\\} \\)",
"\\( M = \\{(x, y) \\mid x^2 = y^2\\}, \\, N = \\{(x, y) \\mid y = x\\} \\)"
] | BCD | 集合与常用逻辑用语 | math |
52011047-1_15 | null | null | null | null | null | Assuming that "Physics is good, mathematics is good" is a true proposition, then the following propositions are correct () | [
"Good physics may not necessarily be good mathematics",
"Good mathematics may not necessarily be good physics",
"Poor mathematics must be bad physics",
"Physics is bad, mathematics is bad"
] | BC | 集合与常用逻辑用语 | math |
50948433-1_16 | iVBORw0KGgoAAAANSUhEUgAAAWIAAAFkCAYAAAAaBTFnAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAEnQAABJ0Ad5mH3gAAETpSURBVHhe7Z0HuBTFtrbPNSdMqBhQBEVUUBG8BsziFSUoAornehRRMKJiDpgJKhzFHDCigooeVDyKAcVAMIFiRBSJegWPmPO5t/7nXf/U2Ay9957ae0L37O99nnpgd1f3THet/qZ61apVf3FCCCHKioRYCCHKjIRYCCHKjIRYCCHKjIRYCCHKjIRYCCHKjIRYCCHKjIRYCCHK... | iVBORw0KGgoAAAANSUhEUgAAAU8AAAF/CAYAAADJt0tfAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAEnQAABJ0Ad5mH3gAADlaSURBVHhe7Z0J3FVz/sfHkC1knaxRlLRhNFSK7KNSSSgZS8oSjSRlNEqWlJRMoaJlEJGtlEjWGq0mWpTRZgnJUpR95vf/v3/d33Ge2733uc+59z733HM/79fr96rnbM99zvmd9/2t39/vjBBCiDIjeQohRAAkTyGECIDkKYQQAZA8hRAiAJKnEEIEQPIUQogASJ5CCBEAyVMI... | iVBORw0KGgoAAAANSUhEUgAAAJoAAACwCAYAAAD3yHdHAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAEnQAABJ0Ad5mH3gAABM3SURBVHhe7Z0JtFXTH8eb1Ku0ECmFpHrpicxUKFSUNBhWJWSMUkloQin0TA2EWg2SWsjUYKoIaUCGhBIpQqJQIrP2///5uee5ve57vfveufudve/vs9ZZ9c65wxm+d+/9G/ZvlzCKYgEVmmIFFZpiBRWaYgUVmmIFFZpiBRWaYgUVmmIFFZpiBRWaYgUVmmIFFZpiBRWaYoV8... | iVBORw0KGgoAAAANSUhEUgAAAYMAAAGVCAYAAAAc652UAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAAEnQAABJ0Ad5mH3gAAFOcSURBVHhe7Z0JvFXj/v9/l26IJHSVKfkhIpkyJZHMRZlK13gRFeGWoZQxyS1zmRLqphtXpJLI0IBKKkOh6C9TVCQkXe7vPv/7/q7zrNbZ9j6dszrD3mt93q/X8zq199rrnP2s53k+z/Ad/scJIYRIPRIDIYQQEgMhhBASAyGEEP9FYiCEEEJiIIQQQmIghBDiv0gMhBBCSAyE... | null | Given\( a > 0 \) and\( a \neq e \), then the image of the function\( f(x) = e^x - a \ln x \) may be () | [
"<image_1>",
"<image_2>",
"<image_3>",
"<image_4>"
] | BCD | 函数与导数 | math |
51626976-1_23 | null | null | null | null | null | Let\( m \) be a real number other than 0, and the function\( f(x) = \begin{cases}
|3x - 1|, & x \leq 2 \\
x^2 - 10x + 24, & x > 2
\end{cases} \), if the function\( F(x) = 2[f(x)]^2 - m f(x) \) has 7 zeros, then the value range of\( m \) is () | [
"\\((-2,0) \\cup (0,16)\\)",
"\\((0,16)\\)",
"\\((0,2)\\)",
"\\((-2,0) \\cup (0, +\\infty)\\)"
] | C | 函数与导数 | math |
51484881-1_23 | null | null | null | null | null | If the function\( y = f(x) \) satisfies a set\( A \) within the domain, and for any\( x \in A \),\( e^x [f(x) - e^x] \) is a constant\( a \), then\( f(x) \) is said to have the property\( M \) on\( A \). Let\( y = g(x) \) be a function with the property of\( M \) on the interval\([-2, 2]\), and for any\( x_1, x_2 \in [... | null | \([-e^4, e^4]\) | 函数与导数 | math |
52105556-1_25 | null | null | null | null | null | In\(\triangle ABC\), the edges bounded by the angles\(A, B, C\) are\(a, b, c\) respectively, and satisfy\[\frac{\sqrt{3}b \sin A}{1 + \cos B} = a\]. Find the value range of\(\sin A \sin C +\sin B \sin C + \sin B \sin A\). | null | \[\left[ \frac{3}{4}, \frac{9}{4} \right] | 三角函数与解三角形 | math |
52105556-1_18 | iVBORw0KGgoAAAANSUhEUgAAAVwAAAD1CAIAAACTENrpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29eUCM6///3z3TNlNN+77SqiRFKrRbUyhUOnZZsuREenOsWRLHkkjkZAmpROFQaFHanfaiRKV935tptvv+/XF/f30IaZmapevxF7Nc12umuZ/367qu1wIhCMIFAAAA/z8YZhsAAABYCyAKAADgO4AoAACA7wCiAAAAvgOIAgAA+A4gCmwPOD8CMBZuZhsAGAiCIEQisb6+vqampr6+3traWlJSEoIgBEHq6+tjYmIM... | null | null | null | null | <image_1>Given the function\( f(x) = A \sin(\omega x + \varphi) (A > 0, \omega > 0, 0 < \varphi < \pi) \), if the partial image of\( f(x) \) and its derivative function\(f'(x) \) is as shown in the figure, then () | [
"\\( f(x + \\pi) = f(x) \\)",
"Function\\( f(x) \\) monotonically decreases on\\(\\left( \\frac{\\pi}{12}, \\frac{7\\pi}{12} \\right)\\)",
"The image of\\(f'(x) \\) is symmetric about the center of the point\\(\\left( -\\frac{\\pi}{6}, 0 \\right)\\)",
"The maximum value of\\( f(x) + f'(x) \\) is\\(\\frac{5}{2... | AB | 三角函数与解三角形 | math |
52105556-1_9 | iVBORw0KGgoAAAANSUhEUgAAASkAAACSCAIAAACWgavXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAf6ElEQVR4nO3dfTxU6f8/8Dkzg3F/N0wlCSMqRJtUbAkV+0FYpa3Ntps2EpUUm0ht5SbtJ1ZWN1t2P25S2BRlLe2ute26aRUpCU1EBuN+7mfO9f3jfH5+PdTupxvmjHE9/9rHOO15D/Oa65zrus51IQAAAgRBEkfEuwAImqJg9iAIHzB7EIQPmD0IwgfMHgThg4zXiVEUBQAgCEIkEgkEAgAAAICiKIlEQhAEO2BkZARFUQ0NDbyK... | iVBORw0KGgoAAAANSUhEUgAAASgAAACfCAIAAADF3dM3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAVEklEQVR4nO3df1AU9f8H8Hvv3R533MFxyGEC8kN+Gr8MJEgEw9TKEPFH41RMYzWWZVTYNGWNozWjzQeVJvuF/dTSavJ3pqZFwEQToKAGchAIJxwXiHDAsnfc7e3u94/9fm4YxT6lt/c+4fX4z9313i/Xe+57971770U8z0sAAO5F4C4AgMkIggcABhA8ADCA4AGAAQQPAAwgeABgAMEDAAMIHgAYQPAAwACCBwAGogeP/69r/jh2... | iVBORw0KGgoAAAANSUhEUgAAARQAAACZCAIAAABhS1KcAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29d3RbZZo/rqvebHXLtuQiuXcnthPSQwqBhBDKQjj0MGdgYGF3FmbZKbsDZ5ZZZnbZAzMLZ5aQ5cDMQMKBAAkhhOAU0h0cO3GJLblKcpGt3vu9vz+ec/Tz10W+uvdKcQZ9/kqurnQfS+/nffrzIhiG0TLIIIPkQb/RAmSQwc2KdJMHw7BgMHj9+vWMxsvgZgdl5EFR1O/322w2g8EwOTkZv+5wOPR6vdPpjLPlwIED... | iVBORw0KGgoAAAANSUhEUgAAARcAAACZCAIAAACKfOmfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29eVwT1/7/z2QhhCVsCatE9n1XAa3KIq4givvWKtb1qrUWaz/9WK92sbe1t9W21qXVarH9VKVaUcCKBRFlR1lkh7AKkoWQANkzc75/zO/mxwPQiySZBJ3nf0xmzjkznNdZ3ud93gcCABjg4OCoAUHXBcDBmfDgKsLBUReMVIQgiFgslslk2GSHg4MlGKmot7c3Nze3paUFm+xwcLBELRWB/6BQKBAEUV1UKBRSqVRl... | null | Part of the image of the function\( f(x) = \omega \sin\left(\omega^2x- \frac{\pi}{6} \right) + 1 \) may be () | [
"<image_1>",
"<image_2>",
"<image_3>",
"<image_4>"
] | ABC | 三角函数与解三角形 | math |
52111625-1_4-3 | iVBORw0KGgoAAAANSUhEUgAAAN0AAADkCAIAAACT07YoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2dZ1gU19vGd2Yb7C4svaMgEBCMBtHYYhSsiQ1LEolRiRKlWGlqFJQioVlQUYJYIsZu8rcQsWADROzEqIgFBOl1e5n2fpjr2ovXCog7M+v8vjm7F3OPc+8pzznneQAMwxg0NCQDJFoADc0boH1JQ0ZYRAvoTjoyJgEAQCtatMd7n5qKj6w7voRhWKFQKJVKGIYBAABBUPM+MAxDURTDMENDQx6PR8X39DZeeWomk4lf... | iVBORw0KGgoAAAANSUhEUgAAAQIAAADgCAIAAABn1v86AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO39Z0AUZ/f/j+/MzhaWpfdeBQSMCKgoKCL2YI3lFlHRaDRyi4oJajRqALu3BWLQoMbYu2DsiootlgRQsCEdpMPussu2ab8H1/ezf/4ISN2C83qky+zMmZ15X+Vc55wLIkmSRkHxZQOr2gAKCtVDyYCCgpIBBQWNhqjagK6BIAgMwzAMa+UYOp3OYDBgmFI+jUajYRiGomhLM0MIghAEYTAYSrdLNfQQGeA4fv369bKy... | null | null | null | As shown in Figure 1<image_1>, fold the regular hexagon\(ABCDEF\) with side length of 2 along\(AD\), so that the plane\(ADE_1F_1\) is perpendicular to the plane\(ABCD\), as shown in Figure 2<image_2>. Point\(M\) is on line segment\(AD\), and\(CD \parallel\) plane\(BMF_1\). Find the cosine value of the dihedral angle\(F... | null | \(-\frac{\sqrt{5}}{5}\) | 空间向量与立体几何 | math |
51878032-1_9 | iVBORw0KGgoAAAANSUhEUgAAARwAAADzCAIAAADb8z5UAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOy9WY9lx5EmaJ/5OXeJJTcm90UqSZRYKqmqq6amZh4KM5jpfus/Vf+lgfkFM0ADjX7oAQaNru4qtUpLlaiFpCjuTGYyMzIj7j3uZvNgZu5+7r1JRlAkBhiES8yIuMs5ftxt/WxxqCpdj+txPb6+wf9fT+B6XI//v41rproe1+NrHtdMdT2ux9c8rpnqelyPr3lcM9X1uB5f87hmqutxPb7mcc1U1+N6fM3jmqmux/X4... | iVBORw0KGgoAAAANSUhEUgAAAPwAAAEHCAIAAAAF67xeAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2dZ0ATWdfHSU9ISCH03ntTwQqiIlgogiKoqNjbrq7d1V111V0b6+q69t5XBRuWdVVQdGWl9470UEMLJISUyfvhPk+evDQxhTq/T+TOzJ07w3/u3Dn3nHMRIpFICQZmOIHs7wbAwPQ1sOhhhh2w6GGGHbDoYYYdsOj7CDabXV1d3d+tgFGCRd93nD59+uXLl/3dChglWPR9RGRk5K+//trfrYD5D+j+bsAQRyQSZWVl... | null | null | null | The Lerot tetrahedron is a very magical "tetrahedron" that can rotate freely between two parallel planes and always remain in contact with both planes, so it can roll back and forth like a ball (see Figure A<image_1>). Using this principle, scientists invented the rotary engine. Lerot tetrahedron is a geometric body su... | [
"The maximum distance between any two points on the surface of Lerot tetrahedron\\(ABCD\\) is greater than 3",
"The cross-sectional area of Lerot tetrahedron\\(ABCD\\) measured by plane\\(ABC\\) is\\(\\frac{9\\sqrt{3}}{4}\\)",
"The sum of the lengths of the intersection lines of the four surfaces of Lerreau tet... | AD | 空间向量与立体几何 | math |
51825787-1_8 | null | null | null | null | null | It is known that\(F\) is the right focus of the ellipse\(C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1(a > b > 0)\),\(P\left(1, \frac{3}{2}\right)\) is the point above\(C\), and that the line\(PF\) and the circle\(O: x^2 + y^2 = 1\) are tangent to the point\(F\). If the two points on\(C\)\(A,B\) satisfy\(PA \perp PB\). Find... | null | \(\frac{\sqrt{183}}{7}\) | 平面解析几何 | math |
51677674-1_3 | iVBORw0KGgoAAAANSUhEUgAAAgMAAAB+CAIAAAA2moU2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOxdd0AU1/be2d3Z3ZnthV0QWGBh6R0ERbCBvUU09qiJsUWN0VghGo0isSYxxljA3qKxJPaCgopSBaVYkQ4ubdnC9pn5/XFf9vGzxSiW9x7fH4muW+7cuXPOvd/5zjkQQRCkdrSjHe1ox/8wyO97AO1oRzva0Y73jHZP0I52tKMd/+ugvu8BtKMd7WjHfzAIgiAIAoIgEokEQVAruv1ffwL/9IGj3RO0ox3taMc/hjXC... | null | null | null | null | <image_1>If you stretch both ends, the following ropes will knot (). | null | 1，2 | 平面解析几何 | math |
52036966-1_9 | null | null | null | null | null | For the function\(f(x)\) defined on\(D\), if there is a real number\(x_0\) such that\(f(x_0) = x_0\), then\(x_0\) is said to be a fixed point of the function\(f(x)\). Then the following conclusions are correct () | [
"The function\\(f(x) = x^\\frac{1}{2}\\) has and only 1 fixed point",
"The function\\(f(x) = 1 - \\lg x\\) has and only 1 fixed point",
"The function\\(f(x) = 2^x - 1\\) has 2 fixed points",
"The function\\(f(x) = 2 \\sin\\left(2x - \\frac{2\\pi}{3}\\right)\\) has 3 fixed points"
] | BCD | 函数与导数 | math |
52035989-1_15 | null | null | null | null | null | The function\(f(x) = (2-a)\ln x - \frac{2a}{x}\)(\(a \in \mathbb{R}\)) is known. If the function\(g(x) = f(x) + \frac{x^2}{e^x} - x^a + \frac{2a}{x}\) has exactly 2 zeros, find the range of values for\(a\). | null | \((-\infty, 2-e)\) | 函数与导数 | math |
52185592-1_36 | null | null | null | null | null | If the line passes through two fixed points $P(x_1,y_1),Q(x_2,y_2)$(where $x_1\neq x_2$), the parametric equation of the line is
\[
\begin{cases}
x = \frac{x_1 + \lambda x_2}{1 + \lambda} \\
y = \frac{y_1 + \lambda y_2}{1 + \lambda}
\end{cases}
\]
($\lambda$is a parameter,$\lambda\neq -1$) Where point $M(x,y)$is any... | [
"The geometric meaning of parameter $\\lambda$is that the number of points $M$directed line segments $PQ$is more than $\\frac{PM}{MQ}$",
"You can use $\\left(\\frac{x_1+\\lambda x_2}{1+\\lambda},\\frac{y_1+\\lambda y_2}{1+\\lambda}\\right)$to represent any point on a line",
"When $\\lambda<0$and $\\lambda\\neq-... | B | 平面向量 | math |
52179167-1_2-2 | null | null | null | null | null | Given a decreasing geometric sequence $\\{a_{n}\\}$, the sum of the first $n$ terms of $\\{a_{n}\\}$ is $S_{n}$. It is known that $\\frac{1}{a_{1}}$, $2S_{2}$, and $8a_{3}$ form an arithmetic sequence, and $3a_2 = a_{1} + 2a_{3}$. The sequence $\\{b_{n}\\}$ satisfies $b_{n+1} = 2b_{n} - 2n + 1$, with $b_{1} = 3$, $n \\... | null | $T_{2n}=\frac{529}{180}-\frac{12n+13}{9\cdot2^{2n-1}}-\frac{1}{4(2n+5)}$ | 数列 | math |
52179167-1_11-1 | null | null | null | null | null | Given that the sequence $\{a_n\}$satisfies $a_1=5$,$a_{n+1}-2a_n = 3^n $($n\in\mathbb{N}^*$), remember $b_n=a_n-3^n$. Let $c_n=\frac{2n+1}{b_n}$, and the sum of the first $n$terms of the sequence $\{c_n\}$is $S_n$. If the inequality $(-1)^n\lambda<S_n+\frac{n}{2^{n-1}}$holds true for all $n\in\mathbb{N}^*$, find the ra... | null | $\left(-\frac{5}{2},\frac{15}{4}\right)$ | 数列 | math |
52179167-1_12-3 | null | null | null | null | null | Given an arithmetic sequence $\\{a_n\\}$ and a geometric sequence $\\{b_n\\}$, with $a_4 = b_1 = 2$, $a_5 = 3(a_4 - a_3)$, and $b_4 = 4(b_3 - b_2)$. For any $m \\in \\mathbb{N}^*$, insert $m$ identical numbers $(-1)^{m+1} \\cdot m$ between $b_k$ and $b_{k+1}$ to form a new sequence. Given a fixed $k$ ($k \\in \\mathbb{... | null | $S_n=\begin{cases}2^{k+2}-2+(k+1)\left(\frac{k^2+4k+2}{2}-n\right),&k\text{is odd}\\2^{k+2}-2+(k+1)\left(n-\frac{k^2+4k+2}{2}\right),&k\text{is even}\end{cases}$ | 数列 | math |
50864643-1_22 | iVBORw0KGgoAAAANSUhEUgAAAVoAAAEsCAIAAAANSOz9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO3dd0BT5/4/8CQQIAl7y957LwUVEUQERYZarFi7rH6traNVr3thlSWCyBQFRUREqihOZIkbUFFcyEZkB7LIPvn9EX9erwMZCSfjef11LxySdyV8znOeieRwOAgAAAAEAgV3AAAABAUoBwAAvAfKAQAA74FyAADAe6AcAADwniTcAQBgLNhsdk1NDYvF+uaVZmZmqqqqExJK6IFyAAglNpvd09NTXFyckZHBYrEUFRU1... | null | null | null | null | <image_1>As shown in the figure, the point $O$is the center of a regular hexagon $ABCDEF$with a side length of 1, and $l$is any straight line passing through the point $O$. Fold the regular hexagon along $l$onto the same plane, then the maximum value of the area of the folded figure is () | null | $6\sqrt{3}-9$ | 等式与不等式 | math |
52206804-1_2-5 | iVBORw0KGgoAAAANSUhEUgAAAPQAAACgCAIAAAC0UtZIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2dd1wUV/f/d2Z2l106AhYWLBQLIggSQVCDvaECiqAkJkYliYnGGJ8kmifd5DHFPFFiwwKi9GrBDgQFQVFBQZAiINKXtixl28z9/nG/v/3yMxbK7k6B9x95xd1l9s7MZ8+ce+455yIAANYQQzARlOwBDDGEuhgS9xCMZUjcQzAWNtkDGEIFdHV1NTQ0dHd3AwAQBEEQBL4OJ1QAAB6PJxAItLS0lG9REDhagiA6Ozur... | null | null | null | null | <image_1>As shown in the figure, it is known that the side length of the diamond $ABCD$is 2, and $\angle A=60^\circ$, and $E,F$are the midpoints of edges $AB and DC$respectively. Fold $\triangle BCF$and $\triangle ADE$along $BF and DE$respectively. If $AC\parallel$Plane $DEBF$is satisfied, then the value range of line ... | [
"$[\\sqrt{3},2\\sqrt{3}]$",
"$[1,2\\sqrt{3}]$",
"$[2,2\\sqrt{3})$",
"$[2,2\\sqrt{3}]$"
] | A | 空间向量与立体几何 | math |
52206804-1_8-1 | iVBORw0KGgoAAAANSUhEUgAAAPsAAACoCAIAAACpCg+oAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29d1xTyff4bSihJiT0IiBIb4IogogNFgv2CnaxoK66NnDtiuVjZ9W1ISoWEEURVEQEFQELiHQUUEB6CaT33Hvz+2OevQ9fRIQQCGLef/AKyS1zkzMzZ86cghEKhQOkSPltkJF0A6RI6VWkEi/l90Iq8VJ+L6QSL+X3Qk7SDZDSI5BIpFevXvH5/AEDBmAwGBkZGSwWSyAQdHV1jYyMVFVVMRiMpNsoGaQS3z9RVVVV... | null | null | null | null | <image_1>As shown in the figure, the quadrangle\(A'B 'C'D'\) is the projection of a square\(ABCD\) with side length 2 on the plane\(\alpha\). The rays\(AA'\),\(BB'\),\(CC'\), and\(DD'\) are parallel to each other, and the angle formed by ray\(AA'\) and plane\(\alpha\) is\(60^\circ\), rotate the square\(ABCD\), keep\dur... | null | \frac{16\sqrt{3}}{3} | 空间向量与立体几何 | math |
50011259-1_4 | iVBORw0KGgoAAAANSUhEUgAAAUgAAAEtCAIAAADlT26NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOydZ1xU1/b3OdM7DEPvvVcRKZZYEFFjYguxJtaoEDRGr4bYFexGvbZco8YaGxpRbAQEEZUqvfdhgBn6zDDD9Hle7PufZy69g3C+L/zgKfvsc+b8ztpl7bUguVyuAgMDM7pADHcFYGBgBh5Y2DAwoxDUcFcApitAR0kqlfL5fA6Hw+fztbS0VFVVIQgCe2traxkMhr6+vra2toqKimJ7WVlZS0uLg4MDAgF/u8ci8K8+... | null | null | null | null | <image_1>As shown in the figure, hyperbolic\The left and right focuses of (x^2 - \frac{y^2}{8} = 1\) are\(F_1\) and\(F_2\) respectively, and the straight line passing through\(F_1\) intersects the two branches of the hyperbola at\(A\) and\(B\) respectively.(\(A\) is on the line segment\(F_1B\)),\(\odot O_1\) and\(\odot... | [
"\\(\\left(\\frac{1}{3}, \\frac{1}{2}\\right)\\)",
"\\(\\left(\\frac{1}{3}, \\frac{2}{3}\\right)\\)",
"\\(\\left(\\frac{1}{2}, \\frac{2}{3}\\right)\\)",
"\\((0, +\\infty)\\)"
] | D | 平面解析几何 | math |
51415429-1_40 | null | null | null | null | null | A shopping mall has launched a shopping lottery promotion. The rules of the event are as follows: ① For every 100 yuan spent by customers in the mall, they can get 1 lottery ticket. ② Each lottery ticket can be held once, that is, one ball is randomly picked up from a box containing 4 white balls and 2 red balls (each ... | null | 7 - 10 \times \left(\frac{4}{5}\right)^{10} + 3 \times \left(\frac{2}{3}\right)^{10} | 计数原理与概率统计 | math |
51415429-1_30 | null | null | null | null | null | Given that events A, B satisfy 0 < P(A) < 1, 0 < P(B) < 1, P(\overline{A}B) - P(A\overline{B}) = 2P (\overline {B}), then P(B)|\overline{A})·P(\overline{A}| The value range of\overline{B}) is | null | \left[0, \frac{2}{3}\right] | 计数原理与概率统计 | math |
51415429-1_29 | iVBORw0KGgoAAAANSUhEUgAAALYAAADbCAIAAABP+PzXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO19Z1xT2fY2OSE06T0QuhQFBRGxAQ5YGMXe+8go4NgRR/R6VWyjjiAqoDggghVULKPiiAUUQQRREUSKSO+dhCQn9f2w/nNeLiKkg8rzwR8mJ2e3Z6+99tprrY3jcrlSAxjA14H0dQUG0N8xQJEB9IIBigygF0j3dQXEji7KFg6H6/mxLx+Ar772w+8e3z9FANxOwP7b5QGMBLh/0eXvHxPfG0UwHnA4HC6Xy2az2Ww2... | iVBORw0KGgoAAAANSUhEUgAAAMgAAADdCAIAAABxIarhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2deVgUV9b/a+tuoOlmkVU2RRhxxR0kiooLiXvUMZO4JLivEc1EUWOiRB2NW/TNJBpcEDU6GmOMUaOg4L4TERREAgoIKMja9Fp17++P87719A9MaJyubpH6/MHzUF3d91TVt+567jkkxpgQETE3lLUNEHkzEYUlIgiisEQEQRSWiCCIwhIRBFFYIoIgCktEEERhiQiCKCwRQWBMOQljDBP0JEmSJGl8pMGDBEEghOqf... | null | null | null | As shown in Figure 1<image_1>, a circle is divided into n (n \geq2) sectors, each sector is colored with one of k colors. Adjacent sectors are required to have different colors. There are a_n = (k-1)^n + (k-1) \times (-1)^n methods. As shown in Figure 2<image_2>, there are 4 different colors of paint. If you paint the ... | null | 201852 | 计数原理与概率统计 | math |
51415429-1_19 | iVBORw0KGgoAAAANSUhEUgAAAN4AAAC3CAIAAADy/ndyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO1dZ0ATabcmBEgIUkJICL2DBUEFARFBRRTRtWAFFzug6KpYwHXdRRQV/URdsWKvWLAXLGBB7AhiAwHFQpNe06bk/jj35vJRQhKiBMzzyzLlZOaZ857+Evh8voIccsgeFDtaADnkaBlyakoIPp8vX3B+KOTUlAQ4jtfX15eVlWEY1tGydFnIqSk2+Hw+m81es2bNvHnzqqqqcBzvaIm6JuTUFA98Ph/DsIyMjFOnTiUl... | null | null | null | null | As shown in the figure<image_1>, a triangle is divided into 9 rooms, and the 2 rooms with common sides are called adjacent rooms. A small ball moves from one room to the adjacent room with equal probability at a time, then () | [
"Put 2 balls in different rooms, and the probability that the rooms are not adjacent is $\\frac{3}{4}$",
"Place $k$balls in different rooms. If the rooms are not adjacent to each other, then $k \\leq 6$",
"There are 6 paths for the ball to depart from room C and reach room H after 4 movements",
"The probabili... | ABD | 计数原理与概率统计 | math |
51415429-1_12 | iVBORw0KGgoAAAANSUhEUgAAAOEAAADGCAIAAACreHCjAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO1dd1gU19fO0tuyHXbpsPSuICJgAcFGMyoSLImCBTuCxt4rGkzs2I3YwIbE3huKCkZBURAF6UXaLtum8f1xvx8PD7K7s7MkYuT9K8E7c+/snDn3nvYeUmtr6w896EE3hsrXXkAPeiAHPTLag+6OHhntQXdHj4z2oLujR0Z70N3RI6M96O7okdEedHf0yGgPujt6ZLQH3R09MtqD7o4eGe1Bd0ePjPagu0Ptay+gWwDD... | iVBORw0KGgoAAAANSUhEUgAAAMYAAAC0CAIAAABAE8TAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29Z0AUSdc27BAUJmeGHIYcVURREBAJIiZWMYEZF8OaFX3UXRXXNSFiRMwJlFXAgIqICCwGzDkBgogSByRN7G6+H/W8fDzIND3ds7fh9vrHWFWerjpdXXXCdUitra3dfuL/QqU5IZFI/6Ys3x+0vrYA3xzkcnlNTQ2CIFgak0gkDoejo6PzU7Ha8FOl/g8QBDl37lx0dDTG9q2trf7+/ps2bdLR0fmXRftuQPr54WsP... | iVBORw0KGgoAAAANSUhEUgAAALcAAAC2CAIAAAAUUIttAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29Z1gUWfM+7ICIhAlMYmbISM4mVBBMICiIEVDBgJgVwxrWrGtcd8WcI0ZEUJKiYsQAYhZRQFQQFJAMk3u6e94P5/rxciHTdPewuP73ub+J55ypPl19QlXdVRSlUtnpvwelUvnly5cRI0awWKzFixdraGhgt0dR9O7du6dPn46JiRkxYoSmpmZHSfrvgPI/CZlMtmDBAoFA8PjxYxRF22yPomhpaamtre3QoUPr6uo6... | iVBORw0KGgoAAAANSUhEUgAAALAAAAC0CAIAAAC7RDEfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29Z1gU2fMGyoCkyRGYIUcFCSIoSJQ1gSAqa8Y/iwmzKOoa0F1zREXUNSAmDJgVI2sCMZBFUBEQJShJ4jBM7G7uh/NcLg8yPd0z6M/dvfVN7D7zdnf16XOq6n2L0NHRofafsY6ODlzXSyAQCATC90T001mf/zWAH2odHR1SqRS7T2hoaGhqav6nfOI/5BAIgjQ0NERFRdXV1WE5vqOjw8XFZd26dWQy+b/jE+r/awA/... | iVBORw0KGgoAAAANSUhEUgAAALQAAAC0CAIAAACyr5FlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO19Z1gT2/e1AUUC6QmhE3pvIgoKqIAKCjYsYO+oKFbutWO91mtDxYq9FxSxgx0VRBFFQUUEpHcIhJQpvB/O8/DwApnMTLjc68///iaemayc7Dlzzt57rU1pamrq8vtZU1PTp0+ffvz4gaIonvFMJtPDw0NNTY1Cofzz6P4rRvkNnQNF0fLycg8Pj9ra2q5duyocD8MwlUo9c+ZM//79VVVVOwXjf8IUT83/mDU1NcEw... | When studying the total probability formula, we transform the study of the occurrence of an event into the analysis of several prerequisites for the occurrence of the situation. This is an important recursive idea. In <image_1>the honeycomb-shaped regular hexagonal map as shown in the figure, the "○" in the upper left ... | [
"<image_2>",
"<image_3>",
"<image_4>",
"<image_5>"
] | BD | 计数原理与概率统计 | math |
45056539-1_4 | iVBORw0KGgoAAAANSUhEUgAAAM8AAAIDCAIAAAAsVt3uAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOydd0AUR/vH2avcwR1HP3oTRHoVK4pYQUEFC5ZYsAVLTDQmaowmlsQSFbtGsURFRSyIJRp7sGBHQIp0BA643svu/v6Y93cvLyICwnEc+/kL7nZn53a/+8zMMzPPA6EoqoeBoRFwnV0BjG4EpjYMzYGpDUNzYGrD0ByY2jA0B6Y2DM1B6OwK/Be1LwZFUYlEUlpaWlRUlJeXV1FRIRaLO+66EAQxGAwnJyd3d3dnZ2dH... | null | null | null | null | The arithmetic operator MOD represents taking the remainder, such as $a \text{MOD} b = c$, which means that the remainder of $a$divided by $b$is $c$. As shown in the figure<image_1>, a program block diagram about taking the remainder. If the value of input $x$is 3, output $x =$() | [
"1",
"3",
"5",
"7"
] | A | 算法与框图 | math |
45056539-1_5 | iVBORw0KGgoAAAANSUhEUgAAAzUAAAC0CAIAAAAiipVbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOydd1wTyfv4M5uQ0EIREEQFETsgIooIytkRbKfSxI5dVGwI3ontsIJixS4qFhALqCBiVxS8A1GQYgdEunQCpOz+/pjPd397SQgBQtGb9x++ZDI782ybfWbmKYAgCBoCgUAgEAgEot2AtbUACAQCgUAgEIh/gfQzBAKBQCAQiPYF0s8QCAQCgUAg2hdIP0MgEAgEAoFoXyD9DIFAIBAIBKJ9gfQzBAKBQCAQiPYFo60F... | null | null | null | null | If you execute the following program diagram<image_1>, the output $S =$() | [
"37",
"46",
"48",
"60"
] | C | 算法与框图 | math |
46737264-1_10 | null | null | null | null | null | Known function $f(x) =| 2x+4| + {2|\frac{2}{a}-x|}$. If $a = [f(x)]_{\min}$, find the solution set for the inequality $f(x-1) \leq2x + 5$. | null | $[0,3]$ | 不等式选讲 | math |
51415427-1_16 | iVBORw0KGgoAAAANSUhEUgAAAQcAAADsCAIAAAD2PfQFAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOx9d1wUV9Q2M1tYWHrvVcACYkMRUSxYEbFjFxVLIrHERqLGGGOLsYAtmmBBUSyxd6WIHRFBpQgKIr2X7Tvt++O87377AhLKLuzqPn/403V37p2Ze+499TkIRVFqKqigghTQjp6ACiooHFRSoYIK9aGSChVUqA+VVKigQn3QO3oCKsgA0i4TBEHgE/iLCq2ASiqUGCRJUhRFURSPxysoKEAQxNraWkNDA0GQqqoqFoul... | null | null | null | null | <image_1>In the regular triangular frustum $ABC_1-ABC$,$AB = 2A_1B_1$,$P, D$are the points on the line segments $B_1C_1, BC$respectively,$O_1, O$are the centers of the upper and lower bases, and $M$is a point within the base $ABC$. The following conclusions are correct () | [
"$AA_1 \\perp BC$",
"If $BD = \\frac{1}{3}BC, AA_1 \\parallel$Plane $B_1DM$, then the trajectory length of point $M$is equal to $\\frac{1}{3}AB$",
"$V_{A-CBB_1C_1} = \\frac{3}{2}V_{A_1-ABC}$",
"When $PD \\perp BC$, the figure formed by the four points $O_1, O, Q, and P$is a right-angled trapezoid"
] | AC | 空间向量与立体几何 | math |
51415427-1_18 | null | null | null | null | null | It is known that the edge length of the regular tetrahedron $A-BCD$is 6, and the points $M$and $N$are the midpoints of $BC and AD$respectively, then the following geometries can be placed into the regular tetrahedron $A-BCD$as a whole () | [
"cone with base on plane $BCD$, base radius $\\sqrt{2}$, and height $2\\sqrt{6}$",
"A cylinder with a base on the plane $BCD$, a base radius of $\\sqrt{2}$, and a height of 1",
"cone with a straight line $MN$, a base radius $\\sqrt{2}$, and a height of 2",
"Cylinder with axis straight $MN$, base radius $\\sqr... | ACD | 空间向量与立体几何 | math |
51415427-1_19 | iVBORw0KGgoAAAANSUhEUgAAAQgAAAD2CAIAAACle451AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2dd1wU1/r/d7axC8hSFEHKSklEVMQgmmiMRGOJ0WtL7IZmEBREEVtULBEBxQsqIBEVQUXB2E000W8sVwWxYqWquHSks33mzO+P87vz2ksx9G3n/YcvnJ2defbMfM7znOc0jCRJGgKB+F+YyjageyFJEgAAAFA8iGEYhmF0Oh3DMOWZhlBpNF8Yd+7cycjI+PDhA0mSbDabyWQaGBgYGxv379/f1tbWzMyMzWbT6XRl... | null | null | null | null | <image_1>As shown in the figure, in a cube $ABCD-A_1B_1C_1D_1$with an edge length of 2, the point $P$in space satisfies $\overriding {AP} = \overriding {AD} + \lambda \overriding {AB} + \mu \overriding {AA_1}$, and $\lambda \in [0,1], \mu \in [0,1]$, then the following statement is correct () | [
"If $\\mu = 1$, then $BP \\perp AD$",
"If $4P = \\sqrt{5}$, the maximum value of $\\lambda + 2\\mu$is $\\frac{\\sqrt{5}}{2}$",
"If $\\lambda = 1$, then the minimum value of the cross-sectional area of the square in plane $BPD_1$is $\\sqrt{6}$",
"If $\\lambda + \\mu = 1$, the minimum value of the tangent value... | ABD | 空间向量与立体几何 | math |
50138377-1_2-5 | null | null | null | null | null | It is known that the general term of the sequence $\{a_n\}$is $a_n = \frac{1}{n^t}$, where $t$is a normal number, and $S_n$is the sum of the first $n$terms of the sequence $\{a_n\}$, then the following statement is incorrect () | [
"$\\exists$constant $m$such that for $\\forall n \\in \\mathbb{Z}^+$there is a necessary and sufficient condition that $S_n < m$is $t > 1$",
"$t < 1$is a sufficient and unnecessary condition for $S_n \\geq \\ln(n+1) (n \\in \\mathbb{Z}^+)$",
"For $\\forall n \\in \\mathbb{Z}^+$,$S_n \\leq2 + \\frac{2}{2^t}$is a... | D | 集合与常用逻辑用语 | math |
50138377-1_4-4 | null | null | null | null | null | It is known that the equation of the moving circle $C$is $(x-\cos \theta)^2 + (y-\sin \theta)^2 = 0$, where $\theta$is a constant, and $\theta \in [\pi, 2\pi)$has the following two propositions:
① There exists $\theta \in [\pi, 2\pi)$, so that circle $C$is tangent to circle $C_1: (x+\cos \theta)^2 + (y+\sin \theta)^2 =... | [
"①② are all true propositions",
"① is the true proposition, ② is the false proposition",
"① is a false proposition, ② is a true proposition",
"①② are all false propositions"
] | C | 集合与常用逻辑用语 | math |
48699793-1_4 | null | null | null | null | null | We know the function $f(x) = x\ln x$. If $x_1, x_2 \in (0, 1)$, then $|f(x_1) - f(x_2)|$and $|x_1 - x_2| ^{\frac{1}{2}} The size relationship of $is ( ）。 | null | $ \leq $ | 函数与导数 | math |
51984964-1_2-2 | null | null | null | null | null | Given $O$ as the coordinate origin, $\overrightarrow{OA}$ and $\overrightarrow{OB}$ as unit vectors, $(\overrightarrow{OA} + \overrightarrow{OB}) \cdot \overrightarrow{OB} = \frac{3}{2}$, point $C$ lies on the fixed line $l: y = x + 2\sqrt{2}$, and the inequality $|\overrightarrow{OA} + \overrightarrow{OB} + \overright... | [
"$(-\\infty, 2 + \\sqrt{3}]$",
"$(-\\infty, 2 - \\sqrt{3}]$",
"$(-\\infty, 2\\sqrt{3}]$",
"$(-\\infty, \\sqrt{3}]$"
] | B | 平面向量 | math |
51984964-1_4-1 | null | null | null | null | null | It is known that the center of the circumscribed circle of $\triangle ABC$is $O$,$H$is the center of gravity of $\triangle ABC$and $|\overline{AB}| = 4, |\overline{AC}|= 6$, then $\overline{OA} \cdot (\overline{HB} + \overline{HC}) =$ _________ | null | \frac{26}{3} | 平面向量 | math |
50692277-1_5-1 | iVBORw0KGgoAAAANSUhEUgAAARcAAAC6CAIAAAALRJ4HAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29Z1wUSfe/TU9PJoNkARUQMaCiYFxMa9rbtGZBVAyIriirrqDiKoZ7zYgKuAbEgDmgqGAOYABFDAioyCA5M8Pkng7Pi/r/5p4HFEFgBqWuF/tx2w5n2vp2VZ065xRCUZQWBAJpBDRNGwCB/PBAFUEgjQWqCAJpLFBFEEhjgSqCQBoLVBEE0ligiiCQxkJX58NIkqQoiiRJGo1Go9EQBNHS0iIIAhxBUVSdxkAgTYX6... | null | null | null | null | Newton, a famous British physicist, used the method of "making a tangent" to find the zero point of the function. As shown in the figure<image_1>, make a tangent to $f(x)$at the point with the abscissa of $x_1$, and the abscissa of the intersection of the tangent and the $x$axis is $x_2$; repeat the above process with ... | [
"$x_{n+1} = \\frac{x_n^2 + 6}{2x_n - 1}$",
"$x_4 = x_1 - \\frac{f(x_1) - f(x_2)}{f'(x_1)} - \\frac{f(x_2) - f(x_3)}{f'(x_2)}$",
"Sequence $\\{a_n\\}$is a decreasing sequence",
"$S_{2025} = 2^{2025} - 1$"
] | ABD | 数列 | math |
50692277-1_6-4 | null | null | null | null | null | If the finite sequence of $n$terms $\{a_n\}$satisfies $a_1 = a_n$,$a_2 = a_{n-1}$,...,$a_n = a_1$, that is,$a_i = a_{n+1-i+1} (i = 1, 2, \cdots, n)$, then the finite sequence $\{a_n\}$is called a "symmetric sequence". It is known that the sum of the first $n$terms of the increasing sequence $\{c_n\}$is $S_n$, and $4S_n... | null | $\sum_{i=1}^n P_i = (n-1)2^{n+1} + n + 2$ | 数列 | math |
51415432-1_16 | null | null | null | null | null | The equation for curve C is given as| x| ^{\frac{1}{2}} + |y| ^{\frac{1}{2}}} = 1, then the following statements are correct () | [
"Curve C is symmetric about the origin",
"If a point on the P(x, y) curve C, then y ≥ 16| x|",
"If curve C and curve (a -|x|)y = 1 has a common point, then a ≥ 1",
"If the point $(\\frac{1}{n+1}, a_n)$ ($n \\in \\mathbb{N}^*$) lies on the curve $C$, then $\\frac{1}{\\sqrt{|a_1|}} + \\frac{1}{\\sqrt{|a_2|}} + ... | ACD | 函数与导数 | math |
49801742-1_4-2 | null | null | null | null | null | Let f(x) = e^{x^3} - x - ax, discuss the number of zeros of f(x). | null | When 0 \leq a < b = \frac{1}{a}e^{\frac{1-2a}{3}}, f(x) has no zero; when a < 0 or a = b, f(x) has one zero; when a > b, f(x) has two zero. | 函数与导数 | math |
51415434-1_16 | null | null | null | null | null | Known curve C:| x| - \frac{y| y|}{4} = 1, P(x_0, y_0) is the point above C, then the following statements are correct () | [
"There is a point P(x_0, y_0) such that 2x_0 - y_0 = -1",
"\\left| 2x_0 - y_0 + \\sqrt{5}\\right| The range of values for is\\left[\\sqrt{5}, 2\\sqrt{2} + \\sqrt{5}\\right]",
"Ruo\\left| 2x_0 - y_0 + m\\right| + \\left| 2x_0 - y_0 + n\\right| The value of is independent of x_0, y_0, and m > 0, n < 0, then the ... | BCD | 平面解析几何 | math |
51415434-1_30 | null | null | null | null | null | Given the hyperbola E: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) with right focus F and left vertex A. Let P be a point on the right branch of E. The chord length intercepted by the line PF on the circle O: $x^2 + y^2 = a^2 + b^2$ is $\\frac{1}{2}\\sqrt{a^2 + b^2}$, and $\\overrightarrow{PA} \\cdot \\o... | null | \frac{8}{5} \text{or} \frac{8}{3} | 平面解析几何 | math |
51415434-1_34 | null | null | null | null | null | Let M be a set of straight lines. If the tangent at any point on C is in M, and any straight line in M is a tangent at a point on C, then C is called the envelope curve of M. The envelope curve of M_1 is known as C_2: x^2 = 4y, and the straight lines l_1, l_2 \in M_2. Let the common points of l_1, l_2 and C_2 be P and ... | null | \frac{1+\sqrt{5}}{2} | 平面解析几何 | math |
51415421-1_27 | null | null | null | null | null | Given the function $f(x) = \\sin(\\omega x + \\frac{\\pi}{3})(\\omega > 0)$ which has exactly one zero in the interval $(\\frac{\\omega}{2}, \\omega)$, and satisfies $f(\\omega) + \\frac{\\sqrt{3}}{2} = 2f(\\frac{\\omega}{2})$. the value of $\\sum_{i=1}^{2025} f(\\frac{i\\omega}{2})=( )$. | null | -\sqrt{3} | 三角函数与解三角形 | math |
51415421-1_30 | null | null | null | null | null | It is known that $a, b, and c$are the opposite sides of the three internal angles $A, B, and C$of acute angle $\triangle ABC$, and the area $S = \frac{a^2 - (b-c)^2}{2}$, then the value range of $\frac{b+2c}{a}$is () | null | \left(\frac{11}{4}, \frac{\sqrt{185}}{4}\right] | 三角函数与解三角形 | math |
51415421-1_33 | null | null | null | null | null | In triangle $ABC$, the opposite sides of corners $A$,$B$, and $C$are $a$,$b$,$c$, and $\tan B = 3\tan A$respectively. When $b = 4$, find the value range of $ac$. | null | $\left(0, 8\sqrt{2}\right]$ | 三角函数与解三角形 | math |
51415421-1_37 | null | null | null | null | null | The function $f(x) = 3x^2 - 8\sin(x + \varphi)$is known, where $|\varphi| \leq \pi$。If $\forall x \geq0 $,$f(x) \geq0 $, find the value range of $\varphi$. | null | $\varphi \in \left[-\pi, \frac{\pi}{6} - \frac{2\sqrt{3}}{3}\right]$ | 三角函数与解三角形 | math |
51415421-1_40 | iVBORw0KGgoAAAANSUhEUgAAAPkAAAD/CAIAAAC8dOfaAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2deSCVaf//3Wc/ODj2nSyJSpRKaVVDmEpEyZIaT5NqapqWqWmZmeaZtqnJ07ROyxTtSFKK1CgGLRItiJLIweFw9u1efn9cz8/jS4nCWdyvv+rgnOs+532u+/P5XJ8FwjBMAwdnAEBQ9AJwcPoJXOs4AwVc6zgDBVzr6gaKonK5HEVRRS9E6cC1rm6w2ewbN26IxWJFL0TpwLWuVggEgt9+++3+/fv4vt4ZXOvqg1wu... | null | null | null | null | As shown in <image_1>, given a line segment $B_1C_1$ with length 2, construct an isosceles triangle $\\triangle A_1B_1C_1$ with $B_1C_1$ as the base and vertex angle $\\theta$ ($0 < \\theta \\leq 90^\\circ$). Take the trisection points $B_2$ and $C_2$ of the side $A_1B_1$ (with $B_2$ closer to $A_1$), and construct an ... | null | $\left[0, \frac{1}{2}\right]$ | 三角函数与解三角形 | math |
51415427-1_33 | iVBORw0KGgoAAAANSUhEUgAAASIAAAC9CAIAAACYm+fDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO29Z0ATWd//zaQQkpAChN6bIogURYog4CqCYMHCrhVX1y7qWi51V9a2665rv1y7a8OKWBBQUUGqIiDSOxaQ3hPSZyb/F+d+cvEgIiAlwfm80skkOWHmO+ecX4UkEokCBgZGf4Ib7AFgYAx9MJlhYPQ7mMwwMPodTGYYGP0OYbAH0C34fP7PP//c0tLC5XIVFBTweDyBQIAgiEgk2tjYzJw508DAgEQiDfYwMTA6B5IL... | null | null | null | null | In $\triangle NBC$,$\angle B = 90^\circ$,$AD \parallel BC$,$NA = CD = 2AB = 2$, as shown in the figure, <image_1>fold $NAD$along $AD$to $\triangle PAD$. If the size of the dihedral corner $P-AD-B$is $120^\circ$. Is there a point $E$on line segment $PD$such that the cosine of the angle formed by plane $ABE$and plane $PD... | null | There is a point $E$, which satisfies the meaning of the question when $DE = \frac{1}{2}DP$or $DE = \frac{4}{5}DP$. | 空间向量与立体几何 | math |
51415427-1_35 | null | null | null | null | null | It is known that the bottom surface $ABCD$of the quadrangular pyramid $S-ABCD$is parallelogram,$SA = AB = SB$,$SB \perp AC$,$\angle ABC = 60^\circ$,$BC = 2$,$AB = 1$. Let $T$be the point on line segment $SD$. Plane $\alpha$passes through points $B$,$T$, and $AC \parallel$Plane $\alpha$. Explore whether there is a point... | null | Yes, the value of $\frac{ST}{SD}$is $\frac{1}{5}$or $\frac{1}{2}$ | 空间向量与立体几何 | math |
51415427-1_38 | iVBORw0KGgoAAAANSUhEUgAAASAAAACWCAIAAADxBcILAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2dd1wU19r4d2Z2l7IsRXov0kGkiBFQYw/RS0xs0ahv1BjBghI0UaNXoliC4r1ibLGLWIk3WJEQY+8FUBAQ6SB92aXsLrtT3j+e353fvoCocXdZcL5/+JHd2TNn5swzzznPeQpCURSLgYFBNaDd3QEGht4MI2AMDCqEETAGBhXCCBgDgwphBIyBQYUwAsbAoEIYAWNgUCGMgDEwqBBGwBgYVAi7i+8oiqIoiiRJ2tsD... | null | null | null | null | <image_1>As shown in the figure, in the pentahedron $ABCDEF$, the side length of the diamond $ABCD$is $2$,$EA = EB = FC = FD = \sqrt{10}$, and $EF > BC$. When the volume of pentahedron $ABCDEF$is the largest, find the cosine of the angle between plane $ABE$and plane $BCFE$. | null | \frac{\sqrt{31}}{31} | 空间向量与立体几何 | math |
51415423-1_14 | null | null | null | null | null | It is known that the infinite sequence\{a_n\} satisfies: a_1=p, a_n+[S_{n-1}]=qn, n≥2, p,q \in \mathbb{R}, where [x] represents the largest integer not exceeding x. Then the following statements are correct () | [
"For any p, q,\\{a_n\\} is not a constant sequence",
"There are positive numbers p, q such that\\{a_n\\} is an increasing sequence",
"For any positive number p, q, there is a positive integer M such that a_M,a_{M+1},a_{M+2},\\cdots are periodic sequences.",
"If\\{a_n\\} is a constant sequence, then there must... | BD | 数列 | math |
51415423-1_17 | null | null | null | null | null | It is known that all terms of the increasing sequence\{a_n\} are positive integers and a_{a_n}= 3n, then () | [
"a_1=3",
"a_n>n",
"a_5=6",
"a_{2023}=81a_{33}"
] | ABD | 数列 | math |
51415423-1_18 | null | null | null | null | null | It is known that the function f(x) with domain (0,+∞) satisfies\frac{f(x)+ xf'(x)}{e^x}=1, f'(1)-1 = 0. The first term of the sequence\{a_n\} is 1, and a_{n+1}f(a_{n+1})-f(a_n)=-1, then () | [
"f(\\ln 2)\\cdot \\ln 2=1",
"f(n)>1",
"a_{2023}<a_{2024}",
"a_n\\geq1"
] | ABC | 数列 | math |
51415423-1_19 | null | null | null | null | null | If the sequence\{f_n\} satisfies f_1=1, f_2=1, and f_{n+2}=f_n+f_{n+1}(n \in \mathbb{N}^*), then the sequence\{f_n\} is called a Fibonacci sequence, also known as the golden section sequence. Fibonacci sequences have direct applications in modern physics, quasicrystal structure, chemistry and other fields. Then the fol... | [
"3f_n=f_{n-2}+f_{n+2}，n \\geq 3，n \\in \\mathbb{N}^*",
"f_1+f_2+f_3+\\cdots+f_{2025}=f_{2026}",
"\\exists n \\in \\mathbb{N}^*, such that f_n, f_{n+1}, f_{n+2} are an equal proportional sequence",
"f_1^2+f_2^2+\\cdots+f_{2025}^2=f_{2025}f_{2026}"
] | ABD | 数列 | math |
51415423-1_21 | null | null | null | null | null | It is known that the positive number column\{$a_n$\} satisfies $a_1=\frac{1}{2},a_{n +1}a_n+a_{n+1}-a_n=4$, if there is a non-zero constant a, so that the sequence $\left\{\frac{a_n+a}{a_n-a}\right\}$is an equal number sequence, and the general term formula of the sequence\{$a_n$\} is $a_n$=( ). | null | \frac{4}{5\cdot(-3)^{n-2}-1}+2 | 数列 | math |
51415423-1_22 | null | null | null | null | null | The general term formula of the given sequence\{a_n\} is a_n=\sqrt{\frac{\sqrt{2}+\sqrt{n +2}}{\sqrt{n+2}+\sqrt{n}}}, and S_n is the sum of the first n terms, then S_7=( ). | null | 4\sqrt{2}+1 | 数列 | math |
51415423-1_40 | null | null | null | null | null | The sequence $\\{a_n\\}$ is an arithmetic sequence with a non-zero common difference, and $a_1 = 1$. It is known that $a_{b_1}, a_{b_2}, a_{b_3}, \\cdots, a_{b_n}$ forms a geometric sequence, with $b_1 = 2$, $b_2 = 6$, and $b_3 = 22$. Let $c_m$ ($m \\in \\mathbb{N}^*$) denote the number of terms in the sequence $\\{a_n... | null | d_{k+1} = 3 \times 4^{\frac{(k-1)(k+2)}{2}} | 数列 | math |
51415436-1_1 | null | null | null | null | null | Let $a$ be a positive number. A geometric sequence $\\{a_n\\}$ with first term $a$ is called a $G$-type sequence if $a_1 + 1$, $a_2 + 2$, $a_3 + 3$ also form a geometric sequence. Given $a_n = a^{2^{n-1}}$ for $n \\in \\mathbb{N}^*$, where $\\frac{1}{2} < a < 1$, and $\\{a_n\\}$ is a $G$-type sequence. Define $b_n = a_... | null | b_{n+1} = 3b_{n+1} - b_n, b_{n+1}^2 - b_n b_{n+2} = 5 | 数列 | math |
51415436-1_8 | null | null | null | null | null | It is known that there are two points $A and B$on the image of the function $y=F(x)$. The equation of the line $AB$is $y=G(x)$. If the line $AB$is exactly a tangent to the curve $y=F(x)$($A,B$are tangent points), and $\forall x \in D$($D$is the domain of $F(x)$)$F(x) \geq G(x)$, then the function $y=F(x)$is called a" t... | null | \left(-\infty, \frac{1}{e}\right) | 数列 | math |
51415436-1_9 | null | null | null | null | null | Given an infinite sequence $\\{a_n\\}$, let $S_n$ denote the sum of the first $n$ terms of $\\{a_n\\}$. For $n, k, m \\in \\mathbb{N}^*$, the following three conditions are provided: \n1. $a_n \\in \\mathbb{N}^*$; \n2. $(n-m)S_{n+m} = (n+m)(S_n - S_m)$; \n3. $a_{n-k} + a_{n-k+1} + \\cdots + a_{n-1} = m a_n$ ($n > k$). ... | null | $a_n=2n$or $a_n=n+2025$ | 数列 | math |
51415436-1_21 | null | null | null | null | null | If the sequence $\{a_n\}$satisfies: For any positive integer $n$, there is a positive integer $k$, such that $|a_{n+1}|=| a_n+k| If $is true, the sequence $\{a_n\}$is called a "normalized sequence of order $k$". Let $S_n$be the sum of the first $n$terms of the sequence $\{a_n\}$. If the sequence $\{a_n\}$is a "16-order... | null | -24 | 数列 | math |
51415436-1_33 | null | null | null | null | null | Known set $M_n=\{x \in \mathbb{N}^*| x \leq 2n\} (n \in \mathbb{N}, n \geq 4)$, if there is a matrix
\[
T=\begin{bmatrix}
a_1 & a_2 & \cdots & a_n \\
b_1 & b_2 & \cdots & b_n
\end{bmatrix}
\]
Satisfy:
① $\{a_1,a_2,\cdots,a_n\} \cup \{b_1,b_2,\cdots,b_n\}=M_n$;
② $a_k-b_k=k (k=1,2,\cdots,n)$。
Then the set $M_n$is calle... | null | $M_6$is not a "good set";$M_5$is a "good set", and there are four good number arrays that satisfy $5 \in \{a_1,a_2,a_3,a_4,a_5\}$:
\[
T_1=\begin{bmatrix}
3 & 8 & 10 & 5 & 9 \\
2 & 6 & 7 & 1 & 4
\end{bmatrix}, \quad
T_2=\begin{bmatrix}
8 & 3 & 5 & 10 & 9 \\
7 & 1 & 2 & 6 & 4
\end{bmatrix}, \quad
T_3=\begin{bmatrix}
4 & ... | 数列 | math |
51415436-1_34 | null | null | null | null | null | Given sequences $\\{a_n\\}$ and $\\{b_n\\}$ where $\\{a_n\\}$ is a geometric sequence with $a_1 = \\frac{1}{2}$, $b_2 = 5$, and the relation $a_1b_n + a_2b_{n-1} + \\cdots + a_nb_1 = 4a_n + b_n - 3$ holds for all $n \\in \\mathbb{N}^*$. Find the sum of all elements in the set $M = \\left\\{x \\left| x^2 - \\left(b_n + ... | null | $6N^2+N+2^{2N+1}-\frac{2}{3} \times 4^{\left[\frac{\log_2(6N-1)+1}{2}\right]}-\frac{4}{3}$ | 数列 | math |
51415436-1_37 | null | null | null | null | null | For a given set $A \subseteq \{(a,b) \mid a \geq0, b \geq0\}$, a set $A$is said to have the property $(K_1, K_2)$if there exist nonnegative real numbers $K_1, K_2$such that: $K_1(1+a^2)(1+b^2) \leq (a+b)(1+ab) \leq K_2(1+a^2)(1+b^2)$holds for any $(a,b)\in A$. If the set $\{(a,b) \mid a \geq 0, b \geq 0, a^3+b^3=2\}$ha... | null | $\frac{\sqrt[3]{2}}{\left(\sqrt[3]{2}\right)^2+1}$ | 数列 | math |
51415436-1_39 | null | null | null | null | null | With the rapid development of China's modernization, the great rejuvenation of the Chinese nation is booming, the happiness index of people's lives is rising, and various indicators related to physical health are receiving more and more attention from the people. The value of the health indicator is known as $a \in K=\... | null | abnormal | 数列 | math |
52488504-1_3 | null | null | null | null | null | In the regular quadrangular prism $ABCD-A_1B_1C_1D_1$,$AA_1=4$,$AB=\sqrt{3}$, point $N$is a moving point (excluding boundaries) on the side surface $BCC_1B_1$, and satisfies $D_1N \perp CN$. Record the angle formed by the straight line $D_1N$and the plane $BCC_1B_1$as $\theta$, then the value range of $\tan \theta$is (... | null | \left(\frac{\sqrt{3}}{4}, \frac{1}{2}\right) \cup \left(\frac{\sqrt{3}}{2}, +\infty\right) | 空间向量与立体几何 | math |
52488504-1_9 | null | null | null | null | null | It is known that the complex number $z_1, z_2$satisfies the condition $|z_1| =2, |z_2| =3, 3z_1-2z_2=2-i$, find the value of $z_1z_2$. | null | -\frac{18}{5} + \frac{24}{5} | 复数 | math |
52488504-1_11 | null | null | null | null | null | Let the parabolic equation be $y^2=2px$, and find the trajectory equation of the center of the regular triangle inscribed in the parabola. | null | 9y^2 = 2px - 8p^2 | 平面解析几何 | math |
52488504-1_2-4 | null | null | null | null | null | There are $n(n\geq3)$countries participating in an international conference and voting is required for decision-making. Voting can only be for or against. The number of votes for each country is a positive integer, and the total number of votes for all countries is $10n+1$. In order to prevent one country from becoming... | null | n+2 | 计数原理与概率统计 | math |
51415438-1_19 | null | null | null | null | null | Given positive integers \(k\) and \(m\) where \(2 \leq m \leq k\), a finite sequence \(\{a_n\}\) is called a \((k, m)\)-sequence if it satisfies both of the following conditions. The minimum number of terms for a \((k, m)\)-sequence is denoted by \(G(k, m)\).\n\nCondition 1: Each term of \(\{a_n\}\) belongs to the set ... | null | 12 | 数列 | math |
52340605-1_2-2 | null | null | null | null | null | Define a class of sets: For the set $ \Omega = \{x_1, x_2, \cdots, x_n\} \ (n \geq2, n \in \mathbb{N}) $, if $ \forall x_i \in \Omega $satisfies $|x_i| < 1 $, then $ \Omega $is called a "unit bounded set"; define an operation in the set $ \Omega $: If $ x_i \in \Omega $,$ x_j \in \Omega $, define $ x_i \otimes x_j = \f... | null | H_k = \{0\} | 集合与常用逻辑用语 | math |
52340605-1_3-07 | null | null | null | null | null | Let n be a positive integer and let the set $ A_n = \left\{ \alpha \mid \alpha = (a_1, a_2, \cdots, a_n),| a_i|\leq1, i = 1, 2, \cdots, n \right\} $, for 2 elements in the set $ A_n $$$ \alpha =(x_1, x_2, \cdots, x_n), \beta = (y_1, y_2, \cdots, y_n) $, if $x_1 + x_2 + \cdots + x_n + y_1 + y_2 + \cdots + y_n = 0 $, the... | null | \frac{2n}{n+3} | 集合与常用逻辑用语 | math |
52340605-1_20 | null | null | null | null | null | It is known that the sequence $\{a_n\}$has a total of $n (n \geq2)$items. For the item $a_t (2 \leq t \leq n)$in $\{a_n\}$, if there is $a_k < a_t$for any $k < t$, then $a_t$is called a "local max item" in $\{a_n\}$, and $M$is the collection of all "local max items" in $\{a_n\}$. If the sequence $\{a_n\}$satisfies $a_1... | null | When $n (n \geq5)$is odd, the maximum number of elements in $ M $is $ \frac{a_n + n - 3}{2} $; when $ n (n \geq6) $is even, the maximum number of elements in $ M $is $ \frac{a_n + n - 4}{2} $. | 集合与常用逻辑用语 | math |
52340605-1_24 | null | null | null | null | null | Given the set $ M = \{1, 2, \cdots, n\} $,$ n \in \mathbb{N}^* $, A and B are non-empty subsets of M. Record the set $ A + B = \left\{ \left( (x + y) \mod n \right) \mid x \in A, y \in B \right\} $. If the positive integer n satisfies: there are non-empty sets A and B, such that the intersection of $ A + B $is empty, a... | null | Positive integers other than 1, 2, 4 | 集合与常用逻辑用语 | math |
52340605-1_26 | null | null | null | null | null | The sum of all elements of the number set M is $\Sigma(M)$(when M is an empty set,$\Sigma(M) = 0$is specified). Given the non-empty finite set $ A \subseteq \mathbb{N}^* $, if for any positive integer $ i \leq \Sigma(A) $, there are always two distinct subsets P, Q of A such that $ i = \Sigma(P) - \Sigma(Q) $, then A i... | null | \frac{3^r - 1}{2} | 集合与常用逻辑用语 | math |
52585165-1_7-7 | null | null | null | null | null | Given non-zero vectors $\\vec{a}$, $\\vec{b}$, $\\vec{c}$ satisfying $(\\vec{a} \\cdot \\vec{b})\\vec{c} = (\\vec{b} \\cdot \\vec{c})\\vec{a}$, with $|\\vec{a}| = |\\vec{c}| = 2$ and $\\vec{b} = (-1, \\sqrt{3})$. If the projection vector of $\\vec{a}$ onto $\\vec{c}$ is $-\\frac{1}{2}\\vec{b}$, find the angle between v... | null | \frac{\pi}{3} \text{or} \frac{2\pi}{3} | 平面向量 | math |
52585163-1_1-6 | iVBORw0KGgoAAAANSUhEUgAAAQgAAADDCAIAAADxzBhrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2deUBM+///Tfs+e02bdqWkmxBZUknZl1wpN4RcZItbwu1yJYQkS6Qo7s2SfJDolltZsqdFSVFK0aJ9mmY72++P9+czv76VtMxMg/P4i5nTOe855/08r/fr/X69Xm8ChmFDcHBw/i9Sg90AHBxJBBcGDk434MLAwekGXBg4ON2ACwMHpxtkBrsBODi9BcMwMIkqJSXyFzpuMXC+GWpra318fOrq6lAUFfW1cGHgfBug... | null | null | null | null | <image_1> Please evaluate the following propositions based on the \"Benz Theorem\":\n① If $P$ is the centroid of $\\triangle ABC$, then $\\overrightarrow{PA} + \\overrightarrow{PB} + \\overrightarrow{PC} = \\vec{0}$;② If $a\\overrightarrow{PA} + b\\overrightarrow{PB} + c\\overrightarrow{PC} = \\vec{0}$ holds, then $P$ ... | null | ①②④⑤ | 三角函数与解三角形 | math |
52585163-1_1-7 | null | null | null | null | null | Point $O$ is a fixed point in plane $\\alpha$, $A$, $B$, $C$ are the three vertices of $\\triangle ABC$ in plane $\\alpha$, where $\\angle B$ and $\\angle C$ are the angles opposite to sides $AC$ and $AB$ respectively. There are four propositions:\n\n① For a moving point $P$ satisfying $\\overrightarrow{OP}=\\overright... | null | 2 | 三角函数与解三角形 | math |
52585163-1_4-6 | null | null | null | null | null | In triangle $ABC$, $AB=6$, $AC=3\\sqrt{5}$. Point $M$ satisfies $\\overrightarrow{AM}=\\frac{1}{5}\\overrightarrow{AB}+\\frac{1}{4}\\overrightarrow{AC}$. The line $DE$ passing through point $M$ intersects sides $AB$ and $AC$ at points $D$ and $E$, respectively, with $\\overrightarrow{AD}=\\frac{1}{\\lambda}\\overrighta... | null | \sqrt[3]{10} | 三角函数与解三角形 | math |
52585163-1_6-6 | iVBORw0KGgoAAAANSUhEUgAAAdcAAAC9CAIAAABu/qAeAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOydd3xV9f3/3+/P55xz987ei5CEACFhiyDDPaHuap211a9aWttqbUXrrIP+1LpbFQe4Z8GBKCqIEDYhYWSRvZO71znn8/n9cQNFrRYwIQmepz7ygHDvOe97cvO67/P+vN+vD3LOQUNDQ0NjiCBDHYCGhobGTxpNhTU0NDSGEk2FNTQ0NIYSTYU1NDQ0hhJNhTU0NDSGEk2FNTQ0NIYSTYU1NDQ0hhJhqAMYLjDGGGOK... | null | null | null | null | As shown in <image_1>, let $P$ be an arbitrary point inside triangle $ABC$, with the sides opposite angles $A$, $B$, and $C$ being $a$, $b$, and $c$ respectively. There always exists an elegant equation: $S_{\\triangle PBC}\\overrightarrow{PA} + S_{\\triangle PAC}\\overrightarrow{PB} + S_{\\triangle PAB}\\overrightarro... | null | ①②④ | 三角函数与解三角形 | math |
52530512-1_5-1 | null | null | null | null | null | Let the function $f(x)=2\sin\left(\omega x-\frac{\pi}{6}\right)-1(\omega>0)$have at least two different zeroes on $[\pi,2\pi]$, then the value range of the real number $\omega$is () | [
"$\\left[\\frac{3}{2},+\\infty\\right)$",
"$\\left[\\frac{3}{2},\\frac{7}{3}\\right]\\cup\\left[\\frac{5}{2},+\\infty\\right)$",
"$\\left[\\frac{13}{6},3\\right]\\cup\\left[\\frac{19}{6},+\\infty\\right)$",
"$\\left[\\frac{1}{2},+\\infty\\right)$"
] | A | 三角函数与解三角形 | math |
52487647-1_7-4 | null | null | null | null | null | The inequality $e^{2x} +3a^2x\geq ae^x(3 + x)$holds for any $x \in [1, +\infty)$, then the range of values for the real number $a$is () | null | \left(-\infty, \frac{e}{3}\right] \cup \left\{\frac{e^3}{3}\right\} | 函数与导数 | math |
52342490-1_3-7 | iVBORw0KGgoAAAANSUhEUgAAANgAAADOCAIAAADTzVs8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAATx0lEQVR4nO3dfUwT9x/A8bs+P0F5KvJUqHVaQTKfYG4S5o/4DDhF3dRlim5qXFxMFjNNNjaj23Rm6twcahxZ0KhgNqdsxocZwQ1xCMqcPFtKBQoo0NLSlj7e3e+Pcx0yKDhp76Sf1x+EHr27r/jutb2HghIEgQBANQbVAwAAgRABXUCIgBYgREALECKgBQgR0AKECGjBp0PUarXXrl3bt2/frVu3XBM7Ozs//fTT3NxcHMddE51O... | iVBORw0KGgoAAAANSUhEUgAAAMgAAADMCAIAAAC5q3vfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAATfElEQVR4nO3daVATBxvA8d0kbORKMOEIcgjIISIoRa0H3ketNmOV8ajVWnuNWEedqVXrOLZer1psx+o4Y8XOOE696qgz2lapiidaAYMo5RIQglwhIUCuTbLH+2ErpZwKWXZJn98Hx9ms8TH+XTeb3Q1K0zQCgKMJuB4AOCcIC7ACwgKsgLAAKyAswAoIq1MURVVUVFy8eHHXrl3Nzc0ty1Uq1dq1a+/fv9965draWq1Wy8WYfEWD... | iVBORw0KGgoAAAANSUhEUgAAAMsAAADDCAIAAACjynIJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAWHElEQVR4nO3de1QTZ/oH8ElCbkMSEg5XExBFQC5qWwUtiiDeaI8Xtl5aoZyuu8puu+WirVVb13W32K57tFal7trqWbM9p+yRsrsgtdsWetUCLRQFReQiYKDcSUhCQpJJ5vfH+ItZIIhkJkPg+fxBk5l35n2KX+aeNwwcxxEAKMOkuwAwzUHCALUgYYhery8vL8/Nzf3iiy+sE00m07Fjx4qLi0c0NpvNHR0dTq/RleEzW3l5eUxM... | iVBORw0KGgoAAAANSUhEUgAAANUAAADOCAIAAAAmM9CMAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAZ1ElEQVR4nO2deVgTZ/7AJwch4U64EQVWMCint9UVa63b+mBx3aKIZW3dVgGt55ZduuvVeresfVrUpVrxLD72ERUt1CpuZVVY6wGiIAShoCAQ7hByzvH7Y/hFjKghmTAx8/384RMmk+98iR/ed+Y9WQRBIIB5WLt2LY/H27FjB92JWC4s8M98NDc3s9lsd3d3uhOxXMA/gE7YdCcAMBrwD6AT8A+gE/APoBPwD6AT8A+gE/APIQii... | null | If $k \in \mathbb{Z}$, then the functional graph of the function $f(x) = x^k e^x$may be () | [
"<image_1>",
"<image_2>",
"<image_3>",
"<image_4>"
] | BCD | 函数与导数 | math |
52465786-1_2-2 | null | null | null | null | null | The function $f(x) = e^{ax-1} - x - a (a > 0)$is known. If the two zeros of the function $f(x)$are $x_1, x_2$, respectively, and $x_1 < x_2$, then $x_2$decreases with the increase of $a$. Is this correct (fill in "Yes" or "No")? | null | Yes | 函数与导数 | math |
52342491-1_4-1 | null | null | null | null | null | It is known that the odd function $f(x)$defined on $\mathbb{R}$satisfies $f(x) = f(2-x)$, and $f'(1) =-1 $. When $x \in [1,2)$,$f(x) = \log_2(x-1) + x$, then the number of roots of the equation $f(x)\left[f(x) + \frac{1}{x}\right] = 0$on the interval $[-2,4]$is () | [
"9",
"10",
"17",
"12"
] | C | 函数与导数 | math |
52320219-1_2-4 | iVBORw0KGgoAAAANSUhEUgAAAPgAAAERCAIAAADZj/2iAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2dd1wU19rHZ3YXdll6h4Wl9yZNlKKAIChFLNhrYso1aqIp9ya5yU1yb3JN8iYxPV5jV+yN3mwUCx3FQu91KUvdOjPn/WMiS2hBWNg237/4zDAzz8785plznnOe58AAAIiAQN4hSdoAAoK5gBA6gUJACJ1AISCETqAQyK3QGxsaBAKBpK0gkBbkU+gdHR2nTp7s6e6WtCEE0oIcCp3H4x0/ejQxIWFwaEjSthBICxRJ... | null | null | null | null | As shown in the figure<image_1>, in the box ABCD-A_{1}B_{1}C_{1}D_{1}, AB=AD=2, AA_{1}=4, and P is the moving point on the line segment CD_{1}, then the following proposition is correct () | [
"B_{1}P Plane A_{1}BD",
"The minimum value for B_{1}P+PD is\\frac{2\\sqrt{170}}{5}",
"The maximum value of the sine value of the angle formed by straight line B_{1}P and plane CDD_{1}C_{1} is\\frac{2}{3}",
"The length of the intersection line between the sphere with B being the center of the sphere and 2\\sqr... | ABD | 空间向量与立体几何 | math |
52320219-1_2-6 | iVBORw0KGgoAAAANSUhEUgAAARcAAADdCAIAAAAetaHlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nOx9d1gT2fe+aaQQAqH3DkFAVBRQUWk2xLVi72tZd+3dte/aWFRsa9d1beiKBVAs2EBBkS4IqDSpoaSQXqbk98d9vnnyAwyRJvjh/cMHJzNz78zcc++557znHIxCoejVgx70oA3Afu8O9KAH3R49UtSDHrQVPVLUgx60Ffjv3YH/USgUCgiCZDIZgiBYLFZLS4tIJPbq1UsqlZJIJAwG87072INvQI8UdTZgGGaz2dXV... | null | null | null | null | As <image_1>shown in the figure, in the combination of a semi-cylinder OOOO_{1} and a quadrangular pyramid A-BCDE, F is the moving point on the semicircular arc BC (excluding B, C), FG is a generatrix of the cylinder, and point A is in the plane where the lower bottom surface of the semi-cylinder is located, OB=2OO_{1}... | null | \frac{\sqrt{30}}{6} | 空间向量与立体几何 | math |
52320219-1_2-18 | iVBORw0KGgoAAAANSUhEUgAAAR0AAACxCAIAAAB2of4NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO2dd1xTZxfHc28SSFhhE0hIAoSlori3qHVVbVWsWme1Wq2iiCKgOGu1Wq2z2latdbfuvVFx4Z642Ruy97rz/eOxKS8uRkDF+/2jn0qSe8+9ub+c85znPOeBSJKkUVBQ2BT4fRtAQVEPoXRFQWF7KF1RUNgeSlcUFLaH0hUFhe2hdEVBYXsoXVFQ2B7G+zaAgsLGEARhMpkwDKswNwtBEIPBsLOzYzAYEATVqg2Urijq... | null | null | null | null | <image_1>In the quadrangular pyramid $P-ABCD$, we know $AD \parallel BC$,$BC = 2AD = 2AB = 4$,$\angle BAD = \frac{\pi}{2}$,$PD = CD$, planar $PCD \perp$planar $PBC$. If $PB = 2$, is there a point $Q$on edge $BC$such that the cosine of the angle between plane $PBD$and plane $PAQ$is $\frac{1}{7}$? If it exists, find the ... | null | There is a midpoint $Q$on edge $BC$, so that the cosine of the angle between plane $PBD$and plane $PAQ$is $\frac{1}{7} | 空间向量与立体几何 | math |
52304004-1_13 | null | null | null | null | null | It is known that in the cube $ABCD-A_1B_1C_1D_1$,$M$,$N$are the midpoints of $CC_1$,$C_1D$respectively, then () | [
"The cosine of the angle formed by the straight line $MN$and $A_1C$is $\\frac{\\sqrt{6}}{3}$",
"The cosine of the angle between plane $BMN$and plane $BC_1D_1$is $\\frac{\\sqrt{10}}{10}$",
"There is a point $Q$on $BC_1$such that $B_1Q \\perp BD_1$",
"There is a point $P$on $B_1D$such that $PA \\parallel \\text... | C | 空间向量与立体几何 | math |
52457403-1_1 | null | null | null | null | null | Given that O is the origin of coordinates, the ellipse is $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 The left and right focuses of (a > b > 0)$are $F_1$,$F_2$respectively,$A$is the upper vertex of the ellipse $C$, and $\triangle AF_1F_2$is an isosceles right triangle with an area of 1. The line $l$intersects the ellipse... | null | \frac{3}{2} | 平面解析几何 | math |
52457403-1_2-2 | iVBORw0KGgoAAAANSUhEUgAAAPYAAADYCAIAAABX1onZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4nO1dd3wVxdqemd1T0xvpPYEEpPcqRcRLE1EEKV4VwfYJinoVL4JyFeEiIFdBRFSkNztKCShCKNKiCQRIAqT3dlJO292Z749JhuWEFsgpSc7zu+YeTn1359l333krJIQAJ5xoueDtLYBDAGOMECKEEEIghPQvAIA9sAB7/rpXCQCEAHSD9zthRyB7C+AQYPzGGEMIEUL0gZzElNYQACJh+gbLb4EAOOnteIBOQ4UCYyyK... | null | null | null | null | <image_1>The curve synthesized from the semi-ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (x \geq0)$and the semi-ellipse $\frac{y^2}{b^2} + \frac{x^2}{c^2} = 1 (x < 0)$is called a "fruit circle", where $a^2 = b^2 + c^2$,$a > b > c > 0$. As shown in the figure,$F$,$F_1$, and $F_2$are the focus of the corresponding ell... | null | When the slope is $k=0$, the midpoint of the chord obtained from the "fruit circle" truncation line always falls on an ellipse; when the slope is $k \neq 0$, the midpoint of the chord obtained from the "fruit circle" truncation line cannot always fall on an ellipse. | 平面解析几何 | math |
52145321-1_8-1 | null | null | null | null | null | It is known that the function $f(x)$is defined on $\mathbb{R}$, and $f(1+x)$is an even function, and $f(2+x^3)$is an odd function. When $0 < x \leq1 $,$f(x) = 2-x$, then () | [
"$f(3) = 1$",
"$f(11) < f(-20)$",
"The solution set for $-\\frac{3}{2} < f(x) \\leq-1$is $\\left\\{x \\left| \\frac{5}{2} + 4k < x < \\frac{7}{2} + 4k, k \\in \\mathbb{Z} \\right.\\ right\\}$",
"$\\sum_{k=1}^{2025} f(k) = 1$"
] | BCD | 函数与导数 | math |
52648716-1_9 | null | null | null | null | null | It is known that all terms of the sequence $\{a_n\}$are positive and satisfy the conditions of $a_1 = 1$,$a_n \neq2 $,$a_{n+1} = \frac{1}{2}a_n(4-a_n)$, remember $b_n = \log_2(2-a_n)$, and the sum of the first $n$terms of the sequence $\{a_n\}$is $S_n$. Is there a $S_n \leq0 $()(fill in "Yes" or" No") | null | Yes | 数列 | math |
52648716-1_23 | null | null | null | null | null | Known columns $\{a_n\}$,$a_1 = 1$,$a_{n+1} = 2a_n - \cos \frac{n\pi}{2} + 2\sin \frac{n\pi}{2} (n \in \mathbb{N}^*)$. Let the sum of the first $n$terms of $\left\{n(a_n - 2^n)\right\}$be $T_n$. If $T_m = 2024 (m \in \mathbb{N}^*)$, find $m$. | null | 4048 or 4047 | 数列 | math |
52530510-1_8-8 | null | null | null | null | null | It is known that f(x) = 2\cos(\omega x + \frac{3\pi}{4}), where\omega > 0. If f(\frac{\pi}{4}) = 0 and the function y = f(x) has a minimum but no maximum in (\frac{\pi}{4}, \frac{\pi}{3}), find the value of\omega. | null | 7 or 15 | 三角函数与解三角形 | math |
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