wehub-resource-sync
31 lines
9.3 KiB
JSON
31 lines
9.3 KiB
JSON
{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Let \\( M = \\{x | x + 2 \\geq 0\\} \\) and \\( N = \\{x | x - 1 < 0\\} \\). Determine \\( M \\cap N \\).", "answer": "\\( \\{x | -2 \\leq x < 1\\} \\)", "license": "", "question_number": "fresh_gaokao.1"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given a complex number with real part -1 and imaginary part \\( \\sqrt{3} \\), determine the complex conjugate \\( z \\) of this number.", "answer": "\\( -1 - \\sqrt{3}i \\)", "license": "", "question_number": "fresh_gaokao.2"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given vectors \\( \\mathbf{a} \\) and \\( \\mathbf{b} \\) where \\( \\mathbf{a} + \\mathbf{b} = (2,3) \\) and \\( \\mathbf{a} - \\mathbf{b} = (-2,1) \\), find the value of \\( |\\mathbf{a}|^2 - |\\mathbf{b}|^2 \\).", "answer": "-1", "license": "", "question_number": "fresh_gaokao.3"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Find the coefficient of \\( x \\) in the expansion of \\( \\left(2x - \\frac{1}{x}\\right)^5 \\).", "answer": "80", "license": "", "question_number": "fresh_gaokao.4"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given the parabola \\( C: y^2 = 8x \\) with focus \\( F \\), and a point \\( M \\) on \\( C \\). If the x-coordinate of \\( M \\) is -3 and its distance from \\( F \\) is 5, find the absolute value of \\( MF \\).", "answer": "4", "license": "", "question_number": "fresh_gaokao.5"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "For triangle \\( ABC \\), given that \\( (a + c)(\\sin A - \\sin C) = b(\\sin A - \\sin B) \\), find the value of \\( \\angle C \\).", "answer": "\\( \\frac{\\pi}{3} \\)", "license": "", "question_number": "fresh_gaokao.6"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "For the given function \\( f(x) = 4^x + \\log_2 x \\), determine the value of \\( f \\left( \\frac{1}{2} \\right) \\).", "answer": "1", "license": "", "question_number": "fresh_gaokao.7"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given the ellipse \\( C \\) with focal points at (-2,0) and (2,0), and the distance between the two focal points being \\( \\sqrt{2} \\), find the equation of the ellipse.", "answer": "\\[ \\frac{x^2}{2} + \\frac{y^2}{2} = 1 \\]", "license": "", "question_number": "fresh_gaokao.8"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Find the solution set for the inequality \\( |x - 2| < 1 \\).", "answer": "(1,3)", "license": "", "question_number": "fresh_gaokao.9"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given vectors \\( \\mathbf{a} = (-2,3) \\) and \\( \\mathbf{b} = (1,2) \\), calculate the dot product \\( \\mathbf{a} \\cdot \\mathbf{b} \\).", "answer": "4", "license": "", "question_number": "fresh_gaokao.10"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given that \\( \\tan \\alpha = 3 \\), determine the value of \\( \\tan 2\\alpha \\).", "answer": "\\( -\\frac{3}{4} \\)", "license": "", "question_number": "fresh_gaokao.11"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "In triangle \\( ABC \\), given that \\( a = 4 \\), \\( b = 5 \\), and \\( c = 6 \\), determine the value of \\( \\sin A \\).", "answer": "\\[ \\sin A = \\frac{\\sqrt{7}}{4} \\]", "license": "", "question_number": "fresh_gaokao.12"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given the equation \\((1 + 2023x)^{100} + (2023 - x)^{100} = a_0 + a_1x + a_2x^2 + ... + a_{100}x^{100}\\), where \\(a_0, a_1, ... a_{100} \\in R\\), \\(0 \\leq k \\leq 100\\) and \\(k \\in N\\). Find the value of \\(k\\) for which \\(a_k < 0\\).", "answer": "49", "license": "", "question_number": "fresh_gaokao.13"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Let set \\( M = \\{-2, -1, 0, 1, 2\\} \\) and \\( N = \\{x|x^2 - x - 6 > 0\\} \\). Find the intersection of sets \\( M \\) and \\( N \\).", "answer": "\\{-2\\}", "license": "", "question_number": "fresh_gaokao.14"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "For the sequence {a_n}, the sum of the first n terms is denoted by \\( S_n \\). Given \\( S_5 = 5S_3 - 4 \\), determine the value of \\( S_4 \\)", "answer": "15", "license": "", "question_number": "fresh_gaokao.15"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given \\( y = (x - 1)^2 + ax + \\sin(x + \\frac{\\pi}{2}) \\), find the value of \\( a \\).", "answer": "2", "license": "", "question_number": "fresh_gaokao.16"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "For \\( x \\) and \\( y \\) satisfying the system of inequalities\n\\[\n\\begin{cases}\n-2x + 3y \\leq 3 \\\\\n3x - 2y \\leq 3 \\\\\nx + y = 1\n\\end{cases}\n\\]\nand given \\( z = 3x + 2y \\), find the maximum value of \\( z \\).", "answer": "15", "license": "", "question_number": "fresh_gaokao.17"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given the set \\( M = \\{-2, -1, 0, 1, 2\\} \\) and \\( N = \\{x | x^2 - x - 6 \\geq 0\\} \\), find the intersection \\( M \\cap N \\).", "answer": "\\{-2\\}", "license": "", "question_number": "fresh_gaokao.18"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given \\( z = \\frac{1 - i}{2 + 2i} \\), find the value of \\( z - \\overline{z} \\).", "answer": "\\(-i\\)", "license": "", "question_number": "fresh_gaokao.19"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "For the function \\( f(x) = \\cos\\omega x - 1 \\) (where \\( \\omega > 0 \\)), within the interval [0, \\( 2\\pi \\)], there are 3 points of inflection. Find the value of \\( \\omega \\).", "answer": "[2, 3)", "license": "", "question_number": "fresh_gaokao.20"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given set \\( A = \\{0, -a\\} \\) and \\( B = \\{1, a - 2, 2a - 2\\} \\), if \\( A \\subseteq B \\), find the value of \\( a \\).", "answer": "1", "license": "", "question_number": "fresh_gaokao.21"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "For the expression \\( (2x^3 - \\frac{1}{x})^6 \\), the coefficient of \\( x^2 \\) in its expansion is ________.", "answer": "60", "license": "", "question_number": "fresh_gaokao.22"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Calculate the magnitude of the complex number:\n|2 + i^2 + 2i^3| = ( )", "answer": "\\( \\sqrt{5} \\)", "license": "", "question_number": "fresh_gaokao.23"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "In a school, there are 4 elective courses related to sports and 4 elective courses related to arts. Students need to choose either 2 or 3 courses out of these 8 courses, with at least one course selected from each category. How many different course selection schemes are there? (Answer in digits)\"", "answer": "64", "license": "", "question_number": "fresh_gaokao.24"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given that \\( \\sin(\u03b1 - \u03b2) = \\frac{1}{3} \\) and \\( \\cos \u03b1 \\sin \u03b2 = \\frac{1}{6} \\), find the value of \\( \\cos(2\u03b1 + 2\u03b2) \\)", "answer": "\\( \\frac{1}{9} \\)", "license": "", "question_number": "fresh_gaokao.25"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Two students, A and B, each choose 2 books to read from a selection of 6 extracurricular books. How many ways can they choose such that exactly one of the books they both selected is the same?", "answer": "120", "license": "", "question_number": "fresh_gaokao.26"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given that \\( a \\in (0,1) \\) and the function \\( f(x) = a^x + (a+1)^x \\) is increasing on the interval \\( (0, +\\infty) \\), determine the range of \\( a \\).", "answer": "\\[ \\left[ \\frac{\\sqrt{5}-1}{2}, -1 \\right) \\]", "license": "", "question_number": "fresh_gaokao.27"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given the sets \\(A = \\{0, -a\\}\\) and \\(B = \\{1, a, 2a - 2\\}\\). If \\(A\\) is a subset of \\(B\\), determine the value of \\(a\\).", "answer": "1", "license": "", "question_number": "fresh_gaokao.28"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given acute angle \\( \\alpha \\), with \\( \\cos \\alpha = \\frac{1+\\sqrt{5}}{4} \\), find the value of \\( \\sin \\frac{\\alpha}{2} \\).", "answer": "\\( \\frac{-1+\\sqrt{5}}{4} \\)", "license": "", "question_number": "fresh_gaokao.29"}
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{"data_topic": "FreshGaokaoMath2023", "data_source": "FreshGaokaoMath2023", "question": "Given vectors \\(a\\) and \\(b\\), such that \\( |a-b| = \\sqrt{3} \\), \\( |a+b| = |2a-b| \\), find the value of \\( |b| \\).", "answer": "\\( \\sqrt{3} \\)", "license": "", "question_number": "fresh_gaokao.30"}
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