| {"question": "What equation is this called if it lies on the graph and goes in a continuous straight line", "options": ["Linear Equation", "Quadratic equation", "Cubic equation", "Inequality"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Given the form of the slope-intercept, what do m and b represent respectively?", "options": ["m is the slope, b is the x-intercept", "m is the y-intercept, b is the slope", "m is the slope, b is the y-intercept", "m is the x-intercept, b is the slope"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Consider the slope-intercept form: y=mx+b. If the slope m is positive, is the line uphill or downhill", "options": ["Downhill", "Vertical", "Uphill", "Horizontal"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Consider the slope-intercept form: y=mx+b. If the slope m is negative, is the line uphill or downhill", "options": ["Uphill", "Downhill", "Vertical", "Horizontal"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Consider the slope-intercept form: y=mx+b. If the slope m is 0, is the line horizontal or vertical?", "options": ["Vertical", "Horizontal", "Uphill", "Downhill"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Consider the slope-intercept form: y=mx+b. If the slope m is undefined, is the line horizontal or vertical?", "options": ["Uphill", "Horizontal", "Vertical", "Downhill"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "To memorize the slope, we use VUX HOY. What does V represent?", "options": ["x=#", "undefined", "vertical line", "Horizontal"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "To memorize the slope, we use VUX HOY. What does U represent?", "options": ["vertical line", "x=#", "undefined", "Horizontal"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "To memorize the slope, we use VUX HOY. What does V represent?", "options": ["vertical line", "undefined", "Horizontal", "x=#"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "To memorize the slope, we use VUX HOY. What does H represent?", "options": ["vertical line", "x=#", "Horizontal", "undefined"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "To memorize the slope, we use VUX HOY. What does O represent?", "options": ["x=#", "slope=0", "Horizontal", "x=0"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "To memorize the slope, we use VUX HOY. What does Y represent?", "options": ["slope=0", "y=#", "y=0", "vertical line"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Given the coordinates of two points$(X_2, Y_2), (X_1, Y_1)$, how to calculate the slope m?", "options": ["$m=(X_2-X_1)/(Y_1-Y_2)$", "$m=(X_1-X_2)/(Y_2-Y_1)$", "$m=(X_2-X_1)/(Y_2-Y_1)$", "$m=(Y_2-X_2)/(Y_1-X_1)$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Consider the quadratic formula. For $ax^2+bx+c=0$,How to represent x with a, b, and c?", "options": ["$x=\\frac{-b\\pm \\sqrt{4ac-b^2}}{2a}$", "$x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}$", "$x=\\frac{-b\\pm \\sqrt{4ac}}{2a}$", "$x=\\frac{-b\\pm \\sqrt{b^2}}{2a}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "What does the discriminant tell as?", "options": ["The number of roots the equation has", "The value of roots the equation has", "The existence of roots the equation has", "The type of roots the equation has"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "What is the formula of the discriminant?", "options": ["$D=b^2$", "$D=4ac$", "$D=4ac-b^2$", "$D=b^2-4ac$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Given the formula of the discriminant: $D=b^2-4ac$. If D>0, how many imaginary roots are there?", "options": ["positive infinity", "2", "1", "0"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Given the formula of the discriminant: $D=b^2-4ac$. If D>0, how many real roots are there?", "options": ["1", "0", "2", "positive infinity"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Given the formula of the discriminant: $D=b^2-4ac$. If D=0, how many real roots are there?", "options": ["2", "1", "positive infinity", "0"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Given the vertex form: $y=a(x-h)^2+k$. How to represent the coordinates of the vertex?", "options": ["(k,h)", "(h,k)", "(h,0)", "(0,k)"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "There are some graphical terms for quadratic functions. What is the highest point of \"y\" in the graph called?", "options": ["Middle", "Minimum", "Maximum", "Extreme point"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "There are some graphical terms for quadratic functions. What is the lowest point of \"y\" in the graph called?", "options": ["Extreme point", "Minimum", "Maximum", "Middle"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Please explain the concept of the domain for f (x).", "options": ["The set of real numbers", "All the values \"y\" for which f(x) is defined.", "All the values \"x\" for which f(x) is defined.", "set of integer"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Please explain the concept of the range for f (x).", "options": ["All the values \"y\" for which f(x) is defined.", "The set of real numbers", "set of integer", "All the values \"x\" for which f(x) is defined."], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Explain the concept of x-intercepts: values that make f(x)=<<<Answer>>>\n and lies on the x-axis.", "options": ["#", "0", "$\\infty$", "$-\\infty$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Explain the concept of y-intercepts: values that make f(x)=<<<Answer>>>\n and lies on the y-axis.", "options": ["0", "1", "$\\infty$", "$-\\infty$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Which axis do we usually use when we are making an equation by looking at the two-dimensional graph?", "options": ["y -axis", "x -axis", "z -axis", "x -axis and y -axis"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "$a^{2}-b^{2}$=", "options": ["All the other options are wrong", "$(a+b)(a-b)$", "$(a+b)(a+b)$", "$(a-b)(a-b)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$(a+b)(a-b)$=", "options": ["$b^{2}$", "$a^{2}-b^{2}$", "$a^{2}+b^{2}$", "$a^{2}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$(x+y)^{2}$=", "options": ["$x^{2}-2xy-y^{2}$", "$x^{2}+2xy+y^{2}$", "$x^{2}-2xy+y^{2}$", "$x^{2}+2xy-y^{2}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$x^{2}+2xy+y^{2}$=", "options": ["$(x-y)^{2}$", "$(x+y)^{2}$", "$(x-y)^{3}$", "$(x+y)^{3}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$(x-y)^{2}$=", "options": ["$x^{2}-2xy-y^{2}$", "$x^{2}+2xy+y^{2}$", "$x^{2}+2xy-y^{2}$", "$x^{2}-2xy+y^{2}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$x^{2}-2xy+y^{2}$=", "options": ["$(x+y)^{3}$", "$(x-y)^{2}$", "$(x-y)^{3}$", "$(x+y)^{2}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "What parts does a complex number contain?", "options": ["two imaginary numbers", "a real number and an imaginary number", "All the other options are wrong", "two real numbers"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "Complex number a+bi, \"a\" is", "options": ["0", "a real number", "All the other options are wrong", "an imaginary number"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "Complex number a+bi, \"b\" is", "options": ["All the other options are wrong", "a real number", "an imaginary number", "0"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "$\\sqrt[]{-1}$=", "options": ["1", "-i", "i", "-1"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "$i^{2}$=", "options": ["1", "i", "-i", "-1"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "$i^{3}$=", "options": ["1", "i", "-1", "-i"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "$i^{4}$=", "options": ["i", "-i", "-1", "1"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "$i^{5}$=", "options": ["-1", "1", "-i", "i"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "What are complex conjugate numbers of $a+bi$?", "options": ["$a-bi$", "$a+bi$", "$abi$", "All the other options are wrong"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "What are complex conjugate numbers of $a-bi$?", "options": ["$a-bi$", "$abi$", "$a+bi$", "All the other options are wrong"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Complex Numbers"} | |
| {"question": "In formula $y=a\\left |x-h\\right |+k$, what is the vertex?", "options": ["(k,h)", "(0,k)", "(h,0)", "(h,k)"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "In formula $y=a\\left |x-h\\right |+k$, what is the axis of symmetry?", "options": ["x=k", "x=h", "y=h", "y=k"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$a^{m}\\times a^{n}= $?", "options": ["$a^{m}+a^{n}$", "$a^{m+n}$", "$a^{m*n}$", "$a^{m^{n} }$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$a^{m}\\times a^{n}= $?", "options": ["$a^{m+n}$", "$a^{m*n}$", "$a^{m-n}$", "$a^{m}+a^{n}$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$\\left ( a^{m} \\right ) ^{n} $=?", "options": ["$a^{m+n}$", "$a^{mn} $", "$a^{m*n}$", "$a^{m}+a^{n}$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "When $a\\ne 0$, $a^{0}=$?", "options": ["a", "-1", "0", "1"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$a^{-4}=$?", "options": ["$1- a^{4} $", "$\\frac{1}{a^{-4} } $", "$1-+a^{4} $", "$\\frac{1}{a^{4} } $"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$\\left ( ab \\right ) ^{n} $?", "options": ["$a^{n}b$", "$ab^{n}$", "$\\left ( a^{n} \\right ) \\left ( b^{n} \\right ) $", "$\\left ( a^{n} \\right ) +\\left ( b^{n} \\right ) $"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "$\\left ( a/b \\right ) ^{n}=$?", "options": ["$\\left ( ab \\right ) ^{n}$", "$a^{n} /b^{n} $", "$a^{n} +b^{n} $", "$a^{n} -b^{n} $"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "A rational exponent has a base that is raised to the power with a <<<>>>?", "options": ["integer", "decimal", "fraction", "irrational number"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "The definition of radicals is that an expression that uses a <<<>>>?", "options": ["square root", "root", "fourth root", "cube root"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "To get rid of the radical in an equation, <<<>>> both sides.", "options": ["square", "cube root", "fourth root", "cube"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Whenever you see a radical in the denominator, you have to rationalize it.To get rid of it, multiply <<<>>> with the same term as that in the denominator.", "options": ["the top and bottom", "neither the top nor the bottom", "only the bottom", "only the top"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Whenever you take the square root of both sides, it ends up being <<<>>>.", "options": ["neither $+$ nor $-$", "only $+$", "$+/-$", "only $-$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Which of the following statements about the definition of an equation is correct?", "options": ["solving one equation for two or more variables", "solving one equation for one variable", "solving two different equations for two or more variables", "solving two different equations for three variables"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Which of the following statements is correct?", "options": ["The functions intersect at only one point, meaning that they have parallel and equal slopes and the same reciprocal.", "The functions intersect at only one point, meaning that they have perpendicular slopes and the same reciprocal.", "The functions intersect at only one point, meaning that they have parallel slopes and opposite reciprocals.", "The functions intersect at only one point, meaning that they have perpendicular slopes and opposite reciprocals."], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following statements is correct?", "options": ["Two equations do not intersect, meaning that they have perpendicular slopes parallel and equal y-intercepts.", "Two equations do not intersect, meaning that they have parallel and equal slopes parallel and equal y-intercepts.", "Two equations do not intersect, meaning that they have parallel and equal slopes parallel but different y-intercepts.", "Two equations do not intersect, meaning that they have perpendicular slopes parallel but different y-intercepts."], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Which of the following statements about infinite solutions is correct?", "options": ["Two equations overlap, meaning that they have the same slopes and different y-intercepts", "Two equations overlap, meaning that they have the same slopes and y-intercepts.", "Two equations overlap, meaning that they have different slopes and y-intercepts", "Two equations overlap, meaning that they have different slopes and the same y-intercepts"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Which of the following statements about exponential equations is incorrect?", "options": ["In the formula $y=ab^{x}$, a is not equal to 0.", "The formula of exponential equations is $y=ab^{x}$.", "The formula of exponential equations is $y=a^{x}b^{x} $.", "In the formula $y=ab^{x}$, b and $b > 0$ is not equal to 1."], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "The initial value is $a$, the annual growth rate is $r$, what is the formula for calculating the value after $t$ years?", "options": ["$P=a(1+t)^{r}$", "$P= (1+t)^{r}$", "$P=a(1+r)^{t}$", "$P=(1+r)^{t}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives"} | |
| {"question": "Which of the following is not a sign of greater than? ", "options": ["above", "at least", "exceeding", "more than"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following is not a sign of greater than or equal? ", "options": ["at least", "ferwer than", "at minimun", "no less than"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following is not a sign of less than? ", "options": ["below", "less than", "at least", "fewer than"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following is not a sign of less than or equal? ", "options": ["at least", "does not exceed", "at most", "no more than"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "When graphing, greater than symbol the line is <<<>>>; the section <<<>>> the line is shaded.", "options": ["[\"dashed\",\"below\"]", "[\"dashed\",\"above\"]", "[\"solid\",\"below\"]", "[\"solid\",\"above\"]"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "When graphing, greater than or equal to symbol the line is <<<>>>; the section <<<>>> the line is shaded.", "options": ["[\"solid\",\"below\"]", "[\"dashed\",\"below\"]", "[\"dashed\",\"above\"]", "[\"solid\",\"above\"]"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "When graphing, less than symbol the line is <<<>>>; the section <<<>>> the line is shaded.", "options": ["[\"dashed\",\"below\"]", "[\"solid\",\"above\"]", "[\"dashed\",\"above\"]", "[\"solid\",\"below\"]"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "When graphing, less than or equal to symbol the line is <<<>>>; the section <<<>>> the line is shaded.", "options": ["[\"solid\",\"above\"]", "[\"dashed\",\"below\"]", "[\"dashed\",\"above\"]", "[\"solid\",\"below\"]"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "When multiplying and dividing by a negative number, we should <<<>>> the <<<>>> symbol.", "options": ["[\"flip\",\"inequality\"]", "[\"not flip\",\"inequality\"]", "[\"flip\",\"equality\"]", "[\"not flip\",\"equality\"]"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} | |
| {"question": "What kind of terms have the same variable and are raised to the same power?", "options": ["Equal Terms", "Combining Like Terms", "Right Terms", "Good Terms"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which is used to find the length of the third side of a right triangle, given the values of two sides already？", "options": ["Pythagorean Theorem", "Right Triangle Theorem", "Newton Theorem", "Right Theorem"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "What is the formula of Pythagorean Theorem?", "options": ["$a^2+b^2 = c$", "$a^2+b^2 = 2c^2$", "${(a+b)}^2 = c^2$", "$a^2+b^2 = c^2$"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "Which are 2 angles whose sum is $180^{\\circ}$？", "options": ["Alternate Exterior Angles", "Complementary Angles", "Alternate Interior Angles", "Supplementary Angles"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "Which are 2 angles opposite of each other and are made by intersecting lines？", "options": ["Alternate Exterior Angles", "Vertical Angles", "Complementary Angles", "Alternate Interior Angles"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "Which are 2 angles lie outside the two parallel lines that are equal in measure if two parallel lines are intersected by a transversal？", "options": ["Alternate Exterior Angles", "Supplementary Angles", "Alternate Interior Angles", "Complementary Angles"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "Which are 2 angles lie within the two parallel lines that are supplementary if two parallel lines are intersected by a transversal？", "options": ["Alternate Exterior Angles", "Same-side Interior Angles", "Same-side Exterior Angles", "Alternate Interior Angles"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "What is the equation formula of a circle, where (h, k) is the center point of the circle, r is the radius?", "options": ["$(x-h)^2+(y-k)^2=2r^2$", "$(x-h)^2+(y-k)^2=r^2$", "$(x+h)^2+(y+k)^2=r^2$", "$(x-h)^2-(y-k)^2=r^2$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Which one is wrong about similar polygons？", "options": ["different shape", "corresponding sides are proportional", "corresponding angles are congruent", "different size"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "What is the formula of arc of a circle？", "options": ["$(2\\pi r) central angle \\cdot 360^{\\circ} = arc length$", "$(2\\pi r) central angle/360^{\\circ} = arc length$", "$(2\\pi r) central angle/180^{\\circ} = arc length$", "$(\\pi r) central angle/360^{\\circ} = arc length$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "What is the distance formula between two points?", "options": ["$d^2 = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$", "$d^2 = (x_2-x_1)^2+(y_2-y_1)^2$", "$d^2 = \\sqrt{(x_2+x_1)^2+(y_2+y_1)^2}$", "$d = \\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "What is the midpoint formula?", "options": ["$midpoint =(x_1+y_1,x_2+y_2)$", "$midpoint =(x_1+x_2,y_1+y_2)$", "$midpoint =(\\frac{x_1+x_2}{2}/2, \\frac{y_1+y_2}{2})$", "$midpoint =((x_1+x_2)/2,(y_1+y_2)/2)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "What is the formula of circumference of the circle?$C$ is the circumference, $r$ is the radius.", "options": ["$2C=\\pi r$", "$C=\\pi r$", "$C=2 \\pi r$", "$C=2r$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "What is the formula of area of a circle?$r$ is the radius.", "options": ["$area = \\pi r^2$", "$area = \\pi r$", "$area = 2\\pi r$", "$area = {\\pi r}^2$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "What is the formula of area of a sector?", "options": ["$A = (\\pi r^2) /360^ {\\circ}$", "$A = (\\pi r^2)measure of the central angle /360^ {\\circ}$", "$A = (\\pi r^2)measure of the central angle /180^ {\\circ}$", "$A = (\\pi r^2)measure of the central angle /90^ {\\circ}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "What is the formula of nscribed angles?", "options": ["$\\angle ABC = measure of intercepted arc$", "$\\angle ABC = 2 \\ast measure of intercepted arc$", "$\\angle ABC = \\frac{1}{2} \\ast measure of intercepted arc$", "$\\angle ABC = 180^{\\circ} \\ast measure of intercepted arc$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "The mean is defined as <<<Answer>>> of the numbers.", "options": ["the average", "the product", "the maximum", "the sum"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "The formula for the mean is represented as<<<Answer>>>.", "options": ["$number of data * sum of all data$", "$sum of all data + number of data$", "$sum of all data / number of data$", "$number of data - sum of all data$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "What is the definition of the median in a data set?", "options": ["The largest number", "The middle number", "The smallest number", "The average of the numbers"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "What is the definition of the mode in a data set?", "options": ["The number that shows up the most", "The average of the numbers", "The middle number", "The smallest number"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "How many modes can a data set have?", "options": ["More than one", "Only one", "No more than two", "None"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "What is the range in a data set?", "options": ["The sum of the highest and lowest values", "The average of the highest and lowest values", "The product of the highest and lowest values", "The difference between the highest and lowest values"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "What does the standard deviation represent in a data set?", "options": ["The sum of the values", "The average of the values", "The range of the values", "The measure of variation of the values from the mean"], "answer": "D", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "What does the term \"correlation\" refer to in statistics?", "options": ["The average of two variables", "The difference between two variables", "The relationship between two variables", "The sum of two variables"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "Correlation is often represented by the symbol ?", "options": ["r", "d", "c", "q"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "When the correlation coefficient \"r\" is greater than zero, this indicates that the two variables are:", "options": ["positively correlated", "Uncorrelated", "Perfectly correlated", "negatively correlated"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "When the correlation coefficient \"r\" is less than zero, this indicates that the two variables are:", "options": ["Negatively correlated", "Perfectly correlated", "Positively correlated", "Uncorrelated"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "When the correlation coefficient \"r\" is equal to zero, it indicates that there is no linear relationship between the two variables, which is also known as:", "options": ["Negative correlation", "No correlation", "Perfect correlation", "Positive correlation"], "answer": "B", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "What is the definition of a ratio?", "options": ["The number of times one number is contained within another", "The product of two numbers", "The sum of two numbers", "The difference between two numbers"], "answer": "A", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "Given two numbers a and b, what is the ratio of a to b?", "options": ["$a + b$", "$b : a$", "$a : b$", "$a * b$"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "What does the term \"proportion\" refer to in mathematics?", "options": ["A geometric figure", "A comparison of two numbers", "A statistical measure of variation", "Two fractions that are equal to each other"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "The form of a proportion is typically expressed as:", "options": ["$(a + b) / c = d$", "$a + b = c + d$", "$ a / b = c / d$", "$a * b = c * d$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "How do you calculate a percentage?", "options": ["By dividing the part by the whole and then multiplying by 100", "By adding the part to the whole and then dividing by 100", "By subtracting the part from the whole and then dividing by 100", "By multiplying the part by the whole and then dividing by 100"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "To determine the percentage change from an old value to a new value, which formula should be used?", "options": ["$(New Value / Old Value) * 100%$", "$(Old Value - New Value) / New Value * 100%$", "$(Old Value + New Value) / 2 * 100%$", "$(New Value - Old Value) / Old Value * 100%$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If the result of the percentage change calculation is positive, what does it signify?", "options": ["The value has increased.", "The value has remained the same.", "The value has decreased.", "The calculation is incorrect."], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} | |
| {"question": "If the result of the percentage change calculation is negative, what does it signify?", "options": ["The calculation is incorrect.", "The value has increased.", "The value has decreased.", "The value has remained the same."], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Applications of Inequalities"} | |
| {"question": "What is the definition of probability?", "options": ["The number of desired outcomes", "The number of all possible outcomes", "The desired outcome divided by all possible outcomes", "The sum of all possible outcomes"], "answer": "C", "topic": "High--Intermediate Probability and Statistics--Foundations of Probability Theory"} | |
| {"question": "What is the correct formula for sin?", "options": ["$Sin = hypotenuse/opposite$", "$Sin = opposite/hypotenuse$", "$Sin = adjacent/hypotenuse$", "$Sin = hypotenuse/adjacent$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "What is the correct formula for cos?", "options": ["$Cos = opposite/hypotenuse$", "$Cos = hypotenuse/opposite$", "$Cos = adjacent/hypotenuse$", "$Cos = hypotenuse/adjacent$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "What is the correct formula for tan?", "options": ["$Tan = hypotenuse/opposite$", "$Tan = opposite/adjacent$", "$Tan = adjacent/hypotenuse$", "$Tan = hypotenuse/adjacent$"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "To convert radians to degrees, you multiply the radians by what factor?", "options": ["$(180°/π)$", "$(π/180°)$", "180° radians", "π radians"], "answer": "A", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "To convert degrees to radians, you multiply the degrees by what factor?", "options": ["π radians", "$(180°/π)$", "180° radians", "$(π/180°)$"], "answer": "D", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "If the elements forming two sets are the same, then these two sets can be called?", "options": ["Identical", "Equal", "Irrelevant", "Similar"], "answer": "B", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "What type of set does $N^{*} $ or $N_{+}$ represent?", "options": ["Set of negative integers", "Set of irrational numbers", "Set of positive integers", "Set of complex numbers"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What letter represents the set of integers?", "options": ["A", "S", "Z", "D"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What letter is used to represent the set of rational numbers?", "options": ["R", "W", "Q", "E"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What does R represent?", "options": ["The set of irrational numbers", "The set of integers", "The set of real numbers", "The set of complex numbers"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following is not a method of representing a set?", "options": ["Interval notation", "Graphical representation", "Analogy", "Description"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "For two sets A and B, if any element of set A is also an element of set B, then set A is called a?", "options": ["Intersection", "Empty set", "Subset", "Proper subset"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Set A is a subset of set B, how is this represented?", "options": ["$A \\in B$", "$A \\le B$", "$A \\subseteq B$", "$A \\approx B$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If a set $A \\sqsubseteq B$, but there exists an element $x \\in B$ and $x \\notin A$, then set A is called what of set B?", "options": ["Proper subset", "Subset", "Intersection", "Union"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Set A is a proper subset of set B, how is this represented?", "options": ["$A \\subseteq B$", "$A \\sqsupseteq B$", "$A \\subsetneeq B$", "$A = B$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is the name of a set that does not contain any elements?", "options": ["Empty set", "Non-empty proper subset", "Intersection", "Union"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is the representation of the empty set?", "options": ["$ \\ni $", "$ \\subset $", "$ \\iff $", "$ \\varnothing $"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is the subset that every set has?", "options": ["Non-empty set", "Complex number set", "Real number set", "Empty set"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If set A contains n elements, how many subsets does set A have?", "options": ["${n}^{2}$", "${n}^{3}$", "$2^{n-1}$", "$2^{n}$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If set A contains n elements, how many proper subsets does set A have?", "options": ["${n}^{3}-1$", "$2^{n-1}$", "${n}^{2}$", "$2^{n}-1$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Generally, the set consisting of all elements that belong to set A or set B is called what? What is it denoted by?", "options": ["Union; $A\\cup B$", "Union; $A\\cap B$", "Intersection; $A\\cup B$", "Intersection: $A\\cap B$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Generally, the set consisting of all elements that belong to set A or set B is called what? What is it denoted by?", "options": ["Union; $A\\cap B$", "Union; $A\\cup B$", "Intersection; $A\\cup B$", "Intersection: $A\\cap B$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "For a set A, the set composed of all elements in the universal set U that do not belong to set A is called the complement of A with respect to U. How is this represented in a formula?", "options": ["${C}_{U}A=\\left \\{ x\\mid x\\in U\\right \\}$", "${C}_{U}A=\\left \\{ x\\mid x\\notin U$, and $x\\in A\\right \\}$", "${C}_{U}A=\\left \\{ x\\mid x\\in U$, and $x\\notin A\\right \\}$", "${C}_{U}A=\\left \\{ x\\mid x\\notin U\\right \\}$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is a proposition?", "options": ["Declarative statement", "All the other options are wrong", "A statement that can be judged true or false", "Inferential statement"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following is not a logical connective?", "options": ["but", "not", "or", "and"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is a simple proposition?", "options": ["A proposition that contains at least one logical connective", "A proposition that contains one logical connective", "A proposition that does not contain logical connectives", "A proposition that contains two logical connectives"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is a compound proposition?", "options": ["All the other options are wrong", "A proposition made up of simple propositions and logical connectors", "A proposition with at least two logical connectors", "Multiple propositions"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If two statements are each other's contrapositives, then these two statements:", "options": ["Have no truth relationship", "One is true and the other is false", "Both true or both false", "Both are false"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If two propositions are either inverses or contrapositives of each other, then these two propositions:", "options": ["are both wrong", "both true or both false", "have no truth-falsehood relationship", "one true and one false"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Generally, if it is known that $p \\Rightarrow q$, then", "options": ["p is a necessary and sufficient condition for q", "All the other options are wrong", "p is a sufficient condition for q, q is a necessary condition for p", "p is a necessary condition for q, q is a sufficient condition for p"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If $p \\Leftrightarrow q$, then", "options": ["All the other options are wrong", "p is a necessary condition for q, q is a sufficient condition for p", "p is a necessary and sufficient condition for q", "p is a sufficient condition for q, q is a necessary condition for p"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "In general, if it is known that $p \\Rightarrow q$, then", "options": ["p is a necessary condition for q, q is a sufficient condition for p", "All the other options are wrong", "p is a necessary and sufficient condition for q", "p is a sufficient condition for q, q is a necessary condition for p"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If $p \\Rightarrow q$ but $q \\nRightarrow p$, then", "options": ["p is a sufficient but not necessary condition for q", "p is a necessary but not sufficient condition for q", "p is a necessary and sufficient condition for q", "All the other options are wrong"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If $p \\nRightarrow q$, but $q \\Rightarrow p$, then", "options": ["All of the above are incorrect", "p is a sufficient but not necessary condition for q", "p is a necessary but not sufficient condition for q", "p is a necessary and sufficient condition for q"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If $p \\Rightarrow q$ and $q \\Rightarrow p$, then p is what kind of condition for q?", "options": ["A necessary and sufficient condition", "A necessary but not sufficient condition", "A sufficient but not necessary condition", "Neither a sufficient nor a necessary condition"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "If $p \\nRightarrow q$ and $q \\nRightarrow p$, then what condition is p to q?", "options": ["Neither sufficient nor necessary condition", "Necessary but not sufficient condition", "Sufficient and necessary condition", "Sufficient but not necessary condition"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following options is not a form of a compound proposition?", "options": ["$p \\land q$", "$p \\lor q$", "$B\\not\\subset A$", "$\\neg p$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Under what condition is the compound statement \"p or q\" false?", "options": ["p is true and q is false", "p is true and q is true", "p is false and q is true", "p is false and q is false"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Under what condition is the compound statement \"p and q\" true?", "options": ["p is true q is false", "p is false q is true", "p is true q is true", "p is false q is false"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is a universal proposition?", "options": ["The proposition represented by the symbol \"$ \\phi $\"", "The proposition represented by the symbol \"$ \\forall $\"", "The proposition represented by the symbol \"$ \\exists $\"", "The proposition represented by the symbol \"$ \\alpha $\""], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following is a universal quantifier?", "options": ["$ \\forall $", "$ \\alpha $", "$ \\phi $", "$ \\exists $"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is a particular proposition?", "options": ["A proposition represented by the symbol \"$ \\phi $\"", "A proposition represented by the symbol \"$ \\alpha $\"", "A proposition represented by the symbol \"$ \\exists $\"", "A proposition represented by the symbol \"$ \\forall $\""], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Which of the following is an existential quantifier?", "options": ["$ \\forall $", "$ \\alpha $", "$ \\exists $", "$ \\phi $"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Given the universal statement p: $\\forall x \\in M , p(x)$, what is its negation $\\neg p$?", "options": ["$\\exists {x}_{0}\\in M, p({x}_{0})$", "$\\forall x \\in M , p(x)$", "$\\forall x \\in M ,\\neg p(x)$", "$\\exists {x}_{0}\\in M,\\neg p({x}_{0})$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Given the propositional function p: $\\exists {x}_{0}\\in M, p({x}_{0})$, what is its negation $\\neg p$?", "options": ["$\\forall x \\in M ,\\neg p(x)$", "$\\exists {x}_{0}\\in M, p({x}_{0})$", "$\\forall x \\in M, p(x)$", "$\\exists {x}_{0}\\in M,\\neg p({x}_{0})$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is the negation of a universal proposition?", "options": ["Existential proposition", "All the other options are wrong", "Existential statement", "Universal proposition"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "What is the negation of an existential proposition?", "options": ["Universal proposition", "Existential proposition", "All the other options are wrong", "Existence proposition"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "A quantity that has both <<<Answer 1>>> and <<<Answer 2>>> is called a vector", "options": ["magnitude, velocity", "force, magnitude", "direction, velocity", "magnitude, direction"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "A line segment with direction is called a directed line segment. What are the three elements that constitute a directed line segment?", "options": ["[\"start point\", \"direction\", \"length\"]", "[\"start point\", \"end point\", \"length\"]", "[\"start point\", \"direction\", \"end point\"]", "[\"end point\", \"direction\", \"length\"]"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "The magnitude of vector $\\vec{AB}$, which is also the length (or norm) of vector $\\vec{AB}$, is denoted by <<<Answer 1>>>; a vector with length zero is called a <<<Answer 2>>>; a vector with length equal to one unit is called a <<<Answer 3>>>.", "options": ["$\\lvert\\vec{AB} \\rvert$, zero vector, unit vector", "$\\lvert \\vec{AB} \\rvert$, unit vector, zero vector", "$\\vec{AB} $, zero vector, unit vector", "$\\lvert \\vec{AB} \\rvert$, parallel vector, unit vector"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sets and Common Logical Terms"} | |
| {"question": "Non-zero vectors that have the same or opposite direction are called <<<Answer>>>.", "options": ["opposite vectors", "unit vectors", "zero vectors", "parallel vectors"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "is parallel to any vector", "options": ["Parallel vector", "Zero vector", "Opposite vector", "Unit vector"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "If the angle between $\\vec{a}$ and $\\vec{b}$ is $\\theta$, then the length of the projection on the top is <<<Answer>>>", "options": ["$ \\lvert \\vec{a} \\rvert \\cdot \\sin \\theta$", "$ \\lvert \\vec{a} \\rvert \\cdot \\cot \\theta$", "$ \\lvert \\vec{a} \\rvert \\cdot \\tan \\theta$", "$ \\lvert \\vec{a} \\rvert \\cdot \\cos \\theta$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Vectors of equal length and the same direction are called <<<Answer>>>", "options": ["zero vector", "unit vector", "opposite vector", "equal vectors"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "$ \\lvert \\vec{a}+\\vec{b} \\rvert $<<<Answer>>>$ \\lvert \\vec{a} \\rvert+\\lvert \\vec{b} \\rvert $", "options": ["$ > $", "$ \\leq $", "$ \\geq $", "$ < $"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "The vector that has the same magnitude as $\\vec{a}$ but in the opposite direction is called the <<<Answer>>> of $\\vec{a}$", "options": ["opposite vector", "zero vector", "unit vector", "equal vector"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "The product of a real number $\\Lambda$ and a vector $a$ is a vector. This operation between a real number and a vector is called scalar multiplication, denoted as: $\\Lambda \\cdot \\vec{a}$. When $\\Lambda$<<<Answer 1>>>$0$, the direction of $\\Lambda \\cdot \\vec{a}$ is the same as the direction of $\\vec{a}$; when $\\Lambda$<<<Answer 2>>>$0$, the direction of $\\Lambda \\cdot \\vec{a}$ is opposite to the direction of $\\vec{a}$.", "options": ["$>$,$<$", "$<$,$>$", "$<$,$<$", "$>$,$>$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Vector $\\vec{a}$ ($\\vec{a} \\neq \\vec{0}$) and $\\vec{b}$ are collinear, how many real numbers $\\Lambda$ exist such that $\\vec{b}=\\Lambda \\cdot \\vec{a}$?", "options": ["One", "Four", "Three", "Two"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "If $ \\vec{e1} $ and $ \\vec{e2} $ are two non-collinear vectors in the same plane, then for any vector $ \\vec{a} $ in this plane, how many pairs of real numbers $\\Lambda1$ and $\\Lambda2$ exist such that $ \\vec{a}=\\Lambda1 \\cdot \\vec{e1}+\\Lambda2 \\cdot \\vec{e2}$?", "options": ["Two pairs", "Three pairs", "Four pairs", "One pair"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "$ \\vec{OC}=\\Lambda \\cdot \\vec{OA}+\\mu \\cdot \\vec{OB}$, If points $A$, $B$, and $C$ are collinear, then what is $\\Lambda + \\mu = $<<<Answer>>>?", "options": ["$1$", "$0$", "$5$", "$2$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "$\\vec{a}=x \\cdot \\vec{i}+y \\cdot \\vec{j}=$<<<Answer>>>", "options": ["$(i,j)$", "$(x,j)$", "$(i,y)$", "$(x,y)$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Let $\\vec{a}=（x_1,y_1)$ and $\\vec{b}=（x_2,y_2)$, then:\n$\\vec{a}+\\vec{b}=$<<<Answer>>>.", "options": ["$(y_1+x_2,x_1+y_2)$", "$(x_1+x_2,y_1+y_2)$", "$(x_1+y_2,y_1+x_2)$", "$(x_1-x_2,y_1-y_2)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Let $\\vec{a}=(x_1,y_1)$,$\\vec{b}=(x_2,y_2)$, then:\n$\\vec{a}-\\vec{b}=$<<<Answer>>>", "options": ["$(x_1-x_2,y_1-y_2)$", "$(x_1+x_2,y_1+y_2)$", "$(x_1+x_2,y_1-y_2)$", "$(x_1-x_2,y_1+y_2)$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Given $\\vec{a}=(x_1,y_1)$,$\\vec{b}=(x_2,y_2)$, then:\n$\\Lambda \\cdot \\vec{a}=$<<<Answer>>>", "options": ["$(\\Lambda \\cdot x_2,\\Lambda \\cdot y_2)$", "$(\\Lambda \\cdot x_2,\\Lambda \\cdot y_1)$", "$(\\Lambda \\cdot x_1,\\Lambda \\cdot y_2)$", "$(\\Lambda \\cdot x_1,\\Lambda \\cdot y_1)$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Given $\\vec{a}= (x_1,y_1)$ and $\\vec{b}= (x_2,y_2)$, then:\n$\\vec{a} \\parallel \\vec{b} \\Leftrightarrow x_1 \\cdot y_2=$<<<Answer>>>.", "options": ["$ x_2 \\cdot y_2 $", "$ y_2 \\cdot x_1 $", "$ x_1 \\cdot y_1 $", "$ x_2 \\cdot y_1 $"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Let $A(x_1,y_1)$ and $B(x_2,y_2)$, then:\n$\\vec{AB}=$<<<Answer>>>.", "options": ["$(x_2-x_1,y_1-y_2)$", "$(x_2-x_1,y_2-y_1)$", "$(x_1-x_2,y_2-y_1)$", "$(x_1-x_2,y_1-y_2)$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Let $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$. Then, the coordinates of the midpoint of line segment $AB$ are <<<Answer>>>.", "options": ["$(\\frac{x_1+x_2}{2},\\frac{y_1+y_2}{2})$", "$(\\frac{x_1+y_2}{2},\\frac{y_1+x_2}{2})$", "$(\\frac{y_1+x_2}{2},\\frac{x_1+y_2}{2})$", "$(\\frac{x_1+x_3}{2},\\frac{y_3+y_2}{2})$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Given $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$, then the centroid coordinates of $\\triangle ABC$ are <<<Answer>>>.", "options": ["$(\\frac{x_1+x_2+x_3}{2},\\frac{y_1+y_2+y_3}{2})$", "$(\\frac{x_1+x_2+y_3}{3},\\frac{y_1+y_2+x_3}{3})$", "$(\\frac{x_1+x_2+x_3}{3},\\frac{y_1+y_2+y_3}{3})$", "$(\\frac{x_1+y_2+x_3}{3},\\frac{y_1+x_2+y_3}{3})$"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |
| {"question": "$\\vec{a}\\cdot \\vec{b}=\\lvert \\vec{a} \\rvert \\cdot \\lvert \\vec{b} \\rvert $<<<Answer>>>", "options": ["$\\cot \\theta$", "$\\sin \\theta$", "$\\tan \\theta$", "$\\cos \\theta$"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "The projection of $\\vec{a}$ on the direction of $\\vec{b}$ is: $\\lvert \\vec{a} \\rvert$<<<Answer>>>(can be negative)", "options": ["$\\cos \\theta$", "$\\tan \\theta$", "$\\cot \\theta$", "$\\sin \\theta$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "$\\vec{a}^2$<<<Answer>>>$\\lvert \\vec{a} \\rvert^2$", "options": ["$\\neq$", "$>$", "$=$", "$<$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "$\\lvert \\vec{a} \\rvert=$\\sqrt{<<<Answer>>>}$", "options": ["$\\vec{a}^3$", "$\\vec{a}^2$", "$\\vec{a}$", "$\\vec{a}^4$"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "$\\vec{a} \\bot \\vec{b} \\Leftrightarrow \\vec{a} \\cdot \\vec{b}=$", "options": ["$0$", "$5$", "$10$", "$1$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "Given $\\vec{a}$=(5,6), $\\vec{b}$=(0,8), then $\\vec{a}*\\vec{b}$ =", "options": ["48", "39", "32", "29"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Plane Vectors"} | |
| {"question": "If from the second term of a sequence, each term and its preceding term have the same constant difference, then this sequence is called what?", "options": ["Geometric sequence", "Decreasing sequence", "Increasing sequence", "Arithmetic sequence"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "Which of the following options is not a formula for the sum of the first n terms of an arithmetic sequence?", "options": ["$ \\frac{\\left ( a_1+a_n \\right ) n}{2} $", "$\\frac{a_n-a_1}{d} +1$", "$ na_1+ \\frac{nd\\left ( n-1 \\right ) }{2} $", "$An^2+Bn$ where $A=\\frac{d}{2},$ B=$a_1-\\frac{d}{2} $"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "In an arithmetic sequence, the subscripts form what kind of sequence?", "options": ["Decreasing sequence", "Increasing sequence", "Geometric sequence", "Arithmetic sequence"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "For an arithmetic sequence ${a_{n}}$ with common difference d, what kind of sequence is ${a_{n}}$ when d is greater than 0?", "options": ["Constant", "Decreasing", "Increasing", "Oscillating"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "In an arithmetic sequence {a_{n}}, $a_{m}=n$, $a_{n}=m$, and $m \\ne n$, then what is $a_{m+n}$?", "options": ["0", "m", "n", "m+n"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "If $S_{m}=S_{n}$ and $m \\ne n$, what is $S_{m+n}$ equal to?", "options": ["$n$", "$m$", "0", "$m+n$"], "answer": "C", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "What is the general term formula for an arithmetic sequence?", "options": ["$a_n = a_1 + (n - 1)d$", "$a_n = a_1 \\cdot r^{n-1}$", "$a_n = \\frac{a_1 + a_n}{2} \\cdot n$", "$a_n = \\frac{a_1 + l}{n}$"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "If a sequence that does not contain 0 from the second term onwards, each term is in the same ratio with its previous term, what is this sequence called?", "options": ["Constant sequence", "Geometric sequence", "Arithmetic sequence", "Monotonically increasing sequence"], "answer": "B", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "In a geometric sequence, what type of sequence is formed by the terms whose indices form an arithmetic sequence?", "options": ["Constant sequence", "Arithmetic sequence", "Monotonically increasing sequence", "Geometric sequence"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "In a geometric sequence with a common ratio of q, if each term is multiplied by a constant &\\lambda $ not equal to zero, what is the common ratio of the sequence now?", "options": ["&\\lambda $", "q", "$q^2$", "$q\\cdot \\lambda $"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "Given a geometric sequence with a common ratio of q and positive terms $\\left \\{ a_n \\right \\}$, what type of sequence is $\\left \\{ \\lg_{}{a_n} \\right \\}$?", "options": ["Geometric sequence", "Neither arithmetic nor geometric sequence", "Both arithmetic and geometric sequence", "Arithmetic sequence"], "answer": "D", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "Given a geometric sequence $\\left \\{ a_n \\right \\}$ with a common ratio of q, what is the common difference of the sequence $\\left \\{ \\lg_{}{a_n} \\right \\}$?", "options": ["$\\lg_{}{q} $", "$q$", "$q_{}^{2} $", "$\\ln_{}{q} $"], "answer": "A", "topic": "High--Intermediate Numbers and Algebra--Sequences"} | |
| {"question": "Regarding the description of conic sections, which of the following is correct?", "options": ["Conic sections are the outlines of a cone cut by a plane", "Conic sections are the outlines of the base of a cone", "Conic sections only include circles and ellipses", "Conic sections are any straight line on a cone"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "When a cone is intersected by a plane, and the angle $\\theta$ formed between the intersecting plane and the axis $PO$ of the cone is less than the angle $\\delta$ formed between $PO$ and the lateral surface, what is the shape of the edge profile formed by the plane cutting through the cone?", "options": ["Parabola", "Ellipse", "Circle", "Hyperbola"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "When a cone is intersected by a plane and the angle $\\theta$ formed by the intersecting plane and the axis of the cone $PO$ is equal to the angle $\\delta$ formed by $PO$ and the lateral surface, what is the curve of the edge profile resulted from the intersection of the plane with the cone?", "options": ["Hyperbola", "Parabola", "Ellipse", "Circle"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "When a cone is intersected by a plane and the angle $\\theta $ between the plane and the cone's axis $PO $ is greater than the angle $\\delta $ between $PO $ and the lateral surface but less than $\\frac{\\pi}{2} $, what is the curve of the edge profile formed by the plane's intersection with the cone?", "options": ["Circle", "Hyperbola", "Ellipse", "Parabola"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "When a cone is intersected by a plane, and the angle $\\theta$ formed between the intersecting plane and the cone's axis $PO$ is equal to $\\frac{\\pi}{2}$, what is the profile of the edge formed by the plane intersecting the cone?", "options": ["Hyperbola", "Parabola", "Ellipse", "Circle"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "If the points equidistant from a fixed point $F$ and a given straight line $l$ form a trajectory, what is this trajectory called?", "options": ["Straight line", "Circle", "Parabola", "Ellipse"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "What kind of trajectory is a parabola?", "options": ["The trajectory of a moving point $P$ in the plane of a parabola whose distances to fixed points $F1$ and $F2$ sum up to a constant (greater than $|F1F2|$)", "All the other options are wrong", "A parabola is the trajectory formed by points that are equidistant to a fixed point $F$ and a fixed line $l$", "A parabola is the trajectory of a point moving in a plane or space in a certain direction and its opposite direction"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "When $p>0$, which standard equation of the parabola is correct?", "options": ["$y=x$", "$x^2=2py$", "$x^2=-2py$", "$y^2=2px$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "When $p<0$, which standard equation of the parabola is correct?", "options": ["$y^2=2px$", "$y^2=-2px$", "$x^2=2py$", "$y=x$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "What is the eccentricity $e$ of a standard parabola?", "options": ["3", "2", "1", "4"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "What are the vertex coordinates of a standard parabola?", "options": ["(2,2)", "(0,0)", "(3,3)", "(1,1)"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "Given the standard parabola equation $y = ax^2 + bx + c$ (where $a \\neq 0$), what is the equation of its axis of symmetry?", "options": ["x = -\\frac{b}{2a}", "y = -\\frac{b}{2a}", "x = \\frac{b}{2a}", "y = \\frac{b}{2a}"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the standard parabola equation, if the value of the focal radius $|MF|$ is the sum of two values, one value being $p/2$, what is the other value?", "options": ["$x_0$", "$x_0+2$", "$x_0+3$", "$x_0+1$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the standard parabola equation, what is the definition of the latus rectum?", "options": ["The chord that passes through the focus of the parabola and makes a 45-degree angle with the axis of symmetry is called the latus rectum", "The chord that passes through the focus of the parabola and is parallel to the axis of symmetry is called the latus rectum", "The chord that passes through the focus of the parabola and is perpendicular to the axis of symmetry is called the latus rectum", "The chord that passes through the focus of the parabola and makes a 60-degree angle with the axis of symmetry is called the latus rectum"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the standard parabola equation, what is the formula for the length of the focal chord?", "options": ["$|AB|=x_1+x_2+p+3$", "$|AB|=x_1+x_2+p+2$", "$|AB|=x_1+x_2+p$", "$|AB|=x_1+x_2+p+1$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the standard equation of a parabola, what is the geometric meaning of the parameter $p$?", "options": ["The parameter $p$ represents the distance from the focus to the directrix, the relationship between the value of $p$ and the size of the opening is uncertain", "The parameter $p$ represents the distance from the focus to the directrix, the value of $p$ is unrelated to the size of the opening", "The parameter $p$ represents the distance from the focus to the directrix, the smaller $p$ is, the wider the opening", "The parameter $p$ represents the distance from the focus to the directrix, the larger $p$ is, the wider the opening"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the standard parabola equation, what is the range of the focal radius?", "options": ["$[p/2，+∞)$", "$[p/2，1)$", "$[p/2，2)$", "$[p/2，0)$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the standard parabola equation, given that $Δ=0$, what is the equation for the relationship between $y'$ and $k$, which is slop of parabola's tnagent line, under the conditions of $x^2=±2py$ and $y=±x^2/2p$?", "options": ["$y?=k+2$", "$y?=k+3$", "$y?=k$", "$y?=k+1$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "What is the formula for the slope of the tangent line to the parabola $y = ax^2 + bx + c$ at the point $(x_0, y_0)$?", "options": ["2ax_0 + b", "$\\frac{b}{2a}$", "$ax_0^2 + bx_0$", "$2a + bx_0$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the standard parabolic equation, when $AB⊥y$, what extreme value does $AB$ take?", "options": ["Maximum value", "Largest value", "Minimum value", "Any value"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "What is the essential geometric meaning of the fraction $Q=\\frac{\\sqrt{(x-a)^2+(y-b)^2} }{\\frac{|Am+By+C|}{\\sqrt{A^2+B^2} } } $?", "options": ["Essentially, this expression represents the distance from point $(m,n)$ to point $(a,b)$", "Essentially, this expression represents the distance from point $(m,n)$ to the line $l: Ax+By+C=0$", "Essentially, this expression represents the ratio of the distance from point $(m,n)$ to point $(a,b)$ and the distance from point $(m,n)$ to the line $l: Ax+By+C=0$", "Essentially, this expression represents the ratio of the distance from the line $l: Ax+By+C=0$ to the distance from point $(m,n)$ to point $(a,b)$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the fraction $Q=\\frac{\\sqrt{(x-a)^2+(y-b)^2}}{\\frac{|Am+By+C|}{\\sqrt{A^2+B^2}}}$, when $0<Q<1$, what is the trajectory of $(x, y)$?", "options": ["Parabola", "Hyperbola", "Straight line", "Ellipse"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the fraction $Q=\\frac{\\sqrt{(x-a)^2+(y-b)^2}}{\\frac{|Am+By+C|}{\\sqrt{A^2+B^2}}}$, when $Q=1$, what is the trajectory of $(x,y)$?", "options": ["Hyperbola", "Straight Line", "Parabola", "Ellipse"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the fraction $Q=\\frac{\\sqrt{(x-a)^2+(y-b)^2} }{\\frac{|Am+By+C|}{\\sqrt{A^2+B^2} } } $, when $Q>1$, what is the trajectory of $(x,y)$?", "options": ["Straight line", "Parabola", "Ellipse", "Hyperbola"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For point $(M,N)$, given $AM^2+BN^2=1$, combined with the definition of trigonometric functions, what is the value of $M$?", "options": ["$\\frac{\\cos\\theta}{\\sqrt{A}}$", "$\\cos\\theta$", "$\\frac{\\sin \\theta}{\\sqrt{B}}$", "$\\sin\\theta$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For point $(M,N)$, given that $AM^2+BN^2=1$, in conjunction with the definition of trigonometric functions, what is the value of $N$?", "options": ["cos?\\theta", "$\\frac{cos?\\theta}{\\sqrt{A}}$", "sin?\\theta", "$\\frac{sin\\theta}{\\sqrt{B}}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the parametric equation of a circle, what is the value of $x$?", "options": ["sin?\\theta", "cos?\\theta", "$b+rsin\\theta$", "$a+rcos?\\theta$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "In the parametric equations of a circle, what is the value of $y$?", "options": ["cos?\\theta", "$a+rcos?\\theta$", "sin?\\theta", "$b+rsin\\theta$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "In the parametric equations of a circle $(x-a)^2 + (y-b)^2 = r^2$, what is the geometric meaning of $a$?", "options": ["Ordinate of the circle center", "Abscissa of the circle center", "Curvature", "Radius"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "In the parametric equation of a circle, what is the geometric meaning of $b$?", "options": ["Radius", "X-coordinate of the center", "Y-coordinate of the center", "Curvature"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "In the parametric equation of a circle, what is the geometric meaning of $r$?", "options": ["Curvature", "Radius", "The ordinate of the center", "The abscissa of the center"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "In the parametric equation of an ellipse with foci on the $x$ axis, what is the expression for $x$?", "options": ["$a^3+cos?\\theta$", "$a^2+cos?\\theta$", "$b+sin\\theta$", "$a+cos?\\theta$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the parametric equation of an ellipse with foci on the x-axis, what is the expression for $y$?", "options": ["$a^2+cos?\\theta$", "$a^3+cos?\\theta$", "$b+sin\\theta$", "$a+cos?\\theta$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the parametric equation of an ellipse with foci on the x-axis, what is the geometric meaning of $a$?", "options": ["Major axis", "Area", "Minor axis", "Projection"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "In the parametric equation of an ellipse with foci on the $x$ axis, what is the geometric significance of $b$?", "options": ["Imaginary axis", "Projection", "Area", "Real axis"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with foci on the x-axis, what is the standard equation?", "options": ["$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with foci on the y-axis, what is the standard equation?", "options": ["$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1(a>b>0)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with foci on the x-axis, what are the ranges of values for x and y?", "options": ["$-b\\leq x\\leq b\\text{ and }-a\\leq y\\leq a$", "$x\\leq-a\\text{ or }x\\geq a\\mathrm{~,~}y\\in R$", "$-a\\leq x\\leq a\\text{ and }-b\\leq y\\leq b$", "$y\\leq-a\\text{ or }y\\geq a\\mathrm{~,~}x\\in R$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with its foci on the y-axis, what are the ranges of values for x and y?", "options": ["$y\\leq-a\\text{ or }y\\geq a\\mathrm{~,~}x\\in R$", "$x\\leq-a\\text{ or }x\\geq a\\mathrm{~,~}y\\in R$", "$-b\\leq x\\leq b\\text{ and }-a\\leq y\\leq a$", "$-a\\leq x\\leq a\\text{ and }-b\\leq y\\leq b$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with its foci on the x-axis, what are the vertices?", "options": ["${A}_1\\left(0,-a\\right)\\text{、A}_2\\left(0,a\\right)$", "$\\begin{aligned}\\mathrm{A}_1&\\left(-a,a\\right)\\text{、A}_2\\left(a,a\\right)\\\\\\\\\\mathrm{B}_1&\\left(0,-b\\right)\\text{、B}_2\\left(0,b\\right)\\end{aligned}$", "$\\begin{aligned}\\mathrm{A}_1&\\left(-a,0\\right)\\text{、A}_2&\\left(a,0\\right)\\\\\\\\\\mathrm{B}_1&\\left(0,-b\\right)\\text{、B}_2&\\left(0,b\\right)\\end{aligned}$", "${A}_1\\left(-a,0\\right)\\text{、A}_2\\left(a,0\\right)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "What is the symmetry of an ellipse?", "options": ["Symmetric about y=$x^2$", "Symmetric about the x-axis, y-axis, and central symmetric about the origin", "Symmetric about y=1", "Symmetric about y=x"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with foci on the x-axis, what are the coordinates of the foci?", "options": ["$F_1(-c,c)$, $F_2(c,0)$", "$F_1(-c,0)$, $F_2(c,0)$", "$F_1(-c,0)$, $F_2(c,c)$", "$F_1(0,-c)$, $F_2(0,c)$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with its foci on the y-axis, what are the foci?", "options": ["$F_1(-c,c)$, $F_2(c,0)$", "$F_1(-c,0)$, $F_2(c,0)$", "$F_1(-c,0)$, $F_2(c,c)$", "$F_1(0,-c)$, $F_2(0,c)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with the standard equation $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$, what is the focal distance $\\left|F_1F_2\\right|=? (c^2=a^2-b^2)$", "options": ["c", "2c", "2b", "2a"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the ellipse with the standard equation $\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$, what is the eccentricity e? (0<e<1)", "options": ["$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2+b^2}{a^2}}=\\sqrt{1+\\frac{b^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2-b^2}{a^2}}=\\sqrt{1-\\frac{c^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2+b^2}{a^2}}=\\sqrt{1+\\frac{c^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a2-b2}{a^2}}=\\sqrt{1-\\frac{b^2}{a^2}}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with foci on the x-axis, what is the equation of the directrix?", "options": ["$y=\\pm\\frac{a^2}c$", "$y=\\pm\\frac{b^2}c$", "$x=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{b^2}c$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with foci on the y-axis, what is the equation of the directrix?", "options": ["$x=\\pm\\frac{a^2}c$", "$y=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{b^2}c$", "$y=\\pm\\frac{b^2}c$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with the standard equation $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$, what is the area of the focal triangle? ($\\theta=\\angle F_1MF_2$)", "options": ["$S_{\\Delta MF_1F_2}=a^2\\tan\\frac{\\theta}{2}$", "$S_{\\Delta MF_1F_2}=b^2\\tan\\frac{\\theta}{2}$", "$S_{\\Delta MF_1F_2}=b^2\\tan\\theta$", "$S_{\\Delta MF_1F_2}=b^2\\cdot\\frac{1}{\\tan\\frac{\\theta}{2}}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse with the standard equation $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$, the chord that passes through a focus and is perpendicular to the major axis is called the latus rectum. What is $\\begin{vmatrix}HH^{\\prime}\\end{vmatrix}=$?", "options": ["$\\frac{b^2}{a}$", "$\\frac{a^2}{b}$", "$\\frac{a}{b}$", "$\\frac{b}{a}$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For an ellipse defined by the standard equation $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1 (a>b>0)$, given two points on the ellipse $A(x_{1,}y_1),B(x_{2,}y_2)$, where k is the slope of the linewhat is the formula for the chord length?", "options": ["$|AB| = \\sqrt{1+k}|x_1-x_2| = \\sqrt{1+k}\\sqrt{(x_1+x_2)^2-4x_1x_2}$", "$|AB| = \\sqrt{1+k^2}|x_1-x_2| = \\sqrt{1+k^2}\\sqrt{(x_1+x_2)^2-2x_1x_2}$", "$|AB| = \\sqrt{1+k^2}|x_1-x_2| = \\sqrt{1+k^2}\\sqrt{(x_1+x_2)^2-4x_1x_2}$", "$|AB| = \\sqrt{1-k^2}|x_1-x_2| = \\sqrt{1-k^2}\\sqrt{(x_1+x_2)^2-4x_1x_2}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "An arbitrary line intersects an ellipse at points A and B, with P being the midpoint of AB. What is the product of the slopes of AB and OP?", "options": ["$-\\frac{a^2}{b^2}$", "$-\\frac{b^2}{a^2}$", "$\\frac{a^2}{b^2}$", "$\\frac{b^2}{a^2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "Methods to find the equation of an ellipse?", "options": ["Analogy method", "Definition method, eliminating parameter method", "Geometric construction method", "Summarization and induction method"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with foci on the x-axis, what is the standard equation?", "options": ["$\\dfrac{y^2}{a^2}-\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{y^2}{a^2}+\\dfrac{x^2}{b^2}=1(a>b>0)$", "$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$", "$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1(a>b>0)$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with foci on the x-axis, what are the ranges of values for x and y?", "options": ["$-b\\leq x\\leq b\\text{ and }-a\\leq y\\leq a$", "$y\\leq-a\\text{ or }y\\geq a\\mathrm{~,~}x\\in R$", "$-a\\leq x\\leq a\\text{ and }-b\\leq y\\leq b$", "$x\\leq-a\\text{ or }x\\geq a\\mathrm{~,~}y\\in R$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the hyperbola with its foci on the y-axis, what are the ranges of values for x and y?", "options": ["$-a\\leq x\\leq a\\text{ and }-b\\leq y\\leq b$", "$y\\leq-a\\text{ or }y\\geq a\\mathrm{~,~}x\\in R$", "$x\\leq-a\\text{ or }x\\geq a\\mathrm{~,~}y\\in R$", "$-b\\leq x\\leq b\\text{ and }-a\\leq y\\leq a$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with foci on the x-axis, what are the vertices?", "options": ["${A}_1\\left(-a,0\\right), {A}_2\\left(a,0\\right)$", "${A}_1\\left(0,-a\\right), {A}_2\\left(0,a\\right)$", "${A}_1\\left(-a,a\\right), {A}_2\\left(a,0\\right)$", "${A}_1\\left(-a,0\\right), {A}_2\\left(a,a\\right)$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with foci on the y-axis, what are the vertices?", "options": ["${A}_1(0,-a), {A}_2(0,a)$", "${A}_1(-a,0), {A}_2(a,a)$", "${A}_1(-a,a), {A}_2(a,0)$", "${A}_1(-a,0), {A}_2(a,0)$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the hyperbola with the standard equation $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$, what is the length of the real axis?", "options": ["b", "2a", "2b", "a"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "What are the symmetries of a hyperbola?", "options": ["Symmetric about y=1", "Symmetric about y=$x^2$", "Symmetric about the x-axis, y-axis, and central symmetric about the origin", "Symmetric about y=x"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the hyperbola with foci on the x-axis, what are the foci?", "options": ["$F_1(-c,0), F_2(c,0)$", "$F_1(-c,c), F_2(c,0)$", "$F_1(-c,0), F_2(c,c)$", "$F_1(0,-c), F_2(0,c)$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with foci on the y-axis, what are the foci?", "options": ["$F_1(-c,c)\\, F_2(c,0)$", "$F_1(-c,0)\\, F_2(c,0)$", "$F_1(-c,0)\\, F_2(c,c)$", "$F_1(0,-c)\\, F_2(0,c)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the hyperbola with the standard equation $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ (a>b>0), what is the focal length $\\left|F_1F_2\\right|=? (c^2=a^2+b^2)$", "options": ["2b", "c", "2c", "2a"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the hyperbola with the standard equation $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ (a>b>0), what is the eccentricity e? (e>1)", "options": ["$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2-b^2}{a^2}}=\\sqrt{1-\\frac{c^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2+b^2}{a^2}}=\\sqrt{1+\\frac{c^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2-b^2}{a^2}}=\\sqrt{1-\\frac{b^2}{a^2}}$", "$e=\\frac ca=\\sqrt{\\frac{c^2}{a^2}}=\\sqrt{\\frac{a^2+b^2}{a^2}}=\\sqrt{1+\\frac{b^2}{a^2}}$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with its foci on the x-axis, what is the equation of the directrix?", "options": ["$y=\\pm\\frac{b^2}c$", "$y=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{b^2}c$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with foci on the y-axis, what is the equation of the directrix?", "options": ["$y=\\pm\\frac{b^2}c$", "$y=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{a^2}c$", "$x=\\pm\\frac{b^2}c$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with its focal points on the x-axis, what is the equation of its asymptotes?", "options": ["$y=\\pm\\frac a b x$", "$y=\\pm\\frac{b^2}c x$", "$y=\\pm\\frac b a x$", "$y=\\pm\\frac{a^2}c x$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with foci on the y-axis, what is the equation of the asymptotes?", "options": ["$y=\\pm\\frac{a^2}c x$", "$y=\\pm\\frac a b x$", "$y=\\pm\\frac b a x$", "$y=\\pm\\frac{b^2}c x$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with its foci on the x-axis, what is the focal radius?", "options": ["$\\begin{gathered}M\\text{ on the right branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=ex_0+a\\\\\\text{Right focus:}\\left|MF_2\\right|=ex_0-a\\end{cases}\\\\M\\text{ on the left branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=-ex_0-a\\\\\\text{Right focus:}\\left|MF_2\\right|=-ex_0+a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ on the upper branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=ey_0+a\\\\\\text{Right focus:}\\left|MF_2\\right|=ey_0-a\\end{cases} \\\\M\\text{ on the lower branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=-ey_0-a\\\\\\text{Right focus:}\\left|MF_2\\right|=-ey_0+a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ on the upper branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=ey_0-a\\\\\\text{Right focus:}\\left|MF_2\\right|=ey_0+a\\end{cases} \\\\M\\text{ on the lower branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=-ey_0+a\\\\\\text{Right focus:}\\left|MF_2\\right|=-ey_0-a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ on the right branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=ex_0-a\\\\\\text{Right focus:}\\left|MF_2\\right|=ex_0+a\\end{cases}\\\\M\\text{ on the left branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=-ex_0+a\\\\\\text{Right focus:}\\left|MF_2\\right|=-ex_0-a\\end{cases}\\end{gathered}$"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with foci on the y-axis, what is the focal radius?", "options": ["$\\begin{gathered}M\\text{ on right branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=ex_0+a\\\\\\text{Right focus:}\\left|MF_2\\right|=ex_0-a\\end{cases}\\\\M\\text{ on left branch}\\begin{cases}\\text{Left focus}:\\left|MF_1\\right|=-ex_0-a\\\\\\text{Right focus:}\\left|MF_2\\right|=-ex_0+a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ on top branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=ey_0-a\\\\\\text{Right focus:}\\left|MF_2\\right|=ey_0+a\\end{cases} \\\\M\\text{ on bottom branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=-ey_0+a\\\\\\text{Right focus:}\\left|MF_2\\right|=-ey_0-a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ on top branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=ey_0+a\\\\\\text{Right focus:}\\left|MF_2\\right|=ey_0-a\\end{cases} \\\\M\\text{ on bottom branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=-ey_0-a\\\\\\text{Right focus:}\\left|MF_2\\right|=-ey_0+a\\end{cases}\\end{gathered}$", "$\\begin{gathered}M\\text{ on right branch}\\begin{cases}\\text{Left focus:}\\left|MF_1\\right|=ex_0-a\\\\\\text{Right focus:}\\left|MF_2\\right|=ex_0+a\\end{cases}\\\\M\\text{ on left branch}\\begin{cases}\\text{Left focus}:\\left|MF_1\\right|=-ex_0+a\\\\\\text{Right focus:}\\left|MF_2\\right|=-ex_0-a\\end{cases}\\end{gathered}$"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the hyperbola with the standard equation $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ (a>b>0), what is the area of the focal triangle?", "options": ["$S_{\\Delta MF_1F_2}=a^2\\tan\\frac\\theta2$", "$S_{\\Delta MF_1F_2}=b^2\\tan\\frac\\theta2$", "$S_{\\Delta MF_1F_2}=b^2\\tan\\theta$", "$S_{\\Delta MF_1F_2}=b^2\\cdot\\frac1{\\tan\\frac\\theta2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the hyperbola with the standard equation $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1$ (a>b>0), the chord that passes through a focus and is perpendicular to the major axis is called a transverse axis. What is the length $\\begin{vmatrix}HH^{\\prime}\\end{vmatrix}=$ of the transverse axis?", "options": ["$\\frac{a^2}b$", "$\\frac{b^2}a$", "$\\frac a b$", "$\\frac b a$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "Discuss the range of point P for the difference problem of $PF_{1},PF_{2}$ on the hyperbola!\nWhen the foci are on the x-axis: $PF_{1}-PF_{2}$ > 0, then point P is on?", "options": ["Upper branch", "Lower branch", "Right branch", "Left branch"], "answer": "C", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For the hyperbola with the standard equation $\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1(a>b>0)$, what is the range of the focal radius?", "options": ["$[b-c,+\\infty)$", "$[c-b,+\\infty)$", "$[a-c,+\\infty)$", "$[c-a,+\\infty)$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "For a hyperbola with the standard equation $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>b>0)$, what is the distance from the focus to the asymptote?", "options": ["$b-a$", "$a$", "$a-b$", "$b$"], "answer": "D", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "Draw lines parallel to the hyperbola $\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ at a point P, the area of the quadrilateral formed by these lines, the hyperbola, and point P, O is?", "options": ["A constant value ab", "A constant value $\\frac{ab}{2}$", "A constant value $\\frac{b}{2}$", "A constant value $\\frac{a}{2}$"], "answer": "B", "topic": "High--Intermediate Geometry--Equations of Conic Sections"} | |
| {"question": "Generally, what is the name of the spatial geometric body formed by translating a plane polygon in a certain direction?", "options": ["Prism", "Cylinder", "Pyramid", "Frustum"], "answer": "A", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "What is the polyhedron called, that is between the base and a plane parallel to the base of a pyramid when it intersects the pyramid?", "options": ["Pyramid", "Frustum", "Cone", "Prism"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "What is the formula for calculating the lateral surface area of a cylinder?", "options": ["$S=4\\pi \\cdot r \\cdot l$", "$S=2\\pi \\cdot r \\cdot l$", "All the other options are wrong", "$S=\\pi \\cdot r \\cdot l$"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "What is the formula for calculating the lateral surface area of a cone?", "options": ["All the other options are wrong", "$S=\\pi \\cdot r \\cdot l$", "$S=4\\pi \\cdot r \\cdot l$", "$S=2\\pi \\cdot r \\cdot l$"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "How is the area of a rhombus calculated?", "options": ["Half the product of the diagonals", "Product of the side lengths", "Product of the diagonals", "Half the product of the side lengths"], "answer": "A", "topic": "High--Intermediate Geometry--Equations of Lines and Circles"} | |
| {"question": "Where are the centers of the circumscribed sphere and inscribed sphere of a cube located?", "options": ["The center of the circumscribed sphere is at the center of the edge, and the center of the inscribed sphere is at the midpoint of the body diagonal", "The center of the circumscribed sphere is at the midpoint of the body diagonal, and the center of the inscribed sphere is at the center of the edge", "Both are at the center of the edge", "Both are at the midpoint of the body diagonal"], "answer": "D", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "What shape are all the faces of a regular tetrahedron?", "options": ["Equilateral triangle", "Ordinary triangle", "Isosceles triangle", "Quadrilateral"], "answer": "A", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "What is the shape of the base of a regular triangular pyramid?", "options": ["Ordinary triangle", "Quadrilateral", "Equilateral triangle", "Isosceles triangle"], "answer": "C", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "If the edge length of a regular tetrahedron is a, what is the radius of its circumscribed sphere?", "options": ["$ \\frac{\\sqrt{3}}{4} a $", "$ \\frac{\\sqrt{6}}{4} a $", "$ \\frac{\\sqrt{6}}{12} a $", "$ \\frac{\\sqrt{3}}{6} a $"], "answer": "B", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "If the edge of a regular tetrahedron has a length of a, then what is the radius of its inscribed sphere?", "options": ["$ \\frac{\\sqrt{6}}{12} a $", "$ \\frac{\\sqrt{3}}{6} a $", "$ \\frac{\\sqrt{3}}{4} a $", "$ \\frac{\\sqrt{6}}{4} a $"], "answer": "A", "topic": "High--Intermediate Geometry--Solid Geometry"} | |
| {"question": "The centroid of a triangle is the intersection point of which three lines?", "options": ["bisectors", "perpendicular bisectors", "medians", "altitudes"], "answer": "C", "topic": "High--Intermediate Geometry--Trigonometric Functions and Solving Triangles"} | |