Question: What equation is this called if it lies on the graph and goes in a continuous straight line

Choices:
- Linear Equation
- Quadratic equation
- Cubic equation
- Inequality

High--Intermediate Geometry--Equations of Lines and Circles


Question: Given the form of the slope-intercept, what do m and b represent respectively?

Choices:
- 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

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

Choices:
- Downhill
- Vertical
- Uphill
- Horizontal

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

Choices:
- Uphill
- Downhill
- Vertical
- Horizontal

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?

Choices:
- Vertical
- Horizontal
- Uphill
- Downhill

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?

Choices:
- Uphill
- Horizontal
- Vertical
- Downhill

High--Intermediate Geometry--Equations of Lines and Circles


Question: To memorize the slope, we use VUX HOY. What does V represent?

Choices:
- x=#
- undefined
- vertical line
- Horizontal

High--Intermediate Geometry--Equations of Lines and Circles


Question: To memorize the slope, we use VUX HOY. What does U represent?

Choices:
- vertical line
- x=#
- undefined
- Horizontal

High--Intermediate Geometry--Equations of Lines and Circles


Question: To memorize the slope, we use VUX HOY. What does V represent?

Choices:
- vertical line
- undefined
- Horizontal
- x=#

High--Intermediate Geometry--Equations of Lines and Circles


Question: To memorize the slope, we use VUX HOY. What does H represent?

Choices:
- vertical line
- x=#
- Horizontal
- undefined

High--Intermediate Geometry--Equations of Lines and Circles


Question: To memorize the slope, we use VUX HOY. What does O represent?

Choices:
- x=#
- slope=0
- Horizontal
- x=0

High--Intermediate Geometry--Equations of Lines and Circles


Question: To memorize the slope, we use VUX HOY. What does Y represent?

Choices:
- slope=0
- y=#
- y=0
- vertical line

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?

Choices:
- $m=(X_2-X_1)/(Y_1-Y_2)$
- $m=(X_1-X_2)/(Y_2-Y_1)$
- $m=(X_2-X_1)/(Y_2-Y_1)$
- $m=(Y_2-X_2)/(Y_1-X_1)$

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?

Choices:
- $x=\frac{-b\pm \sqrt{4ac-b^2}}{2a}$
- $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
- $x=\frac{-b\pm \sqrt{4ac}}{2a}$
- $x=\frac{-b\pm \sqrt{b^2}}{2a}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: What does the discriminant tell as?

Choices:
- 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

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: What is the formula of the discriminant?

Choices:
- $D=b^2$
- $D=4ac$
- $D=4ac-b^2$
- $D=b^2-4ac$

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?

Choices:
- positive infinity
- 2
- 1
- 0

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?

Choices:
- 1
- 0
- 2
- positive infinity

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?

Choices:
- 2
- 1
- positive infinity
- 0

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?

Choices:
- (k,h)
- (h,k)
- (h,0)
- (0,k)

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?

Choices:
- Middle
- Minimum
- Maximum
- Extreme point

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?

Choices:
- Extreme point
- Minimum
- Maximum
- Middle

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Please explain the concept of the domain for f (x).

Choices:
- 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

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Please explain the concept of the range for f (x).

Choices:
- 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.

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
Explain the concept of x-intercepts: values that make f(x)=<<<Answer>>>
 and lies on the x-axis.

Choices:
- 0
- $\infty$
- $-\infty$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question:
Explain the concept of y-intercepts: values that make f(x)=<<<Answer>>>
 and lies on the y-axis.

Choices:
- 0
- 1
- $\infty$
- $-\infty$

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?

Choices:
- y -axis
- x -axis
- z -axis
- x -axis and y -axis

High--Intermediate Geometry--Equations of Lines and Circles


Choices:
- All the other options are wrong
- $(a+b)(a-b)$
- $(a+b)(a+b)$
- $(a-b)(a-b)$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $(a+b)(a-b)$=

Choices:
- $b^{2}$
- $a^{2}-b^{2}$
- $a^{2}+b^{2}$
- $a^{2}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Choices:
- $x^{2}-2xy-y^{2}$
- $x^{2}+2xy+y^{2}$
- $x^{2}-2xy+y^{2}$
- $x^{2}+2xy-y^{2}$


Choices:
- $(x-y)^{2}$
- $(x+y)^{2}$
- $(x-y)^{3}$
- $(x+y)^{3}$


Choices:
- $x^{2}-2xy-y^{2}$
- $x^{2}+2xy+y^{2}$
- $x^{2}+2xy-y^{2}$
- $x^{2}-2xy+y^{2}$


Choices:
- $(x+y)^{3}$
- $(x-y)^{2}$
- $(x-y)^{3}$
- $(x+y)^{2}$


Question: What parts does a complex number contain?

Choices:
- two imaginary numbers
- a real number and an imaginary number
- All the other options are wrong
- two real numbers

High--Intermediate Numbers and Algebra--Complex Numbers


Question: Complex number a+bi, "a" is

Choices:
- 0
- a real number
- All the other options are wrong
- an imaginary number

High--Intermediate Numbers and Algebra--Complex Numbers


Question: Complex number a+bi, "b" is

Choices:
- All the other options are wrong
- a real number
- an imaginary number
- 0

High--Intermediate Numbers and Algebra--Complex Numbers


Choices:
- 1
- -i
- i
- -1

High--Intermediate Numbers and Algebra--Complex Numbers


Choices:
- 1
- i
- -i
- -1


Choices:
- 1
- i
- -1
- -i


Choices:
- i
- -i
- -1
- 1


Choices:
- -1
- 1
- -i
- i


Question: What are complex conjugate numbers of $a+bi$?

Choices:
- $a-bi$
- $a+bi$
- $abi$
- All the other options are wrong

High--Intermediate Numbers and Algebra--Complex Numbers


Question: What are complex conjugate numbers of $a-bi$?

Choices:
- $a-bi$
- $abi$
- $a+bi$
- All the other options are wrong

High--Intermediate Numbers and Algebra--Complex Numbers


Question: In formula $y=a\left |x-h\right |+k$, what is the vertex?

Choices:
- (k,h)
- (0,k)
- (h,0)
- (h,k)

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?

Choices:
- x=k
- x=h
- y=h
- y=k

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $a^{m}\times  a^{n}= $?

Choices:
- $a^{m}+a^{n}$
- $a^{m+n}$
- $a^{m*n}$
- $a^{m^{n} }$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $a^{m}\times  a^{n}= $?

Choices:
- $a^{m+n}$
- $a^{m*n}$
- $a^{m-n}$
- $a^{m}+a^{n}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $\left ( a^{m}  \right ) ^{n} $=?

Choices:
- $a^{m+n}$
- $a^{mn} $
- $a^{m*n}$
- $a^{m}+a^{n}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: When $a\ne 0$, $a^{0}=$?

Choices:
- a
- -1
- 0
- 1

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Choices:
- $1- a^{4} $
- $\frac{1}{a^{-4} } $
- $1-+a^{4} $
- $\frac{1}{a^{4} } $


Question: $\left ( ab \right ) ^{n} $?

Choices:
- $a^{n}b$
- $ab^{n}$
- $\left ( a^{n}  \right ) \left ( b^{n} \right ) $
- $\left ( a^{n}  \right ) +\left ( b^{n} \right ) $

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: $\left ( a/b \right ) ^{n}=$?

Choices:
- $\left ( ab \right ) ^{n}$
- $a^{n} /b^{n} $
- $a^{n} +b^{n} $
- $a^{n} -b^{n} $

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 <<<>>>?

Choices:
- integer
- decimal
- fraction
- irrational number

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: The definition of radicals is that an expression that uses a <<<>>>?

Choices:
- square root
- root
- fourth root
- cube root

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: To get rid of the radical in an equation, <<<>>> both sides.

Choices:
- square
- cube root
- fourth root
- cube

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.

Choices:
- the top and bottom
- neither the top nor the bottom
- only the bottom
- only the top

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Whenever you take the square root of both sides, it ends up being <<<>>>.

Choices:
- neither $+$ nor $-$
- only $+$
- only $-$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Which of the following statements about the definition of an equation is correct?

Choices:
- 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

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Which of the following statements is correct?

Choices:
- 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.

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following statements is correct?

Choices:
- 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.

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Which of the following statements about infinite solutions is correct?

Choices:
- 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

High--Intermediate Geometry--Equations of Lines and Circles


Question: Which of the following statements about exponential equations is incorrect?

Choices:
- 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.

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?

Choices:
- $P=a(1+t)^{r}$
- $P= (1+t)^{r}$
- $P=a(1+r)^{t}$
- $P=(1+r)^{t}$

High--Intermediate Numbers and Algebra--Elementary Functions and Their Derivatives


Question: Which of the following is not a sign of greater than?

Choices:
- above
- at least
- exceeding
- more than

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following is not a sign of greater than or equal?

Choices:
- at least
- ferwer than
- at minimun
- no less than

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following is not a sign of less than?

Choices:
- below
- less than
- at least
- fewer than

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following is not a sign of less than or equal?

Choices:
- at least
- does not exceed
- at most
- no more than

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.

Choices:
- ["dashed","below"]
- ["dashed","above"]
- ["solid","below"]
- ["solid","above"]

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.

Choices:
- ["solid","below"]
- ["dashed","below"]
- ["dashed","above"]
- ["solid","above"]

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.

Choices:
- ["dashed","below"]
- ["solid","above"]
- ["dashed","above"]
- ["solid","below"]

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.

Choices:
- ["solid","above"]
- ["dashed","below"]
- ["dashed","above"]
- ["solid","below"]

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: When multiplying and dividing by a negative number, we should <<<>>> the <<<>>> symbol.

Choices:
- ["flip","inequality"]
- ["not flip","inequality"]
- ["flip","equality"]
- ["not flip","equality"]

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: What kind of terms have the same variable and are raised to the same power?

Choices:
- Equal Terms
- Combining Like Terms
- Right Terms
- Good Terms

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？

Choices:
- Pythagorean Theorem
- Right Triangle Theorem
- Newton Theorem
- Right Theorem

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: What is the formula of Pythagorean Theorem?

Choices:
- $a^2+b^2 = c$
- $a^2+b^2 = 2c^2$
- ${(a+b)}^2 = c^2$
- $a^2+b^2 = c^2$

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: Which are 2 angles whose sum is $180^{\circ}$？

Choices:
- Alternate Exterior Angles
- Complementary Angles
- Alternate Interior Angles
- Supplementary Angles

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: Which are 2 angles opposite of each other and are made by intersecting lines？

Choices:
- Alternate Exterior Angles
- Vertical Angles
- Complementary Angles
- Alternate Interior Angles

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？

Choices:
- Alternate Exterior Angles
- Supplementary Angles
- Alternate Interior Angles
- Complementary Angles

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？

Choices:
- Alternate Exterior Angles
- Same-side Interior Angles
- Same-side Exterior Angles
- Alternate Interior Angles

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?

Choices:
- $(x-h)^2+(y-k)^2=2r^2$
- $(x-h)^2+(y-k)^2=r^2$
- $(x+h)^2+(y+k)^2=r^2$
- $(x-h)^2-(y-k)^2=r^2$

High--Intermediate Geometry--Equations of Lines and Circles


Question: Which one is wrong about similar polygons？

Choices:
- different shape
- corresponding sides are proportional
- corresponding angles are congruent
- different size

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: What is the formula of arc of a circle？

Choices:
- $(2\pi r) central angle \cdot 360^{\circ} = arc length$
- $(2\pi r) central angle/360^{\circ} = arc length$
- $(2\pi r) central angle/180^{\circ} = arc length$
- $(\pi r) central angle/360^{\circ} = arc length$

High--Intermediate Geometry--Equations of Lines and Circles


Question: What is the distance formula between two points?

Choices:
- $d^2 = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
- $d^2 = (x_2-x_1)^2+(y_2-y_1)^2$
- $d^2 = \sqrt{(x_2+x_1)^2+(y_2+y_1)^2}$
- $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

High--Intermediate Geometry--Equations of Lines and Circles


Question: What is the midpoint formula?

Choices:
- $midpoint =(x_1+y_1,x_2+y_2)$
- $midpoint =(x_1+x_2,y_1+y_2)$
- $midpoint =(\frac{x_1+x_2}{2}/2, \frac{y_1+y_2}{2})$
- $midpoint =((x_1+x_2)/2,(y_1+y_2)/2)$

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.

Choices:
- $2C=\pi r$
- $C=\pi r$
- $C=2 \pi r$
- $C=2r$

High--Intermediate Geometry--Equations of Lines and Circles


Question: What is the formula of area of a circle?$r$ is the radius.

Choices:
- $area = \pi r^2$
- $area = \pi r$
- $area = 2\pi r$
- $area = {\pi r}^2$

High--Intermediate Geometry--Equations of Lines and Circles


Question: What is the formula of area of a sector?

Choices:
- $A = (\pi r^2) /360^ {\circ}$
- $A = (\pi r^2)measure of the central angle /360^ {\circ}$
- $A = (\pi r^2)measure of the central angle /180^ {\circ}$
- $A = (\pi r^2)measure of the central angle /90^ {\circ}$

High--Intermediate Geometry--Equations of Lines and Circles


Question: What is the formula of nscribed angles?

Choices:
- $\angle ABC = measure of intercepted arc$
- $\angle ABC = 2 \ast measure of intercepted arc$
- $\angle ABC = \frac{1}{2} \ast measure of intercepted arc$
- $\angle ABC = 180^{\circ} \ast measure of intercepted arc$

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: The mean is defined as <<<Answer>>> of the numbers.

Choices:
- the average
- the product
- the maximum
- the sum

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: The formula for the mean is represented as<<<Answer>>>.

Choices:
- $number of data * sum of all data$
- $sum of all data + number of data$
- $sum of all data / number of data$
- $number of data - sum of all data$

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What is the definition of the median in a data set?

Choices:
- The largest number
- The middle number
- The smallest number
- The average of the numbers

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What is the definition of the mode in a data set?

Choices:
- The number that shows up the most
- The average of the numbers
- The middle number
- The smallest number

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: How many modes can a data set have?

Choices:
- More than one
- Only one
- No more than two
- None

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What is the range in a data set?

Choices:
- 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

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What does the standard deviation represent in a data set?

Choices:
- 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

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What does the term "correlation" refer to in statistics?

Choices:
- The average of two variables
- The difference between two variables
- The relationship between two variables
- The sum of two variables

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: Correlation is often represented by the symbol ?

Choices:
- r
- d
- c
- q

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:

Choices:
- positively correlated
- Uncorrelated
- Perfectly correlated
- negatively correlated

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:

Choices:
- Negatively correlated
- Perfectly correlated
- Positively correlated
- Uncorrelated

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:

Choices:
- Negative correlation
- No correlation
- Perfect correlation
- Positive correlation

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What is the definition of a ratio?

Choices:
- 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

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: Given two numbers a and b, what is the ratio of a to b?

Choices:
- $a + b$
- $b : a$
- $a : b$
- $a * b$

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What does the term "proportion" refer to in mathematics?

Choices:
- A geometric figure
- A comparison of two numbers
- A statistical measure of variation
- Two fractions that are equal to each other

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: The form of a proportion is typically expressed as:

Choices:
- $(a + b) / c = d$
- $a + b = c + d$
- $ a / b = c / d$
- $a * b = c * d$

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: How do you calculate a percentage?

Choices:
- 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

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?

Choices:
- $(New Value / Old Value) * 100%$
- $(Old Value - New Value) / New Value * 100%$
- $(Old Value + New Value) / 2 * 100%$
- $(New Value - Old Value) / Old Value * 100%$

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?

Choices:
- The value has increased.
- The value has remained the same.
- The value has decreased.
- The calculation is incorrect.

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: If the result of the percentage change calculation is negative, what does it signify?

Choices:
- The calculation is incorrect.
- The value has increased.
- The value has decreased.
- The value has remained the same.

High--Intermediate Numbers and Algebra--Applications of Inequalities


Question: What is the definition of probability?

Choices:
- 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

High--Intermediate Probability and Statistics--Foundations of Probability Theory


Question: What is the correct formula for sin?

Choices:
- $Sin = hypotenuse/opposite$
- $Sin = opposite/hypotenuse$
- $Sin = adjacent/hypotenuse$
- $Sin = hypotenuse/adjacent$

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: What is the correct formula for cos?

Choices:
- $Cos = opposite/hypotenuse$
- $Cos = hypotenuse/opposite$
- $Cos = adjacent/hypotenuse$
- $Cos = hypotenuse/adjacent$

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: What is the correct formula for tan?

Choices:
- $Tan = hypotenuse/opposite$
- $Tan = opposite/adjacent$
- $Tan = adjacent/hypotenuse$
- $Tan = hypotenuse/adjacent$

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: To convert radians to degrees, you multiply the radians by what factor?

Choices:
- $(180°/π)$
- $(π/180°)$
- 180° radians
- π radians

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: To convert degrees to radians, you multiply the degrees by what factor?

Choices:
- π radians
- $(180°/π)$
- 180° radians
- $(π/180°)$

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?

Choices:
- Identical
- Equal
- Irrelevant
- Similar

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: What type of set does $N^{*} $ or $N_{+}$ represent?

Choices:
- Set of negative integers
- Set of irrational numbers
- Set of positive integers
- Set of complex numbers

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What letter represents the set of integers?

Choices:
- A
- S
- Z
- D

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What letter is used to represent the set of rational numbers?

Choices:
- R
- W
- Q
- E

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What does R represent?

Choices:
- The set of irrational numbers
- The set of integers
- The set of real numbers
- The set of complex numbers

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following is not a method of representing a set?

Choices:
- Interval notation
- Graphical representation
- Analogy
- Description

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?

Choices:
- Intersection
- Empty set
- Subset
- Proper subset

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Set A is a subset of set B, how is this represented?

Choices:
- $A \in B$
- $A \le  B$
- $A \subseteq B$
- $A \approx  B$

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?

Choices:
- Proper subset
- Subset
- Intersection
- Union

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Set A is a proper subset of set B, how is this represented?

Choices:
- $A \subseteq B$
- $A \sqsupseteq B$
- $A \subsetneeq B$
- $A =  B$

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is the name of a set that does not contain any elements?

Choices:
- Empty set
- Non-empty proper subset
- Intersection
- Union

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is the representation of the empty set?

Choices:
- $ \ni $
- $ \subset $
- $ \iff $
- $ \varnothing $

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is the subset that every set has?

Choices:
- Non-empty set
- Complex number set
- Real number set
- Empty set

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: If set A contains n elements, how many subsets does set A have?

Choices:
- ${n}^{2}$
- ${n}^{3}$
- $2^{n-1}$
- $2^{n}$

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?

Choices:
- ${n}^{3}-1$
- $2^{n-1}$
- ${n}^{2}$
- $2^{n}-1$

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?

Choices:
- Union; $A\cup B$
- Union; $A\cap B$
- Intersection; $A\cup B$
- Intersection: $A\cap B$

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?

Choices:
- Union; $A\cap B$
- Union; $A\cup B$
- Intersection; $A\cup B$
- Intersection: $A\cap B$

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?

Choices:
- ${C}_{U}A=\left \{ x\mid x\in U\right \}$
- ${C}_{U}A=\left \{ x\mid x\notin U$, and $x\in A\right \}$
- ${C}_{U}A=\left \{ x\mid x\in U$, and $x\notin A\right \}$
- ${C}_{U}A=\left \{ x\mid x\notin U\right \}$

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is a proposition?

Choices:
- Declarative statement
- All the other options are wrong
- A statement that can be judged true or false
- Inferential statement

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following is not a logical connective?

Choices:
- but
- not
- or
- and

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is a simple proposition?

Choices:
- 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

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is a compound proposition?

Choices:
- 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

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: If two statements are each other's contrapositives, then these two statements:

Choices:
- Have no truth relationship
- One is true and the other is false
- Both true or both false
- Both are false

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:

Choices:
- are both wrong
- both true or both false
- have no truth-falsehood relationship
- one true and one false

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Generally, if it is known that $p \Rightarrow q$, then

Choices:
- 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

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: If $p \Leftrightarrow q$, then

Choices:
- 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

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: In general, if it is known that $p \Rightarrow q$, then

Choices:
- 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

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: If $p \Rightarrow q$ but $q \nRightarrow p$, then

Choices:
- 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

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: If $p \nRightarrow q$, but $q \Rightarrow p$, then

Choices:
- 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

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?

Choices:
- A necessary and sufficient condition
- A necessary but not sufficient condition
- A sufficient but not necessary condition
- Neither a sufficient nor a necessary condition

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?

Choices:
- Neither sufficient nor necessary condition
- Necessary but not sufficient condition
- Sufficient and necessary condition
- Sufficient but not necessary condition

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following options is not a form of a compound proposition?

Choices:
- $p \land q$
- $p \lor q$
- $B\not\subset A$
- $\neg p$

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Under what condition is the compound statement "p or q" false?

Choices:
- 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

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Under what condition is the compound statement "p and q" true?

Choices:
- p is true q is false
- p is false q is true
- p is true q is true
- p is false q is false

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is a universal proposition?

Choices:
- 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 $"

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following is a universal quantifier?

Choices:
- $ \forall $
- $ \alpha $
- $ \phi $
- $ \exists $

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is a particular proposition?

Choices:
- 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 $"

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Which of the following is an existential quantifier?

Choices:
- $ \forall $
- $ \alpha $
- $ \exists $
- $ \phi $

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 $\neg p$?

Choices:
- $\exists {x}_{0}\in M, p({x}_{0})$
- $\forall x \in M , p(x)$
- $\forall x \in M ,\neg p(x)$
- $\exists {x}_{0}\in M,\neg p({x}_{0})$

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 $\neg p$?

Choices:
- $\forall x \in M ,\neg p(x)$
- $\exists {x}_{0}\in M, p({x}_{0})$
- $\forall x \in M, p(x)$
- $\exists {x}_{0}\in M,\neg p({x}_{0})$

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is the negation of a universal proposition?

Choices:
- Existential proposition
- All the other options are wrong
- Existential statement
- Universal proposition

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: What is the negation of an existential proposition?

Choices:
- Universal proposition
- Existential proposition
- All the other options are wrong
- Existence proposition

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

Choices:
- magnitude, velocity
- force, magnitude
- direction, velocity
- magnitude, direction

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?

Choices:
- ["start point", "direction", "length"]
- ["start point", "end point", "length"]
- ["start point", "direction", "end point"]
- ["end point", "direction", "length"]

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>>>.

Choices:
- $\lvert\vec{AB} \rvert$, 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

High--Intermediate Numbers and Algebra--Sets and Common Logical Terms


Question: Non-zero vectors that have the same or opposite direction are called <<<Answer>>>.

Choices:
- opposite vectors
- unit vectors
- zero vectors
- parallel vectors

High--Intermediate Numbers and Algebra--Plane Vectors


Question: is parallel to any vector

Choices:
- Parallel vector
- Zero vector
- Opposite vector
- Unit vector

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>>>

Choices:
- $ \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$

High--Intermediate Numbers and Algebra--Plane Vectors


Question: Vectors of equal length and the same direction are called <<<Answer>>>

Choices:
- zero vector
- unit vector
- opposite vector
- equal vectors

High--Intermediate Numbers and Algebra--Plane Vectors


Question: $ \lvert \vec{a}+\vec{b} \rvert $<<<Answer>>>$ \lvert \vec{a} \rvert+\lvert \vec{b} \rvert $

Choices:
- $ \leq $
- $ \geq $

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 $\vec{a}$

Choices:
- opposite vector
- zero vector
- unit vector
- equal vector

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 $\vec{a}$.

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 $\vec{b}=\Lambda \cdot \vec{a}$?

Choices:
- One
- Four
- Three
- Two

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}$?

Choices:
- Two pairs
- Three pairs
- Four pairs
- One pair

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>>>?

Choices:
- $1$
- $0$
- $5$
- $2$

High--Intermediate Numbers and Algebra--Plane Vectors


Question: $\vec{a}=x \cdot \vec{i}+y \cdot \vec{j}=$<<<Answer>>>

Choices:
- $(i,j)$
- $(x,j)$
- $(i,y)$
- $(x,y)$

High--Intermediate Numbers and Algebra--Plane Vectors


Question:
Let $\vec{a}=（x_1,y_1)$ and $\vec{b}=（x_2,y_2)$, then:
$\vec{a}+\vec{b}=$<<<Answer>>>.

Choices:
- $(y_1+x_2,x_1+y_2)$
- $(x_1+x_2,y_1+y_2)$
- $(x_1+y_2,y_1+x_2)$
- $(x_1-x_2,y_1-y_2)$

High--Intermediate Numbers and Algebra--Plane Vectors


Question:
Let $\vec{a}=(x_1,y_1)$,$\vec{b}=(x_2,y_2)$, then:
$\vec{a}-\vec{b}=$<<<Answer>>>

Choices:
- $(x_1-x_2,y_1-y_2)$
- $(x_1+x_2,y_1+y_2)$
- $(x_1+x_2,y_1-y_2)$
- $(x_1-x_2,y_1+y_2)$

High--Intermediate Numbers and Algebra--Plane Vectors


Question:
Given $\vec{a}=(x_1,y_1)$,$\vec{b}=(x_2,y_2)$, then:
$\Lambda \cdot \vec{a}=$<<<Answer>>>

Choices:
- $(\Lambda \cdot x_2,\Lambda \cdot y_2)$
- $(\Lambda \cdot x_2,\Lambda \cdot y_1)$
- $(\Lambda \cdot x_1,\Lambda \cdot y_2)$
- $(\Lambda \cdot x_1,\Lambda \cdot y_1)$

High--Intermediate Numbers and Algebra--Plane Vectors


Question:
Given $\vec{a}= (x_1,y_1)$ and $\vec{b}= (x_2,y_2)$, then:
$\vec{a} \parallel \vec{b} \Leftrightarrow x_1 \cdot y_2=$<<<Answer>>>.

Choices:
- $ x_2 \cdot y_2 $
- $ y_2 \cdot x_1 $
- $ x_1 \cdot y_1 $
- $ x_2 \cdot y_1 $

High--Intermediate Numbers and Algebra--Plane Vectors


Question:
Let $A(x_1,y_1)$ and $B(x_2,y_2)$, then:
$\vec{AB}=$<<<Answer>>>.

Choices:
- $(x_2-x_1,y_1-y_2)$
- $(x_2-x_1,y_2-y_1)$
- $(x_1-x_2,y_2-y_1)$
- $(x_1-x_2,y_1-y_2)$

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>>>.

Choices:
- $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$
- $(\frac{x_1+y_2}{2},\frac{y_1+x_2}{2})$
- $(\frac{y_1+x_2}{2},\frac{x_1+y_2}{2})$
- $(\frac{x_1+x_3}{2},\frac{y_3+y_2}{2})$

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>>>.

Choices:
- $(\frac{x_1+x_2+x_3}{2},\frac{y_1+y_2+y_3}{2})$
- $(\frac{x_1+x_2+y_3}{3},\frac{y_1+y_2+x_3}{3})$
- $(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$
- $(\frac{x_1+y_2+x_3}{3},\frac{y_1+x_2+y_3}{3})$

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles


Question: $\vec{a}\cdot \vec{b}=\lvert \vec{a} \rvert \cdot \lvert \vec{b} \rvert $<<<Answer>>>

Choices:
- $\cot \theta$
- $\sin \theta$
- $\tan \theta$
- $\cos \theta$

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)

Choices:
- $\cos \theta$
- $\tan \theta$
- $\cot \theta$
- $\sin \theta$

High--Intermediate Numbers and Algebra--Plane Vectors


Question: $\vec{a}^2$<<<Answer>>>$\lvert \vec{a} \rvert^2$

Choices:
- $\neq$

High--Intermediate Numbers and Algebra--Plane Vectors


Question: $\lvert \vec{a} \rvert=$\sqrt{<<<Answer>>>}$

Choices:
- $\vec{a}^3$
- $\vec{a}^2$
- $\vec{a}$
- $\vec{a}^4$

High--Intermediate Numbers and Algebra--Plane Vectors


Question: $\vec{a} \bot \vec{b} \Leftrightarrow \vec{a} \cdot \vec{b}=$

Choices:
- $0$
- $5$
- $10$
- $1$

High--Intermediate Numbers and Algebra--Plane Vectors


Question: Given $\vec{a}$=(5,6), $\vec{b}$=(0,8), then $\vec{a}*\vec{b}$ =

Choices:
- 48
- 39
- 32
- 29

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?

Choices:
- Geometric sequence
- Decreasing sequence
- Increasing sequence
- Arithmetic sequence

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?

Choices:
- $ \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} $

High--Intermediate Numbers and Algebra--Sequences


Question: In an arithmetic sequence, the subscripts form what kind of sequence?

Choices:
- Decreasing sequence
- Increasing sequence
- Geometric sequence
- Arithmetic sequence

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?

Choices:
- Constant
- Decreasing
- Increasing
- Oscillating

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 $a_{m+n}$?

Choices:
- 0
- m
- n
- m+n

High--Intermediate Numbers and Algebra--Sequences


Question: If $S_{m}=S_{n}$ and $m \ne n$, what is $S_{m+n}$ equal to?

Choices:
- $n$
- $m$
- 0
- $m+n$

High--Intermediate Numbers and Algebra--Sequences


Question: What is the general term formula for an arithmetic sequence?

Choices:
- $a_n = a_1 + (n - 1)d$
- $a_n = a_1 \cdot r^{n-1}$
- $a_n = \frac{a_1 + a_n}{2} \cdot n$
- $a_n = \frac{a_1 + l}{n}$

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?

Choices:
- Constant sequence
- Geometric sequence
- Arithmetic sequence
- Monotonically increasing sequence

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?

Choices:
- Constant sequence
- Arithmetic sequence
- Monotonically increasing sequence
- Geometric sequence

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?

Choices:
- &\lambda $
- q
- $q^2$
- $q\cdot \lambda $

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 $\left \{ \lg_{}{a_n}  \right \}$?

Choices:
- Geometric sequence
- Neither arithmetic nor geometric sequence
- Both arithmetic and geometric sequence
- Arithmetic sequence

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 $\left \{ \lg_{}{a_n}  \right \}$?

Choices:
- $\lg_{}{q} $
- $q$
- $q_{}^{2} $
- $\ln_{}{q} $

High--Intermediate Numbers and Algebra--Sequences


Question: Regarding the description of conic sections, which of the following is correct?

Choices:
- 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

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?

Choices:
- Parabola
- Ellipse
- Circle
- Hyperbola

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?

Choices:
- Hyperbola
- Parabola
- Ellipse
- Circle

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?

Choices:
- Circle
- Hyperbola
- Ellipse
- Parabola

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?

Choices:
- Hyperbola
- Parabola
- Ellipse
- Circle

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?

Choices:
- Straight line
- Circle
- Parabola
- Ellipse

High--Intermediate Geometry--Equations of Conic Sections


Question: What kind of trajectory is a parabola?

Choices:
- 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

High--Intermediate Geometry--Equations of Conic Sections


Question: When $p>0$, which standard equation of the parabola is correct?

Choices:
- $y=x$
- $x^2=2py$
- $x^2=-2py$
- $y^2=2px$

High--Intermediate Geometry--Equations of Conic Sections


Question: When $p<0$, which standard equation of the parabola is correct?

Choices:
- $y^2=2px$
- $y^2=-2px$
- $x^2=2py$
- $y=x$

High--Intermediate Geometry--Equations of Conic Sections


Question: What is the eccentricity $e$ of a standard parabola?

Choices:
- 3
- 2
- 1
- 4

High--Intermediate Geometry--Equations of Conic Sections


Question: What are the vertex coordinates of a standard parabola?

Choices:
- (2,2)
- (0,0)
- (3,3)
- (1,1)

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?

Choices:
- x = -\frac{b}{2a}
- y = -\frac{b}{2a}
- x = \frac{b}{2a}
- y = \frac{b}{2a}

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?

Choices:
- $x_0$
- $x_0+2$
- $x_0+3$
- $x_0+1$

High--Intermediate Geometry--Equations of Conic Sections


Question: In the standard parabola equation, what is the definition of the latus rectum?

Choices:
- 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

High--Intermediate Geometry--Equations of Conic Sections


Question: In the standard parabola equation, what is the formula for the length of the focal chord?

Choices:
- $|AB|=x_1+x_2+p+3$
- $|AB|=x_1+x_2+p+2$
- $|AB|=x_1+x_2+p$
- $|AB|=x_1+x_2+p+1$

High--Intermediate Geometry--Equations of Conic Sections


Question: In the standard equation of a parabola, what is the geometric meaning of the parameter $p$?

Choices:
- 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

High--Intermediate Geometry--Equations of Conic Sections


Question: In the standard parabola equation, what is the range of the focal radius?

Choices:
- $[p/2，+∞)$
- $[p/2，1)$
- $[p/2，2)$
- $[p/2，0)$

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 $y=±x^2/2p$?

Choices:
- $y?=k+2$
- $y?=k+3$
- $y?=k$
- $y?=k+1$

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 $(x_0, y_0)$?

Choices:
- 2ax_0 + b
- $\frac{b}{2a}$
- $ax_0^2 + bx_0$
- $2a + bx_0$

High--Intermediate Geometry--Equations of Conic Sections


Question: In the standard parabolic equation, when $AB⊥y$, what extreme value does $AB$ take?

Choices:
- Maximum value
- Largest value
- Minimum value
- Any value

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} } } $?

Choices:
- 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 $(a,b)$

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 $(x, y)$?

Choices:
- Parabola
- Hyperbola
- Straight line
- Ellipse

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 $(x,y)$?

Choices:
- Hyperbola
- Straight Line
- Parabola
- Ellipse

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 $(x,y)$?

Choices:
- Straight line
- Parabola
- Ellipse
- Hyperbola

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 $M$?

Choices:
- $\frac{\cos\theta}{\sqrt{A}}$
- $\cos\theta$
- $\frac{\sin \theta}{\sqrt{B}}$
- $\sin\theta$

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 $N$?

Choices:
- cos?\theta
- $\frac{cos?\theta}{\sqrt{A}}$
- sin?\theta
- $\frac{sin\theta}{\sqrt{B}}$

High--Intermediate Geometry--Equations of Conic Sections


Question: In the parametric equation of a circle, what is the value of $x$?

Choices:
- sin?\theta
- cos?\theta
- $b+rsin\theta$
- $a+rcos?\theta$

High--Intermediate Geometry--Equations of Lines and Circles


Question: In the parametric equations of a circle, what is the value of $y$?

Choices:
- cos?\theta
- $a+rcos?\theta$
- sin?\theta
- $b+rsin\theta$

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 $a$?

Choices:
- Ordinate of the circle center
- Abscissa of the circle center
- Curvature
- Radius

High--Intermediate Geometry--Equations of Lines and Circles


Question: In the parametric equation of a circle, what is the geometric meaning of $b$?

Choices:
- Radius
- X-coordinate of the center
- Y-coordinate of the center
- Curvature

High--Intermediate Geometry--Equations of Lines and Circles


Question: In the parametric equation of a circle, what is the geometric meaning of $r$?

Choices:
- Curvature
- Radius
- The ordinate of the center
- The abscissa of the center

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 $x$?

Choices:
- $a^3+cos?\theta$
- $a^2+cos?\theta$
- $b+sin\theta$
- $a+cos?\theta$

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 $y$?

Choices:
- $a^2+cos?\theta$
- $a^3+cos?\theta$
- $b+sin\theta$
- $a+cos?\theta$

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 $a$?

Choices:
- Major axis
- Area
- Minor axis
- Projection

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 $b$?

Choices:
- Imaginary axis
- Projection
- Area
- Real axis

High--Intermediate Geometry--Equations of Conic Sections


Question: For an ellipse with foci on the x-axis, what is the standard equation?

Choices:
- $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1(a>b>0)$
- $\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1(a>b>0)$
- $\dfrac{y^2}{a^2}-\dfrac{x^2}{b^2}=1(a>b>0)$
- $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1(a>b>0)$

High--Intermediate Geometry--Equations of Conic Sections


Question: For an ellipse with foci on the y-axis, what is the standard equation?

Choices:
- $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1(a>b>0)$
- $\dfrac{y^2}{a^2}-\dfrac{x^2}{b^2}=1(a>b>0)$
- $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1(a>b>0)$
- $\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1(a>b>0)$

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?

Choices:
- $-b\leq x\leq b\text{ and }-a\leq y\leq a$
- $x\leq-a\text{ or }x\geq a\mathrm{~,~}y\in R$
- $-a\leq x\leq a\text{ and }-b\leq y\leq b$
- $y\leq-a\text{ or }y\geq a\mathrm{~,~}x\in R$

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?

Choices:
- $y\leq-a\text{ or }y\geq a\mathrm{~,~}x\in R$
- $x\leq-a\text{ or }x\geq a\mathrm{~,~}y\in R$
- $-b\leq x\leq b\text{ and }-a\leq y\leq a$
- $-a\leq x\leq a\text{ and }-b\leq y\leq b$

High--Intermediate Geometry--Equations of Conic Sections


Question: For an ellipse with its foci on the x-axis, what are the vertices?

Choices:
- ${A}_1\left(0,-a\right)\text{、A}_2\left(0,a\right)$
- $\begin{aligned}\mathrm{A}_1&\left(-a,a\right)\text{、A}_2\left(a,a\right)\\\\\mathrm{B}_1&\left(0,-b\right)\text{、B}_2\left(0,b\right)\end{aligned}$
- $\begin{aligned}\mathrm{A}_1&\left(-a,0\right)\text{、A}_2&\left(a,0\right)\\\\\mathrm{B}_1&\left(0,-b\right)\text{、B}_2&\left(0,b\right)\end{aligned}$
- ${A}_1\left(-a,0\right)\text{、A}_2\left(a,0\right)$

High--Intermediate Geometry--Equations of Conic Sections


Question: What is the symmetry of an ellipse?

Choices:
- 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

High--Intermediate Geometry--Equations of Conic Sections


Question: For an ellipse with foci on the x-axis, what are the coordinates of the foci?

Choices:
- $F_1(-c,c)$, $F_2(c,0)$
- $F_1(-c,0)$, $F_2(c,0)$
- $F_1(-c,0)$, $F_2(c,c)$
- $F_1(0,-c)$, $F_2(0,c)$

High--Intermediate Geometry--Equations of Conic Sections


Question: For an ellipse with its foci on the y-axis, what are the foci?

Choices:
- $F_1(-c,c)$, $F_2(c,0)$
- $F_1(-c,0)$, $F_2(c,0)$
- $F_1(-c,0)$, $F_2(c,c)$
- $F_1(0,-c)$, $F_2(0,c)$

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 $\left|F_1F_2\right|=? (c^2=a^2-b^2)$

Choices:
- c
- 2c
- 2b
- 2a

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)

Choices:
- $e=\frac ca=\sqrt{\frac{c^2}{a^2}}=\sqrt{\frac{a^2+b^2}{a^2}}=\sqrt{1+\frac{b^2}{a^2}}$
- $e=\frac ca=\sqrt{\frac{c^2}{a^2}}=\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{1-\frac{c^2}{a^2}}$
- $e=\frac ca=\sqrt{\frac{c^2}{a^2}}=\sqrt{\frac{a^2+b^2}{a^2}}=\sqrt{1+\frac{c^2}{a^2}}$
- $e=\frac ca=\sqrt{\frac{c^2}{a^2}}=\sqrt{\frac{a2-b2}{a^2}}=\sqrt{1-\frac{b^2}{a^2}}$

High--Intermediate Geometry--Equations of Conic Sections


Question: For an ellipse with foci on the x-axis, what is the equation of the directrix?

Choices:
- $y=\pm\frac{a^2}c$
- $y=\pm\frac{b^2}c$
- $x=\pm\frac{a^2}c$
- $x=\pm\frac{b^2}c$

High--Intermediate Geometry--Equations of Conic Sections


Question: For an ellipse with foci on the y-axis, what is the equation of the directrix?

Choices:
- $x=\pm\frac{a^2}c$
- $y=\pm\frac{a^2}c$
- $x=\pm\frac{b^2}c$
- $y=\pm\frac{b^2}c$

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? ($\theta=\angle F_1MF_2$)

Choices:
- $S_{\Delta MF_1F_2}=a^2\tan\frac{\theta}{2}$
- $S_{\Delta MF_1F_2}=b^2\tan\frac{\theta}{2}$
- $S_{\Delta MF_1F_2}=b^2\tan\theta$
- $S_{\Delta MF_1F_2}=b^2\cdot\frac{1}{\tan\frac{\theta}{2}}$

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 $\begin{vmatrix}HH^{\prime}\end{vmatrix}=$?

Choices:
- $\frac{b^2}{a}$
- $\frac{a^2}{b}$
- $\frac{a}{b}$
- $\frac{b}{a}$

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?

Choices:
- $|AB| = \sqrt{1+k}|x_1-x_2| = \sqrt{1+k}\sqrt{(x_1+x_2)^2-4x_1x_2}$
- $|AB| = \sqrt{1+k^2}|x_1-x_2| = \sqrt{1+k^2}\sqrt{(x_1+x_2)^2-2x_1x_2}$
- $|AB| = \sqrt{1+k^2}|x_1-x_2| = \sqrt{1+k^2}\sqrt{(x_1+x_2)^2-4x_1x_2}$
- $|AB| = \sqrt{1-k^2}|x_1-x_2| = \sqrt{1-k^2}\sqrt{(x_1+x_2)^2-4x_1x_2}$

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?

Choices:
- $-\frac{a^2}{b^2}$
- $-\frac{b^2}{a^2}$
- $\frac{a^2}{b^2}$
- $\frac{b^2}{a^2}$

High--Intermediate Geometry--Equations of Conic Sections


Question: Methods to find the equation of an ellipse?

Choices:
- Analogy method
- Definition method, eliminating parameter method
- Geometric construction method
- Summarization and induction method

High--Intermediate Geometry--Equations of Conic Sections


Question: For a hyperbola with foci on the x-axis, what is the standard equation?

Choices:
- $\dfrac{y^2}{a^2}-\dfrac{x^2}{b^2}=1(a>b>0)$
- $\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1(a>b>0)$
- $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1(a>b>0)$
- $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1(a>b>0)$

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?

Choices:
- $-b\leq x\leq b\text{ and }-a\leq y\leq a$
- $y\leq-a\text{ or }y\geq a\mathrm{~,~}x\in R$
- $-a\leq x\leq a\text{ and }-b\leq y\leq b$
- $x\leq-a\text{ or }x\geq a\mathrm{~,~}y\in R$

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?

Choices:
- $-a\leq x\leq a\text{ and }-b\leq y\leq b$
- $y\leq-a\text{ or }y\geq a\mathrm{~,~}x\in R$
- $x\leq-a\text{ or }x\geq a\mathrm{~,~}y\in R$
- $-b\leq x\leq b\text{ and }-a\leq y\leq a$

High--Intermediate Geometry--Equations of Conic Sections


Question: For a hyperbola with foci on the x-axis, what are the vertices?

Choices:
- ${A}_1\left(-a,0\right), {A}_2\left(a,0\right)$
- ${A}_1\left(0,-a\right), {A}_2\left(0,a\right)$
- ${A}_1\left(-a,a\right), {A}_2\left(a,0\right)$
- ${A}_1\left(-a,0\right), {A}_2\left(a,a\right)$

High--Intermediate Geometry--Equations of Conic Sections


Question: For a hyperbola with foci on the y-axis, what are the vertices?

Choices:
- ${A}_1(0,-a), {A}_2(0,a)$
- ${A}_1(-a,0), {A}_2(a,a)$
- ${A}_1(-a,a), {A}_2(a,0)$
- ${A}_1(-a,0), {A}_2(a,0)$

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?

Choices:
- b
- 2a
- 2b
- a

High--Intermediate Geometry--Equations of Conic Sections


Question: What are the symmetries of a hyperbola?

Choices:
- 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

High--Intermediate Geometry--Equations of Conic Sections


Question: For the hyperbola with foci on the x-axis, what are the foci?

Choices:
- $F_1(-c,0), F_2(c,0)$
- $F_1(-c,c), F_2(c,0)$
- $F_1(-c,0), F_2(c,c)$
- $F_1(0,-c), F_2(0,c)$

High--Intermediate Geometry--Equations of Conic Sections


Question: For a hyperbola with foci on the y-axis, what are the foci?

Choices:
- $F_1(-c,c)\, F_2(c,0)$
- $F_1(-c,0)\, F_2(c,0)$
- $F_1(-c,0)\, F_2(c,c)$
- $F_1(0,-c)\, F_2(0,c)$

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 $\left|F_1F_2\right|=? (c^2=a^2+b^2)$

Choices:
- 2b
- c
- 2c
- 2a

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)

Choices:
- $e=\frac ca=\sqrt{\frac{c^2}{a^2}}=\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{1-\frac{c^2}{a^2}}$
- $e=\frac ca=\sqrt{\frac{c^2}{a^2}}=\sqrt{\frac{a^2+b^2}{a^2}}=\sqrt{1+\frac{c^2}{a^2}}$
- $e=\frac ca=\sqrt{\frac{c^2}{a^2}}=\sqrt{\frac{a^2-b^2}{a^2}}=\sqrt{1-\frac{b^2}{a^2}}$
- $e=\frac ca=\sqrt{\frac{c^2}{a^2}}=\sqrt{\frac{a^2+b^2}{a^2}}=\sqrt{1+\frac{b^2}{a^2}}$

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?

Choices:
- $y=\pm\frac{b^2}c$
- $y=\pm\frac{a^2}c$
- $x=\pm\frac{a^2}c$
- $x=\pm\frac{b^2}c$

High--Intermediate Geometry--Equations of Conic Sections


Question: For a hyperbola with foci on the y-axis, what is the equation of the directrix?

Choices:
- $y=\pm\frac{b^2}c$
- $y=\pm\frac{a^2}c$
- $x=\pm\frac{a^2}c$
- $x=\pm\frac{b^2}c$

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?

Choices:
- $y=\pm\frac a b x$
- $y=\pm\frac{b^2}c x$
- $y=\pm\frac b a x$
- $y=\pm\frac{a^2}c x$

High--Intermediate Geometry--Equations of Conic Sections


Question: For a hyperbola with foci on the y-axis, what is the equation of the asymptotes?

Choices:
- $y=\pm\frac{a^2}c x$
- $y=\pm\frac a b x$
- $y=\pm\frac b a x$
- $y=\pm\frac{b^2}c x$

High--Intermediate Geometry--Equations of Conic Sections


Question: For a hyperbola with its foci on the x-axis, what is the focal radius?

Choices:
- $\begin{gathered}M\text{ on the right branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=ex_0+a\\\text{Right focus:}\left|MF_2\right|=ex_0-a\end{cases}\\M\text{ on the left branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=-ex_0-a\\\text{Right focus:}\left|MF_2\right|=-ex_0+a\end{cases}\end{gathered}$
- $\begin{gathered}M\text{ on the upper branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=ey_0+a\\\text{Right focus:}\left|MF_2\right|=ey_0-a\end{cases} \\M\text{ on the lower branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=-ey_0-a\\\text{Right focus:}\left|MF_2\right|=-ey_0+a\end{cases}\end{gathered}$
- $\begin{gathered}M\text{ on the upper branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=ey_0-a\\\text{Right focus:}\left|MF_2\right|=ey_0+a\end{cases} \\M\text{ on the lower branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=-ey_0+a\\\text{Right focus:}\left|MF_2\right|=-ey_0-a\end{cases}\end{gathered}$
- $\begin{gathered}M\text{ on the right branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=ex_0-a\\\text{Right focus:}\left|MF_2\right|=ex_0+a\end{cases}\\M\text{ on the left branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=-ex_0+a\\\text{Right focus:}\left|MF_2\right|=-ex_0-a\end{cases}\end{gathered}$

High--Intermediate Geometry--Equations of Conic Sections


Question: For a hyperbola with foci on the y-axis, what is the focal radius?

Choices:
- $\begin{gathered}M\text{ on right branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=ex_0+a\\\text{Right focus:}\left|MF_2\right|=ex_0-a\end{cases}\\M\text{ on left branch}\begin{cases}\text{Left focus}:\left|MF_1\right|=-ex_0-a\\\text{Right focus:}\left|MF_2\right|=-ex_0+a\end{cases}\end{gathered}$
- $\begin{gathered}M\text{ on top branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=ey_0-a\\\text{Right focus:}\left|MF_2\right|=ey_0+a\end{cases} \\M\text{ on bottom branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=-ey_0+a\\\text{Right focus:}\left|MF_2\right|=-ey_0-a\end{cases}\end{gathered}$
- $\begin{gathered}M\text{ on top branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=ey_0+a\\\text{Right focus:}\left|MF_2\right|=ey_0-a\end{cases} \\M\text{ on bottom branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=-ey_0-a\\\text{Right focus:}\left|MF_2\right|=-ey_0+a\end{cases}\end{gathered}$
- $\begin{gathered}M\text{ on right branch}\begin{cases}\text{Left focus:}\left|MF_1\right|=ex_0-a\\\text{Right focus:}\left|MF_2\right|=ex_0+a\end{cases}\\M\text{ on left branch}\begin{cases}\text{Left focus}:\left|MF_1\right|=-ex_0+a\\\text{Right focus:}\left|MF_2\right|=-ex_0-a\end{cases}\end{gathered}$

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?

Choices:
- $S_{\Delta MF_1F_2}=a^2\tan\frac\theta2$
- $S_{\Delta MF_1F_2}=b^2\tan\frac\theta2$
- $S_{\Delta MF_1F_2}=b^2\tan\theta$
- $S_{\Delta MF_1F_2}=b^2\cdot\frac1{\tan\frac\theta2}$

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?

Choices:
- $\frac{a^2}b$
- $\frac{b^2}a$
- $\frac a b$
- $\frac b a$

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!
When the foci are on the x-axis: $PF_{1}-PF_{2}$ > 0, then point P is on?

Choices:
- Upper branch
- Lower branch
- Right branch
- Left branch

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?

Choices:
- $[b-c,+\infty)$
- $[c-b,+\infty)$
- $[a-c,+\infty)$
- $[c-a,+\infty)$

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?

Choices:
- $b-a$
- $a$
- $a-b$
- $b$

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?

Choices:
- A constant value ab
- A constant value $\frac{ab}{2}$
- A constant value $\frac{b}{2}$
- A constant value $\frac{a}{2}$

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?

Choices:
- Prism
- Cylinder
- Pyramid
- Frustum


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?

Choices:
- Pyramid
- Frustum
- Cone
- Prism


Question: What is the formula for calculating the lateral surface area of a cylinder?

Choices:
- $S=4\pi \cdot r \cdot l$
- $S=2\pi \cdot r \cdot l$
- All the other options are wrong
- $S=\pi \cdot r \cdot l$


Question: What is the formula for calculating the lateral surface area of a cone?

Choices:
- All the other options are wrong
- $S=\pi \cdot r \cdot l$
- $S=4\pi \cdot r \cdot l$
- $S=2\pi \cdot r \cdot l$


Question: How is the area of a rhombus calculated?

Choices:
- Half the product of the diagonals
- Product of the side lengths
- Product of the diagonals
- Half the product of the side lengths

High--Intermediate Geometry--Equations of Lines and Circles


Question: Where are the centers of the circumscribed sphere and inscribed sphere of a cube located?

Choices:
- 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


Question: What shape are all the faces of a regular tetrahedron?

Choices:
- Equilateral triangle
- Ordinary triangle
- Isosceles triangle
- Quadrilateral


Question: What is the shape of the base of a regular triangular pyramid?

Choices:
- Ordinary triangle
- Quadrilateral
- Equilateral triangle
- Isosceles triangle


Question: If the edge length of a regular tetrahedron is a, what is the radius of its circumscribed sphere?

Choices:
- $ \frac{\sqrt{3}}{4} a $
- $ \frac{\sqrt{6}}{4} a $
- $ \frac{\sqrt{6}}{12} a $
- $ \frac{\sqrt{3}}{6} a $


Question: If the edge of a regular tetrahedron has a length of a, then what is the radius of its inscribed sphere?

Choices:
- $ \frac{\sqrt{6}}{12} a $
- $ \frac{\sqrt{3}}{6} a $
- $ \frac{\sqrt{3}}{4} a $
- $ \frac{\sqrt{6}}{4} a $


Question: The centroid of a triangle is the intersection point of which three lines?

Choices:
- bisectors
- perpendicular bisectors
- medians
- altitudes

High--Intermediate Geometry--Trigonometric Functions and Solving Triangles