2022 CSAT Mathematics (Elective - Geometry)

What is the value of ?

For the function , what is the value of ?

For the arithmetic sequence , Find the value of .

The graph of the function is shown as follows. What is the value of ?

For a sequence with first term 1, where for all natural numbers : Find the value of .

How many integers make the equation have three distinct real roots?

For where , if , what is the value of ?

When the line bisects the area of the region enclosed by the curve and the line , what is the value of the constant ?

The line intersects the graphs of the two functions at points and respectively. When , what is the value of the constant ?

For a cubic function , when the tangent line at point on the curve coincides with the tangent line at point on the curve , what is the value of ?

For a positive number , there is a function defined on the set : As shown in the figure, there is a line passing through three points , , and on the graph of the function . Let be the point other than where the line passing through point and parallel to the -axis intersects the graph of the function . When triangle is an equilateral triangle, what is the area of triangle ? (Note: is the origin.)

A function continuous on the set of all real numbers satisfies for all real numbers . Given that the maximum value of function is 1 and the minimum value is 0, find the value of .

For two constants , the -intercept of the line passing through the two points on the coordinate plane is equal to the -intercept of the line passing through the two points . For the function , when , what is the value of ?

A point moves on a number line, and its position at time is given by two constants as: When the velocity of point at time satisfies , which of the following statements from the options are correct? A. B. There exists in the open interval such that . C. If for all where , then there exists in the open interval such that .

There are two circles with centers respectively and radius length . As shown in the figure, there are three distinct points on circle and a point on circle , where the three points and the three points are each on a straight line. Let . The following is the process of finding the ratio of the lengths of segments and when and . Since , we have and from we get , so . At this time, since , triangles and are congruent. Let Since , we have , and since , we have . In triangle , since , by the cosine law, . Since , . Let the expressions that fit in (A) and (C) above be and respectively, and let the number that fits in (B) be . What is the value of ?

Find the value of .

For the function , given that and , find the value of .

For the sequence , Find the value of .

Find the maximum value of the real number such that the function is increasing on the entire set of real numbers.

A function that is differentiable on the set of all real numbers satisfies the following conditions. (a) On the closed interval , . (b) For some constants , on the interval , . Find the value of .

The sequence satisfies the following conditions. (a) (b) For all natural numbers , (c) Find the value of . [4 points]

Let be a cubic function with leading coefficient and let be the number of real roots of the equation in the closed interval for a real number . The function satisfies the following conditions. (A) For all real numbers , . (B) Find the value of .

In coordinate space, let point be the reflection of point across the plane, and let point be the reflection of point across the plane. What is the length of line segment ?

What is the length of the major axis of the hyperbola where one focus has coordinates ? (where is a positive number)

In the coordinate plane, let be the acute angle formed by the two lines What is the value of ?

There is a point in the first quadrant on the ellipse with foci . Let be the circle that is simultaneously tangent to both lines and , has its center on the -axis, and has a negative -coordinate for its center. Let be the center of circle . If the area of quadrilateral is 72, what is the length of the radius of circle ?

As shown in the figure, there is a cube with edge length 4. If the midpoint of segment is , what is the area of triangle ?

For two positive numbers , let be the focus of the parabola , and let be the focus of the parabola . When the line segment intersects the two parabolas at points and respectively, we have and . What is the value of ?

In the coordinate plane, for parallelogram where , , and , point satisfies the following conditions. (a) where (b) For point moving on a circle centered at point and passing through point , let and be the maximum and minimum values of , respectively. If , find the value of . (Note: and are rational numbers.)

In coordinate space, there is a sphere with center and passing through point : Let be a point moving on the circle formed by the intersection of sphere and plane , and let be a point moving on sphere . Let and be the orthogonal projections of points and onto the -plane, respectively. For the two points and that maximize the area of triangle , the area of the orthogonal projection of triangle onto plane is . Find the value of . (Note that is the origin, the three points are not collinear, and and are coprime natural numbers.)