October 2024 National Unified Academic Achievement Test (Grade 11)

When a polynomial function satisfies , what is the value of ?

What is the period of the function ?

Given that function satisfies for all real numbers , what is the value of ?

For an arithmetic sequence with a positive first term and common difference , ifwhat is the value of ?

Let be the sum of the first terms of the sequence . When , what is the value of ?

Given that the sequence satisfies for all natural numbers , what is the value of ?

In the coordinate plane, for a point P on the line with positive -coordinate, let be the measure of the angle represented by the ray . When the ray representing angle and the ray representing angle coincide, what is the -coordinate of point P? (Note: O is the origin, and the positive direction of the -axis is the initial side.)

For a natural number , let be the number of real th roots of . Find the value of .

When two natural numbers and greater than and less than satisfy what is the maximum value of ?

As shown in the figure, for two real numbers and greater than , the curve and the line intersect at two distinct points A and B. Let C be the point where the line passing through point B with slope meets the curve , and let D be the point where the line meets the y-axis. When and , what is the value of ? (Note: The x-coordinate of point B is greater than the x-coordinate of point A.)

For a sequence with first term , if the sequence satisfies for all natural numbers , what is the sum of all values of such that ?

Find the value of when the arc length of a sector with radius and area is .

Let be an arithmetic sequence with a positive integer common difference. Let be the sum of the first terms from the first term to the -th term. For some natural number , if holds, find the value of .

For two positive numbers and a quadratic function with leading coefficient , let be a function defined on the set by where the function is continuous at . For a real number , let be the number of intersection points between the graph of and the line . The function satisfies the following conditions: (a) For any two positive numbers , if , then . (b) The function is discontinuous only at , , where , and , . Find the value of .