June 2023 National Unified Academic Achievement Test (Grade 11)

What is the value of ?

What is the value of ?

What is the area of a sector with radius length and central angle measure ?

When , what is the solution to the equation ?

In triangle ABC where , , and , what is the length of segment BC?

For two constants , when the graph of the function looks like the figure, what is the value of ? (Note that the line is an asymptote of the function's graph.)

The graph of the function is translated parallel by units in the direction of the -axis, then reflected with respect to the line . This results in a graph that coincides with the graph of the function . What is the value of the constant ?

For where and , what is the value of ?

For three constants , when the graph of the function is as shown in the figure, what is the value of ? (where )

There is a triangle ABC inscribed in a circle with radius 4. When the perimeter of this triangle is 12, what is the value of ?

How many integers satisfy the inequality ?

For two natural numbers that satisfy the equation what is the value of ?

Given a function defined on . Let A be the point where the graph of function intersects the -axis, and let B and C be the two points where it intersects the line . When triangle ABC is an equilateral triangle, what is the value of ?

In the coordinate plane, for a point P on the curve , let be the measure of the angle represented by the ray OP. When , what is the length of line segment OP? (Here, O is the origin, and the positive direction of the x-axis is taken as the initial line.)

As shown in the figure, there are two curves and three points . For a real number , let the points where the two curvesmeet the line be respectively, and the points where they meet the line be respectively. When the area of quadrilateral equals the area of triangle , what is the value of ?

For a natural number , let be the maximum value of the function on the interval . For natural numbers , what is the sum of all values of such that is irrational?

Find the value of .

Find the solution to the equation .

When the periods of the two functions and are equal, find the value of the positive number .

When the graph of the function passes through the point , find the value of the constant .

Find the value of for the real number that satisfies the equation

For two subsets of the set of all natural numbers , given that and the sum of all elements in set is , find the sum of all elements in set . (Note that , , are three distinct natural numbers.)

For a natural number , when , let be the sum of all real roots of the equation with respect to : Find the value of .

As shown in the figure, there is a triangular piece of paper ABC with . For point D on segment BC, point E on segment AB, and point F on segment AC, the paper is folded along segment EF so that point A coincides with point D. When the ratio of the radii of the circumcircles of triangle BDE and triangle DCF is , the length of segment DF is . Find the value of . (Note: The thickness of the paper is not considered, and and are relatively prime natural numbers.)

For the function (where is a real number) and a positive real number , define the function asFind the sum of all values of when the number of intersection points between the graph of the function and the line is and .