November 2023 National Unified Academic Achievement Test (Grade 11)

What is the value of ?

When the four numbers form an arithmetic sequence in this order, what is the value of ?

The graph of function is shown in the figure. What is the value of this expression?

In the interval , how many points do the graph of the function and the line intersect?

What is the minimum value of the natural number such that the curve intersects with the curve in the first quadrant?

When , what is the value of ?

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

For a polynomial function , let function be defined as If , what is the value of ?

Two sequences , satisfy for all natural numbers . When , what is the value of ?

For a real number where , let be the distance between the two points where the graph of the function and the line intersect. Find the value of .

In triangle ABC where as shown in the figure, let M be the midpoint of segment AB and N be the point that divides segment AC in the ratio 3:5. Given that and the area of the circumcircle of triangle AMN is , what is the area of triangle ABC?

For all sequences where all terms are natural numbers and satisfy the following conditions, let and be the maximum and minimum values of , respectively. Find the value of . (a) (b) For all natural numbers :

Find the value of .

Find the length of the radius of a sector whose central angle measures and whose arc length is .

For the function , find the value of .

Two points P and Q move on a number line. Their positions at time are given byrespectively. When the directions of motion of both points P and Q change simultaneously at time , find the value of . (where and are constants.)

Find the sum of all natural number values of such that an arithmetic sequence with common difference satisfies the following conditions. (a) (b) For all natural numbers , .

For two constants , the function on the closed interval has a maximum value of and a minimum value of . When , find the value of . (where and are relatively prime natural numbers.)