October 2024 National Unified Academic Achievement Test (Grade 12) – Calculus

What is the value of ?

For the function , what is the value of ?

For where and , what is the value of ?

What is the value of ?

For the function to be continuous on the set of all real numbers, what is the sum of all constant values of ?

Let be the sum of the first terms of a geometric sequence with positive common ratio. Given that find the value of .

When a line passing through the point with slope bisects the area of the region enclosed by the graph of the function , the -axis, and the two lines and , what is the value of the constant ?

On the coordinate plane, there are two points and . When the point that externally divides the line segment in the ratio lies on the line , what is the value of the positive number ?

At time , two points and simultaneously start from the origin and move along a number line. Their velocities at time are respectivelyAfter starting, for a positive number such that the two points and meet exactly once, what is the distance traveled by point from time to time ?

As shown in the figure, for quadrilateral ABCD inscribed in a circle with Let , , then . If the intersection point of line segments AC and BD is E, what is the length of line segment AE? (Given that )

For a quartic function with leading coefficient , the function is differentiable on the entire set of real numbers, and the equation of the tangent line at point on the curve is . What is the sum of all distinct real values of such that ?

For a sequence where all terms are natural numbers, satisfying the following condition for all natural numbers : Find the sum of all values of such that .

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

For the function , find the value of .

For a sequence and constant , given that find the value of .

For two natural numbers and , the function is defined as: For a real number , let be the number of distinct real roots of the equation with respect to . When the function satisfies the following conditions, find the minimum value of . (a) The range of function is . (b) The number of natural numbers for which is .

What is the value of ?

What is the value of ?

For all natural numbers such that the sequence converges, when what is the value of ? (Here, and are constants.)

As shown in the figure, there is a solid figure with its base being the region enclosed by the curve , the -axis, and the two lines , . When this solid figure is cut by planes perpendicular to the -axis, all cross-sections are squares. What is the volume of this solid figure?

For the function (where is a constant) and constant , consider the function If is continuous on the set of all real numbers and has an inverse function, what is the value of ?

Let the function and the function intersect at points whose -coordinates are positive. When all such positive -coordinates are arranged in ascending order, let denote the -th number. What is the value?

There is a line passing through the point with positive slope and a curve . When the angle between line and the positive direction of the -axis is , let be the -coordinate of the point where line intersects the curve in the first quadrant. Given that , we have . Find the value of . (Here, is a constant and are integers.)

For two constants , when the function satisfies the following conditions, find the value of . (a) The number of real numbers that satisfy is . (b)