October 2024 National Unified Academic Achievement Test (Grade 12) — Probability and Statistics

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 .

How many ways are there to arrange the 4 letters in a row?

Two events and are independent and What is the value of ?

What is the coefficient of in the expansion of the polynomial ?

A company produces reusable cups where the weight of each cup follows a normal distribution with mean and standard deviation . When cups are randomly selected from this company's production and the sample mean is , the confidence interval for the population mean is . What is the value of ? (Note: The unit of weight is , and when is a random variable following a standard normal distribution, calculate using .)

There is a box containing balls. Each ball has one of the numbers , , or written on it. Two balls are randomly drawn simultaneously from this box, and let be the random variable representing the product of the two numbers on the drawn balls. When the random variable satisfies what is the value of ?

Point P is at the origin of a number line. A bag contains 4 cards with the numbers 1, 2, 3, 4 written on them, one number per card. The following trial is performed using this bag. Randomly draw one card from the bag, check the number written on the card, and then put it back in the bag. If the confirmed number k is odd, move point P in the positive direction by k units, and if it is even, move point P in the negative direction by k units. After repeating this trial 4 times, when the coordinate of point P is 0 or greater, the probability that the product of the four confirmed numbers is odd is . Find the value of . (Here, and are relatively prime natural numbers.)