2025 Academic Year College Scholastic Ability Test – Probability and Statistics

What is the value of ?

For the function , what is the value of ?

For a geometric sequence where both the first term and common ratio are positive , if it satisfies what is the value of ?

When the function is continuous on the set of all real numbers, what is the value of the constant ?

For the function , what is the value of ?

When , what is the value of ?

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

For two real numbers , , what is the value of ?

For the function , when , what is the value of the positive number ?

For the function defined on the closed interval , find the minimum value of for ordered pairs of natural numbers such that the function has a maximum value of at .

A point P moves along a number line starting at time . The position of point P at time is given by What is the acceleration of point P at the time when the direction of motion changes after starting?

Given a sequence with and an arithmetic sequence with that satisfy for all natural numbers , find the value of .

A cubic function with leading coefficient satisfies For the origin and point , let be the point where line segment intersects the curve , other than point . Let be the area enclosed by the curve , the -axis, and line segment , and let be the area enclosed by the curve and line segment . Find the value of .

As shown in the figure, in triangle ABC, point D is taken on segment AB such that , and let O be the circle centered at point A and passing through point D, and let E be the point where circle O intersects segment AC. Given that , and the ratio of the areas of triangle ADE and triangle ABC is . When the radius of the circumcircle of triangle ABC is , what is the maximum value of the area of triangle PBC for a point P on circle O? (Note: )

For a constant and a quadratic function with a negative leading coefficient, the function satisfies the following conditions. (a) The function is differentiable on the entire set of real numbers. (b) The number of distinct real roots of the equation with respect to is . What is the value of ?

Find the value of the real number that satisfies the equation

Given that the sequence satisfies for all natural numbers , find the value of .

Let be the -coordinate of the intersection point of the curve and the line . A function defined on the set of all real numbers satisfies the following conditions. For all real numbers such that , and . Find the value of .

For all sequences where all terms are integers and satisfy the following conditions, find the sum of all possible values of . (a) For all natural numbers , (b) The minimum value of the natural number such that is .

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

For two events and , given that what is the value of ?

A sample of size 256 is randomly drawn from a population that follows the normal distribution N(m, 2²). Using the sample mean, a 95% confidence interval for m is obtained as a ≤ m ≤ b. What is the value of b - a? (Given that when Z is a random variable following the standard normal distribution, P(|Z| ≤ 1.96) = 0.95.)

A survey was conducted on the preferences for subject A and subject B among 16 students in a class. Each student who participated in the survey chose one of subject A or subject B. 9 students chose subject A and 7 students chose subject B. When 3 students are randomly selected from the 16 students who participated in the survey, what is the probability that at least one of the selected 3 students chose subject B?

There is a bag containing 5 cards with the numbers written on them, one number per card. A trial is performed where one card is randomly drawn from the bag, the number on the card is recorded, and then the card is put back. This trial is repeated 3 times, and let be the mean of the three numbers obtained. If , what is the value of the positive number ?

Random variable follows normal distribution and random variable follows normal distribution satisfying the following conditions. For all real numbers : and . When the value of obtained using the following standard normal distribution table is , find the value of . (Note that and are positive numbers.)