2022 CSAT Mathematics (Elective: Probability and Statistics)

What is the value of ?

For the function , what is the value of ?

For the arithmetic sequence , Find the value of .

The graph of the function is shown as follows. What is the value of ?

For a sequence with first term 1, where for all natural numbers : Find the value of .

How many integers make the equation have three distinct real roots?

For where , if , what is the value of ?

When the line bisects the area of the region enclosed by the curve and the line , what is the value of the constant ?

The line intersects the graphs of the two functions at points and respectively. When , what is the value of the constant ?

For a cubic function , when the tangent line at point on the curve coincides with the tangent line at point on the curve , what is the value of ?

For a positive number , there is a function defined on the set : As shown in the figure, there is a line passing through three points , , and on the graph of the function . Let be the point other than where the line passing through point and parallel to the -axis intersects the graph of the function . When triangle is an equilateral triangle, what is the area of triangle ? (Note: is the origin.)

A function continuous on the set of all real numbers satisfies for all real numbers . Given that the maximum value of function is 1 and the minimum value is 0, find the value of .

For two constants , the -intercept of the line passing through the two points on the coordinate plane is equal to the -intercept of the line passing through the two points . For the function , when , what is the value of ?

A point moves on a number line, and its position at time is given by two constants as: When the velocity of point at time satisfies , which of the following statements from the options are correct? A. B. There exists in the open interval such that . C. If for all where , then there exists in the open interval such that .

There are two circles with centers respectively and radius length . As shown in the figure, there are three distinct points on circle and a point on circle , where the three points and the three points are each on a straight line. Let . The following is the process of finding the ratio of the lengths of segments and when and . Since , we have and from we get , so . At this time, since , triangles and are congruent. Let Since , we have , and since , we have . In triangle , since , by the cosine law, . Since , . Let the expressions that fit in (A) and (C) above be and respectively, and let the number that fits in (B) be . What is the value of ?

Find the value of .

For the function , given that and , find the value of .

For the sequence , Find the value of .

Find the maximum value of the real number such that the function is increasing on the entire set of real numbers.

A function that is differentiable on the set of all real numbers satisfies the following conditions. (a) On the closed interval , . (b) For some constants , on the interval , . Find the value of .

The sequence satisfies the following conditions. (a) (b) For all natural numbers , (c) Find the value of . [4 points]

Let be a cubic function with leading coefficient and let be the number of real roots of the equation in the closed interval for a real number . The function satisfies the following conditions. (A) For all real numbers , . (B) Find the value of .

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

If random variable follows a binomial distribution and , what is the value of ?

Find the number of all ordered quintuples of natural numbers satisfying the following conditions: (Condition 1) (Condition 2)

There is a bag containing 10 cards, each with a natural number from 1 to 10 written on it. When randomly drawing 3 cards simultaneously from this bag, what is the probability that the smallest number among the three natural numbers written on the drawn cards is 4 or less, or 7 or more?

The driving range per charge of electric vehicles produced by a certain automobile company follows a normal distribution with mean and standard deviation . When 100 electric vehicles produced by this automobile company are randomly sampled and the sample mean of their driving range per charge is , the 95% confidence interval for the population mean is . When 400 electric vehicles produced by this automobile company are randomly sampled and the sample mean of their driving range per charge is , the 99% confidence interval for the population mean is . Given that and , find the value of . (Note: The unit of driving range is , and when is a random variable following a standard normal distribution, calculate using and .)

For two sets , find the number of functions from to that satisfy the following conditions: (a) For every element in set , . (b) The range of function has exactly 3 elements.

Two continuous random variables and have ranges and , and their probability density functions are and respectively. The graph of the probability density function of random variable is shown in the figure. For all such that : When this condition is satisfied, . Find the value of . (Here, and are relatively prime natural numbers.) [4 points]

There is a basket containing at least 10 white balls and 10 black balls each, and an empty pocket. The following experiment is performed using one die. Roll the die once If the number shown is 5 or greater, put 2 white balls from the basket into the pocket, If the number shown is 4 or less, put 1 black ball from the basket into the pocket. When this experiment is repeated 5 times, let be the number of white balls and black balls in the pocket after the -th trial, respectively. When , the probability that there exists a natural number such that is . Find the value of . (Note: and are relatively prime natural numbers.)