2023 CSAT Mathematics (Elective: Probability and Statistics)

What is the value of ?

For a geometric sequence with a positive common ratio that satisfies find the value of .

For a polynomial function , let function be defined as Given that and , what is the value of ?

Given that and , what is the value of ?

The function has a local maximum at and a local minimum at . Find the value of . (where are constants.)

For an arithmetic sequence where all terms are positive and the first term equals the common difference, if is satisfied, what is the value of ?

What is the -intercept of the tangent line drawn from the point to the curve ?

The function has a maximum value of 7 and a minimum value of 3 on the closed interval . Find the value of . (where are constants.)

Let be the area of the region enclosed by the two curves and the -axis, and let be the area of the region enclosed by the two curves and the line . When , what is the value of the constant ? (Given that )

As shown in the figure, quadrilateral is inscribed in a circle and What is the length of the radius of this circle?

A function continuous on the set of all real numbers satisfies the following condition. When , . (where is a natural number) The function defined on the open interval has a minimum value of 0 at . Find the value of .

For a natural number , let be the number of natural numbers such that there exists an integer among the -th roots of . Find the value of .

For a polynomial function , function is defined as follows: For the function , which of the following statements in are correct? A. B. Function is continuous on the set of all real numbers. C. If function is decreasing on the closed interval and , then function has a minimum value on the set of all real numbers.

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 real number that satisfies the equation

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

For two sequences , Find the value of .

Find the number of integers such that the equation has exactly 2 distinct positive real roots.

A point P moving on a number line has velocity and acceleration at time that satisfy the following conditions. (a) When , . (b) When , . Find the distance traveled by point P from time to .

For a natural number , let the function be defined as For a real number , let be the number of distinct real roots of the equation in . Find the sum of all natural numbers such that the maximum value of the function is 4.

Given that is a cubic function with leading coefficient 1 and is a function continuous on the set of all real numbers, find the value of when the following conditions are satisfied: (a) For all real numbers : (b) The minimum value of function is . (c)

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

From the numbers , allowing repetition, 4 numbers are chosen and arranged in a row to form four-digit natural numbers. How many of these numbers are odd numbers that are 4000 or greater?

A box contains 5 white masks and 9 black masks. When randomly drawing 3 masks simultaneously from this box, what is the probability that at least one of the drawn 3 masks is a white mask?

A bag contains 1 white ball with the number 1, 1 white ball with the number 2, 1 black ball with the number 1, and 3 black balls with the number 2. From this bag, 3 balls are randomly drawn simultaneously. Let be the event that among the 3 drawn balls, there is 1 white ball and 2 black balls, and let be the event that the product of all the numbers written on the 3 drawn balls equals 8. Find the value of .

The volume of one shampoo bottle produced by a company follows a normal distribution . A 95% confidence interval for using the sample mean obtained from randomly selecting 16 shampoo bottles from this company is . When a 99% confidence interval for using the sample mean obtained from randomly selecting bottles from this company is , what is the minimum natural number such that is at most 6? (Note that the unit of volume is mL, and when is a random variable following a standard normal distribution, calculate using and .)

The range of values for the continuous random variable is , and the graph of the probability density function of is as shown in the figure. Given that , find the value of . (Note: , , are constants.)

For the set is a natural number less than or equal to 10, find the number of functions that satisfy the following conditions. (a) For all natural numbers less than or equal to 9, . (b) When , , and when , . (c)