2021 CSAT Mathematics (Type Na)

What is the value of ?

For a geometric sequence with first term , if , what is the value of ?

What is the value of ?

What is the maximum value of the function ?

Two events and are independent, and Find the value of .

For the function , what is the value of ?

How many natural numbers satisfy the inequality ?

When rolling a single die three times and letting the numbers that come up be in order, what is the probability that ?

What is the x-intercept of the line that is perpendicular to the tangent line at point on the curve and passes through point ?

For two sequences , Find the value of .

A population follows a normal distribution . A random sample of size 16 is drawn from this population, and the sample mean is denoted as . What is the value of ?

The sequence has and satisfies for all natural numbers . What is the value of ?

For the set , find the number of functions that satisfy the following condition:

A point moves on a number line with velocity at time given by If the distance traveled by point from time to is 25, what is the value of the constant ?

There are 6 students including three students . Find the number of ways these 6 students can sit around a circular table at equal intervals satisfying the following conditions. (Note: Arrangements that coincide when rotated are considered the same.) (a) and are adjacent. (b) and are not adjacent.

When , what is the sum of all solutions to the equation ?

Two polynomial functions satisfy For the function , what is the value of ?

For a real number , define the function as Let the function What is the maximum value of such that has exactly one extremum?

Find the coefficient of in the expansion of the polynomial .

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

Find the value of .

Find the positive value of such that the curve and the line intersect at exactly 2 points.

Given the function Find the value of when this function is continuous on the set of all real numbers. (Here, and are constants.)

Find the area of the region enclosed by the curve and the line .

There is a triangle with and . When the radius of the circumcircle of triangle is 7, let the length of segment be . Find the value of .

There is a bag containing 5 balls with the numbers written on them, one on each ball. Using this bag and one die, a trial is conducted to obtain a score according to the following rules. Randomly draw one ball from the bag. If the number written on the drawn ball is 3, roll the die 3 times and use the sum of the three numbers shown as the score. If the number written on the drawn ball is 4, roll the die 4 times and use the sum of the four numbers shown as the score. The probability of obtaining a score of 10 points in one trial is . Find the value of . (where and are coprime natural numbers.)

Function is a cubic function with leading coefficient 1, and function is a linear function. Define function as: If function is differentiable on the entire set of real numbers and , find the value of .