2021 CSAT Mathematics (Type Ga)

What is the value of ?

What is the value of ?

For where , if , what is the value of ?

For two events , Find the value of .

How many natural numbers satisfy the inequality ?

A sample of size 16 is randomly drawn from a population that follows a normal distribution . If the sample mean is denoted as , what is the value of ?

For the function , if the maximum value and minimum value are and respectively, what is the value of ?

What is the area of the region enclosed by the curve , the -axis, and the two lines ?

There are 5 cards with letters written on them (one letter per card) and 4 cards with numbers written on them (one number per card). When all 9 cards are randomly arranged in a row using each card exactly once, what is the probability that the card with letter A has cards with numbers on both sides immediately adjacent to it?

There is a triangle where and . When the radius of the circumscribed circle of triangle is 7, what is the length of segment ?

What is the value of ?

As shown in the figure, there is a rectangle with . Let be the point that divides the segment in the ratio , and place a point inside the rectangle such that and , then draw triangle . Let be the figure obtained by coloring the quadrilateral . In figure , draw a rectangle with vertices at point , point on segment , point on segment , and point on segment , where . Using the same method as obtaining figure , draw triangle in rectangle and color quadrilateral to obtain figure . Continuing this process, if is the area of the colored region in the -th figure , what is the value of ?

For a differentiable function on , If a differentiable function on satisfies the following conditions, what is the value of ? (a) For all real numbers , . (b)

Point is at the origin of the coordinate plane. Using a single die, the following trial is performed. Roll the die once, and if the number shown is 2 or less, move point by 3 units in the positive direction of the -axis, 3 or more, move point by 1 unit in the positive direction of the -axis. This trial is repeated 15 times, and let be the random variable representing the distance between the moved point and the line . What is the value of ?

For a real number , let the function be defined as What is the sum of all values of such that ?

There is a bag containing 5 balls with the numbers written on them, one number per ball. Using this bag and one die, a trial is conducted to obtain a score according to the following rules: Draw one ball randomly 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 that appear 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 that appear as the score. What is the probability that the score obtained from performing this trial once is 10 points?

For the function , let be a function with domain of all real numbers and range , and let be a natural number satisfying the following conditions. Find the value of . The function is continuous on the set of all real numbers, and

The sequence satisfies and the following conditions for all natural numbers . (a) (b) When , what is the value of ?

Find the coefficient of in the expansion of .

For the function , find the value of .

As shown in the figure, in right triangle with , let and be the points where a circle centered at with radius 1 intersects line segments and , respectively. Let be the trisection point of arc that is closer to point , and let be the point where line intersects line segment . When , let be the area of the common region inside triangle and outside sector , and let be the area of sector . Find the value of . (where )

For an arithmetic sequence with first term 3, given that , find the value of .

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

Find the number of natural numbers such that the value of is a natural number less than or equal to 40.

For two constants , let the function be For the inverse function of the function , when the composite function satisfies the following conditions, find the value of . (a) The function is differentiable on the set of all real numbers. (b)

Find the number of ways to distribute 6 black hats and 6 white hats to four students according to the following rules without any remaining hats. (Note that hats of the same color are indistinguishable.) (a) Each student receives at least 1 hat. (b) Student receives at least 4 black hats. (c) Only 2 students including receive more black hats than white hats.

For a cubic function with leading coefficient 1, a function defined on the set of all real numbers satisfies the following conditions. (a) In the interval , the number of values of where function has local maxima is 3, and all these local maximum values are equal. (b) The maximum value of function is and the minimum value is 0. When , find the value of . (Here, and are rational numbers.)