2020 CSAT Mathematics (Type Ga)

For two vectors , what is the sum of all components of the vector ?

What is the value of ?

In coordinate space, if a point on the -axis with coordinates is equidistant from two points and , what is the value of ?

What is the coefficient of in the expansion of ?

What is the slope of the tangent line to the curve at the point ?

A bag contains 3 white balls and 4 black balls. When 4 balls are randomly drawn simultaneously from this bag, what is the probability of getting 2 white balls and 2 black balls?

What is the value of ?

The position of a point moving on the coordinate plane at time is given by What is the minimum speed of point for ?

In isosceles triangle where , let and . If , what is the value of ?

How many integers are there such that the curve has an inflection point?

As shown in the figure, for a positive number , consider the region bounded by the curve , the -axis, the -axis, and the line as the base, and a solid whose cross-sections perpendicular to the -axis are all squares. If the volume of this solid is , what is the value of ?

As shown in the figure, let the ellipse have foci at two points and . Let be the point where the ellipse intersects the -axis with positive -coordinate. Let be the point where the line intersects the line , and let be the point where the line intersects the ellipse with positive -coordinate. If the difference between the perimeter of triangle and the perimeter of triangle is 4, find the area of triangle . (Given that and )

Let A be the point where the graph of the exponential function intersects the line . For point , what is the product of all values of such that line and line are perpendicular to each other? (Note: is the origin.)

Find the number of all ordered pairs of non-negative integers that satisfy the following conditions: (a) (b) and .

There is an equilateral triangle with side length 10 in a plane. For a point that satisfies , when the length of segment is minimized, what is the area of triangle ?

Four distinct points on a circle satisfy the following conditions. Find the value of . (a) (b)

When flipping a coin 7 times, what is the probability that the following conditions are satisfied? (a) Heads appears 3 or more times. (b) There is at least one case where heads appears consecutively.

For the function , find the value of .

A random variable follows a binomial distribution and . Find the value of .

In the coordinate plane, let be a circle centered at point on the curve and tangent to the -axis. Let be the point where circle touches the -axis, and let be the point where the circle intersects line segment . If , find the value of . (Here, is the origin, and are integers.)

When rolling a die 5 times, let be the number of times an odd number appears. When flipping a coin 4 times, let be the number of times heads appears. If the probability that equals 3 is , find the value of . (where and are relatively prime natural numbers.)

For the function , suppose function is differentiable and satisfies Find the value of .

As shown in the figure, there is a rhombus-shaped paper with side length 4 and . Let and be the midpoints of sides and respectively. The paper is folded along the three line segments to form a tetrahedron . The area of the orthogonal projection of triangle onto plane is . Find the value of . (Note: The thickness of the paper is not considered, is the point where the three points coincide when the paper is folded, and and are coprime natural numbers.)

From the numbers , allowing repetition, select five numbers that satisfy the following conditions, then arrange them in a row to find the total count of all possible five-digit natural numbers. (a) Each odd number is either not selected or selected exactly once. (b) Each even number is either not selected or selected exactly twice.

In coordinate space, for two points , let be two planes that contain the line segment and are tangent to the sphere . When the points of tangency of the two planes with the sphere are respectively, the volume of tetrahedron is . Find the value of . (where and are relatively prime natural numbers.)

For a positive real number , let be the value of real number such that the curve meets the curve at exactly one point. Find the value of .