June 2023 National Unified Academic Achievement Test (Grade 10)

What is the value of ? (where )

For two polynomials and , what is the simplified form of ?

When the polynomial in is divisible by , what is the value of the constant ?

When the equation is an identity in , what is the value of ? (where are constants.)

When the solution to the inequality is , what is the value of ?

How many natural numbers are there such that the graph of the quadratic function and the line do not intersect?

What is the value of ?

When , what is the value of ? (where )

For the system of equations , when the solution is , what is the value of ?

When one root of the quadratic equation in is , what is the value of ? (where are real numbers and )

When a quadratic polynomial with leading coefficient satisfies the following conditions, what is the value of ? (a) The remainder when is divided by is . (b) The remainder when is divided by is .

Let be two distinct complex roots of the cubic equation in : When , what is the value of ? (where is a real number)

An ideal gas is a hypothetical gas where there are no attractive or repulsive forces between molecules and the size of molecules can be ignored. When the volume of an ideal gas in a steel container is , the number of moles is , the absolute temperature is , and the pressure is , the following relationship holds: Steel containers and contain ideal gases with volumes and respectively. The number of moles of the ideal gas in steel container is times the number of moles of the ideal gas in steel container , and the pressure of the ideal gas in steel container is times the pressure of the ideal gas in steel container . When the absolute temperatures of the ideal gases in steel containers and are the same, what is the value of ?

As shown in the figure, let the line intersect the graphs of the two quadratic functions and at points and respectively. For the two points and , what is the minimum value of the area of quadrilateral ?

As shown in the figure, let the x-coordinates of the two points A and B where the graph of the quadratic function and the line intersect be and , respectively. Let H be the foot of the perpendicular from point B to the x-axis, and let C be the foot of the perpendicular from point A to line segment BH. When , what is the value of ? (Note: )

The following is a process to find all values of such that the quartic equation in has four distinct integer solutions for natural number . Let . Letting , the given equation becomes and by the quadratic formula, . Therefore, from or , we get or or or . For equation to have integer solutions, must be a natural number. Therefore, all values of for which equation has four distinct integer solutions are (나), (다). If the expression suitable for (가) above is , and the numbers suitable for (나), (다) are respectively, what is the value of ? (Given that )

As shown in the figure, there is a right triangle ABC with segment AB as the hypotenuse. When the foot of the perpendicular from point C to segment AB is H, and the area of triangle ABC is . When , what is the value of ? (Given that )

Find the coefficient of in the expansion of the polynomial .

When the solution to the inequality with respect to is , find the value of . (Here, and are constants.)

When the polynomial is divided by , let the remainder be . Find the value of . (where are constants.)

Let and be the two distinct imaginary roots of the quadratic equation . Find the value of . (Here, and are the complex conjugates of and , respectively.)

The following shows part of the process of finding the quotient and remainder when the cubic polynomial is divided by using synthetic division.Find the remainder when is divided by . (Here, are constants.)

As shown in the figure, the graph of the quadratic function intersects the line at only one point A. Let B be the point where the graph of the quadratic function intersects the y-axis, H be the foot of the perpendicular from point A to the x-axis, and C be the point where line segment OA and line segment BH intersect. When the area of triangle BOC is and the area of triangle ACH is , we have . Find the value of . (Here, O is the origin, and p and q are relatively prime natural numbers.)

Find the maximum value of when two natural numbers not exceeding satisfy where .

Two quadratic functions satisfy the following conditions. (a) The graph of function intersects the -axis only at one point . (b) The solution to the inequality is . (c) For all real numbers , . When the number of integers such that the equation has no real roots is , find the maximum value of .