2022 Dandae Affiliated High School — 1st Year, 1st Semester: Final Exam Past Questions

Find the sum of the values of all real numbers such that the distance between the two points and is .

Let the four roots of the quadratic equation be and , find the value of .

Find the number of integers that satisfy the system of inequalities

Find the sum of the values of all integers such that the quadratic inequality holds for all real numbers .

Find the largest integer when the cubic equation has only one real root.

There are two points and as shown in the figure below. For the straight line , find the value of for the point on the straight line such that .

For the two straight lines which of the following is correct? (Note that is a real number.) a. When , the two straight lines and are parallel to each other. b. The straight line always passes through the point , regardless of the value of . c. The value of for two straight lines and to be parallel exists.

If the solution of the quadratic inequality is , find the solution of the quadratic inequality .

Let be a natural number less than or equal to such that the polynomial is defined as: Choose one that has all the correct statements among below. A. B. The number of different real roots of the equation is . C. The number of such that all roots of are integers is .

Let be the tangent line at point on the graph of the quadratic function in the coordinate plane, as shown in the figure, and let be the straight line that passes through point and is perpendicular to straight line . Let be the point where straight line meets the -axis and be the point where straight line meets the graph of the quadratic function that is not at point . Let be the area of the triangle , find the value of .

The curve and the straight line meet at two points and . Find the value of the constant when the -coordinate of the center of the line segment is . (Note that the -coordinate of the point is smaller than the -coordinate of the point ).

Find the value of for the range such that there exists a real number and satisfying the system of equations

Let be the midpoint of a line segment of length and let be the point that internally divide the line segment by . When the three points that are not on the straight line satisfy the following conditions, a) Point externally divides the line segment by . b) Point internally divides the line segment by a . c) Line and line are perpendicular. find the area of the triangle when .

For the system of inequalities with respect to : Let (where ) be the number of integer solutions that satisfy the inequalities and let be the maximum value of the real number for that . Choose the correct statement(s) from the following. (Note, ) (i). (ii). The minimum value of such that becomes an integer is . (iii). The sum of all for which is an integer,up to , is .

All the roots of the cubic equation for are natural numbers. Given two integers and such that and , find the number of ordered pairs .

Let be the intersection of the two lines and , and let be the points where the two lines and meet the -axis, respectively. Let be a point that moves along the circumscribed circle of the triangle with the three points as its vertices, then the point is the centroid of the triangle with respect to the point in the second quadrant. Also, the area of the triangle is times the area of the triangle . Let be the intersection of the bisector of and the line segment . When the sum of the and coordinates of the centroid of triangle is , find the value of . (Note that are natural numbers that are mutually prime).

There are integers satisfying the inequalities . Let be the smallest value of and be the largest value of . Find the value of . (Note that and are real numbers).

For the triangle with vertices , and , there is a point that moves along the line segment , as shown in the figure. (Note that ) The point is where the straight line passing through the point and parallel to meets the line segment . Let be the point where the straight line passing through the point and parallel to meets the line segment , and let be the point where the straight line passing through the point and parallel to meets the line segment . For the trapezoid , find its largest area.

For two natural numbers and a real number , find the sum of the values of that satisfies the equation

Find the equation of the straight line .

Find the distance from the point to the line .

Find the area of the triangle .

Find the solution to the inequality .

Find the solution to the inequality .

Find the solution to the inequality.