October 2024 National Unified Academic Achievement Test (Grade 10)

What is the distance between the two points and on the coordinate plane?

For the universal set and its subset , what is the product of all elements in the set ?

What is the y-intercept of the line obtained by translating the line by unit in the direction of the -axis and units in the direction of the -axis?

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

When the solution of the system of equations is , , what is the value of ?

A polynomial is divisible by , and when is divided by , the remainder is . When is divided by , let the remainder be . What is the value of ?

For a non-real complex number , when , what is the value of ? (Here, is the complex conjugate of .)

For two points A(3,0) and B(0,a) on the coordinate plane, when the point that externally divides line segment AB in the ratio 2:3 lies on the circle , what is the value of ?

Let be the circle with center at the origin that has minimum area among all circles that intersect with the line . When the area of circle is , what is the value of the positive constant ?

Three sets , , satisfy the following conditions. (a) (b) When , what is the value of ?

What is the sum of all real values of such that there exists exactly one real number that satisfies the system of inequalities?

On the circle , there are two distinct points and . Among the points on circle , there are two distinct points and that satisfy and . When the area of quadrilateral is , what is the value of ?

A quadratic function with leading coefficient satisfies the following conditions. (a) There exist two distinct integers such that . (b) All natural numbers such that the product of the maximum and minimum values of function on is not equal to the value of are . When the minimum value of function is , what is the value of ?

For a positive number that is not , let the line intersect the graphs of the two functions and at points and , respectively, and let the line intersect the -axis at point . What is the sum of the maximum and minimum values of that satisfy ?

Let be a circle with diameter , where and are two distinct points on the curve . Given that the area of circle is and a line passing through point with slope is tangent to circle , find the -intercept of line .

For a positive number , there is a rectangular parallelepiped where , . Let be the point that divides segment in the ratio , and let be the point that divides segment in the ratio . From the rectangular parallelepiped , a triangular prism is cut out to create cross-section . Let be the volume of the quadrangular prism and be the volume of the triangular prism . When , what is the length of segment ?

Find the value of when the line passing through the two points and on the coordinate plane is parallel to the line .

Find the number of natural numbers such that the equation has complex roots.

For two conditions about real number : Find the minimum value of real number such that is a necessary condition for .

When the polynomial is factored as , find the value of . (Here, , , and are constants.)

In the coordinate plane, let and be the centers of two different circles that are tangent to both lines and and pass through the point . Find the length of the line segment .

Two quadratic polynomials satisfy the following conditions. (a) For all real numbers (b) When , find the value of .

As shown in the figure, two points A and B are placed on circle with center such that . For point C on segment , let be the circle with diameter AC, and for point D on segment , let be the circle with diameter BD. Let the centers of circles and be and , respectively. Given that the area of quadrilateral is 34 and , the sum of the areas of the three circles , , and is . Find the value of . (Note that point C is neither point A nor point , and point D is neither point B nor point .)