September 2024 National Unified Academic Achievement Test (Grade 10)

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

For the complex number , what is the value of ? (where and is the complex conjugate of )

When the equation is an identity in , what is the value of for the two constants and ?

When the distance between two points A(1,3) and B(2,a) on the coordinate plane is , what is the value of the positive number ?

For two points A(1,2) and B(a,b) on the coordinate plane, if the point that divides line segment AB internally in the ratio 1:2 has coordinates (2,3), what is the value of a+b?

The quadratic equation in has two distinct roots and . When , what is the value of the constant ?

When the solution to the quadratic inequality is , what is the value of for the two constants and ?

When the y-intercept of a line passing through the point and perpendicular to the line is , what is the value of the constant ?

When the polynomial is factored as , what is the value of for the two constants ?

There is a line that passes through the point and has slope . When the distance between the origin and line is , what is the value of positive ?

When the quadratic equation in always has a double root regardless of the value of real number , what is the value of for the two constants ?

What is the sum of all real values of such that the cubic equation in has exactly distinct real roots?

A polynomial satisfies the following conditions. (a) When is divided by , the quotient and remainder are equal. (b) is divisible by . When the remainder of dividing by is , what is the value of ?

When the polynomial in is divided by , the remainder is . Find the value of the constant .

Find the number of all integers that satisfy the system of inequalities

When the line is translated by units in the direction of the -axis, and this translated line is tangent to the graph of the quadratic function , find the value of the constant .

As shown in the figure, let point on the coordinate plane be reflected across the line to get point , and let point be reflected across the -axis to get point . When the radii of the circumcircles of triangles and are and respectively, . Find the value of for the constant . (Note: is the origin.)

A quadratic function with a positive leading coefficient has a graph that intersects the -axis at two points and , and intersects the -axis at point . Let be the vertex of the graph of the quadratic function , and let and be the feet of the perpendiculars dropped from points and to line , respectively. When quadrilateral is a square, find the value of .

For two real numbers and , there is a quadratic function . There are distinct circles whose centers lie on the graph of the function and are simultaneously tangent to the line and the -axis. When the -coordinates of the centers of these three circles are , , and respectively, the three real numbers , , and satisfy the following conditions. (a) (b) The -coordinate of the centroid of the triangle with vertices at , , is . Find the value of .