June 2024 National Unified Academic Achievement Test (Grade 10)

What is the value of ? (where )

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

What is the remainder when the polynomial is divided by ?

When the solution to the quadratic inequality in is , what is the value of the constant ?

Given and , find the value of .

What is the minimum value of the natural number such that the quadratic equation has two distinct complex roots?

What is the remainder when is divided by ?

When two polynomials and in are both divisible by , what is the value of ? (where and are real numbers)

For the cubic equation , let and be the two distinct complex roots. What is the value of ?

Let the solutions of the system of equations for : be or . What is the maximum value of the natural number such that and are two distinct real numbers?

As shown in the figure, the graph of the quadratic function and the line meet at only one point A. For the two points B and C where the graph of the quadratic function meets the x-axis, what is the area of triangle ABC? (where is a constant.)

When the polynomial in is divided by , the quotient is and the remainder is . When is divisible by , what is the value of ? (Here, and are constants.)

For a real number , let the complex number be . When is a negative real number, find the number of natural numbers such that where is the complex conjugate of , and .

For the quadratic function on the interval , the following conditions are satisfied. (a) The function has a minimum value at . (b) The maximum value of the function is . What is the value of ? (where and are constants.)

When the cubic equation in has three distinct real roots , , , what is the sum of all real values of such that ?

Find the coefficient of in the expansion of the polynomial .

When the two roots of the quadratic equation with respect to are and , find the value of . (Here, and are constants.)

For a complex number , when the equation holds, find the value of . (Here, is the complex conjugate of , and .)

Let and be the two roots of the quadratic equation in . Find the value of the real number that satisfies

Find the sum of all real values of such that when the solution to the system of inequalities for is , the condition is satisfied.

For a quadratic polynomial and a linear polynomial , when is divided by , the quotient is and the remainder is . Find the value of .

As shown in the figure, there is a sector OAB with radius 1 and central angle of 90°. For a point C on arc AB, draw a circle with diameter BC. Let P be the point closer to B among the points where a line passing through the midpoint of segment BC and parallel to line OB intersects the circle. When BC = x, let S(x) be the area of triangle OAP. If the maximum value of S(x) is q/p, find the value of p + q. (where 0 < x < √2, and p and q are coprime natural numbers.)

Two quadratic functions and satisfy the following conditions. (a) For all real numbers , . (b) The minimum value of real number such that the maximum value of function on equals the minimum value of function on is , and the maximum value is . (c) The sum of all real roots of the equation is negative. When and , find the value of .