March 2023 National Unified Academic Achievement Test (Grade 10)

What is the value of ?

What is the coefficient of the linear term in the polynomial ?

In the right triangle ABC as shown in the figure, where , , , what is the length of segment AB?

What is the y-intercept of the line passing through the two points and on the coordinate plane?

For two points A and B on a circle, when the length of arc AB is of the circumference of the circle, what is the measure of the inscribed angle subtended by arc AB?

When a rectangular prism with a square base of side length has a volume of , what is the surface area of this rectangular prism?

When the graphs of two linear equations and intersect at the point , what is the value of ? (where and are constants)

As shown in the figure, point A(a, b) in the first quadrant lies on the graph of the quadratic function . Given that point B is where this quadratic function's graph intersects the y-axis, and the area of triangle OAB is 4, find the value of . (Note: O is the origin.)

A student departs from home and travels at a speed of on the way there, and returns home along the same route at a speed of . If the total travel time must be 2 hours or less, what is the maximum distance the student can travel from home until returning home?

For four distinct points A(1,1), B(8,1), C(6,4), D(a,b) on the graph of the quadratic function y=f(x), when , what is the value of ?

For two natural numbers , what is the minimum value of the real number such that the polynomial can be factored as ?

As shown in the figure, for rhombus ABCD and a point E outside this rhombus, given that ∠ADE = 72° and line CD perpendicularly bisects segment BE, what is the measure of angle CEB? (Given that 0° < ∠ADC < 72°)

Let A and B be the vertices of the graphs of the two quadratic functions and , respectively. For the point C where the graph of the quadratic function intersects the y-axis, if the area of quadrilateral OACB is 7, what is the value of the positive number ? (where O is the origin.)

As shown in the figure, let the graph of the inverse proportion intersect with the graphs of two direct proportions at points and respectively in the first quadrant. Let be the point where a line passing through point and parallel to the -axis intersects with the graph of the direct proportion . Given that the area of triangle is and the -coordinate of point is twice the -coordinate of point , find the value of the real number . (Given that )

As shown in the figure, there is a sector OAB with center O and central angle of 120°. For two points C and D on arc AB where ∠AOC = ∠DOB = 30°, let E be the intersection point of segment OC and segment AD. When the perpendicular bisector of segment OD intersects segment OB at point F, and BF = (2√3)/3, what is the area of triangle ODE?

When one root of the quadratic equation is , find the value of the constant .

When the solution of the system of linear equations is , find the value of .

As shown in the figure, there is triangle ABC with and . For point D on the line passing through point C and parallel to line AB, and point E on segment AB, we have . Let F be the intersection point of segment DE and segment AC, and . Find the value of x. (Note that and point E is not point A.)

When rolling a single die twice and letting the numbers that come up be and in order, find the number of all ordered pairs such that is a divisor of .

For three real numbers , when the median of the following data is , the mean is , and the mode is , find the value of .

For two primes with , find the value of when the number of natural numbers satisfying is .

In the figure, triangle ABC has point D on segment AC and point E on line BD such that DE : DA : DB = 1 : 2 : 4. A line through point D parallel to line BC intersects segments AB and EC at points F and G respectively. Given that FD = 2, DG = 1, and the area of triangle AFD is 3, find the value of p + q when the area of triangle EDG is q/p. (Note: point E is outside triangle ABC, and p and q are coprime natural numbers.)

As shown in the figure, there is a circle that circumscribes triangle where . Let be the point other than where the line passing through point and perpendicular to line meets circle , and let be the point where segment and segment intersect. When the tangent to circle at point and the tangent at point meet at point , we have . If , find the value of . (Here, are integers.)