High School Grade 1, 1st Semester Midterm Exam: Past Questions

For two polynomials and , find the value of when and .

Find the largest natural number such that the quadratic equation has two different real roots.

Factorize the polynomial .

Find the coefficient of in the expansion of the polynomial for when the coefficient of is .

When the cubic polynomial for is divided by , the remainder is . Find the value of . (where and are real numbers)

Let and be the two roots of the quadratic equation for . For the two roots of the equation, find the largest real number such that the value of is an integer.

Find the value of for a complex number satisfying .

For all real numbers , the following equation holds. Find the value of .

For all natural number , find the sum of the values of such that the polynomial is factorizable over the range of natural numbers.

For two polynomials and of the same order and two real numbers and , we have for all . When , , find the value of . (Note that all coefficients are real numbers)

For two natural numbers and less than or equal to , find the number of ordered pairs such that is a negative real number. (Note that ).

For two quadratic functions and , we can write the function as Find a range of values of such that the graphs of the straight lines and meet at four different points.

For two integers , a quadratic function and a linear function satisfy the following conditions. (a) The maximum value of the function is . (b) The graph of the function and the graph of the function meet at two points and . (c) For any integer such that , the number of all ordered pairs of such that the integer satisfies the inequality is . Find the value of when the sum of the largest and smallest values of the real numbers satisfying the equation is .

Let a polynomial and a polynomial satisfy the following conditions (a) The quotient of divided by is , and the remainder is . (b) . (c) The graph of meets the straight line at a single point. Find the value of .

For two polynomials and whose coefficients are real, find the value of when the polynomial has as a factor.

For two real numbers and , we have Find the value of . (Note that )

Consider a quadratic function with a coefficient of and a quadratic function with a coefficient of for their highest order terms. When the equation has different real roots and , the equation has different real roots and . Find the value of for which the equation has different real roots. (Note that )