June 2024 High School Senior Mock Exam – Calculus

What is the value of ?

For the function , what is the value of ?

For the sequence , given that and , what is the value of ?

The graph of function is shown in the figure. What is the value of ?

For the function , what is the value of ?

What is the sum of all real values of such that the equation has exactly 2 distinct real roots?

For a geometric sequence where , given that what is the value of ?

For the function when the function is continuous on the entire set of real numbers, what is the value of the constant ?

When the area of the circumcircle of triangle ABC satisfying the following conditions is , what is the area of triangle ABC? (a) (b)

A cubic function with leading coefficient and satisfies When the -intercept of the tangent line to the curve at the point is , what is the value of ? (Note: is a constant.)

As shown in the figure, let A be a point in the first quadrant on the curve . Let B be the point where a line through point A parallel to the y-axis intersects the curve . Let C be the point where a line through point A parallel to the x-axis intersects the curve , and let D be the point where a line through point C parallel to the y-axis intersects the curve . When , what is the area of quadrilateral ABCD?

Let be the area of the region enclosed by the curve , the line , and the -axis, and let be the area of the region enclosed by the curve and the two lines and . When , what is the value of the constant ? (Given that )

What is the sum of all natural numbers that satisfy the following condition? The number of natural numbers such that is positive is .

For a cubic function with leading coefficient and a constant , the function satisfies the following conditions. (a) The function is increasing and differentiable on the set of all real numbers. (b) For all real numbers , and . What is the minimum value of ?

Find the value of real number that satisfies the equation .

For a function , given that and , find the value of .

Given that , find the value of the constant .

A point P starts from the origin at time t=0 and moves along a number line. The velocity v(t) of point P at time t (t ≥ 0) is given by: When the position of point P at the time when its direction of motion changes for the second time is 1, find the value of the positive constant k.

For two natural numbers not exceeding , let be the set of points where the graph of the function defined on the open interval intersects the line , and let be the sets of points where it intersects the lines respectively. For the ordered pairs such that , let be the maximum value of and be the minimum value. Find the value of .

The sequence satisfies , and for all natural numbers : Find the product of all values of such that .

What is the value of ?

When the sequence satisfies what is the value of ?

For a positive number , let the curve intersect the two lines and at points and respectively, and let be the foot of the perpendicular from point to the -axis. When the area of triangle is denoted as , what is the value of ?

When the function is for a real number , let be the minimum value of that satisfies . When the function is discontinuous only at , what is the value of ? (where is a constant.)

For the function (where is a constant) and two positive numbers , , the function is differentiable on the entire set of real numbers. When , find the value of . (where , are rational numbers, and is an irrational number.)

Let the functions and intersect at points whose -coordinates, arranged in ascending order, are denoted as for the -th value. Find the value of