June 2024 High School Senior Mock CSAT – Geometry

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 y-intercept of the tangent line to the ellipse at the point ? (where is a positive number)

In the coordinate plane, for two vectors , when vector satisfies what is the minimum value of ?

For the hyperbola , let a line passing through one focus and parallel to the -axis intersect the hyperbola at two points and respectively. If one asymptote of the hyperbola has the equation and , find the value of . (Here, and are positive numbers.)

As shown in the figure, let F and F' be the two foci of an ellipse whose vertices are the midpoints P, Q, R, S of the four sides of rectangle ABCD. A parabola with focus at point F and directrix along line AB passes through three points F', Q, and S. When the area of rectangle ABCD is , what is the length of segment FF'?

In the coordinate plane, there is a curve and four points , , , . For all points on the curve where the absolute value of the -coordinate is less than or equal to , . When point on the curve is in the first quadrant and , find the perimeter of triangle . (Here, and are positive numbers.)

There is a hyperbola with two foci at and , and the length of the major axis is . For a point on the hyperbola such that , point satisfies Find the maximum value of for point .