A group of students consists of 5 boys and 6 girls. How many ways can you choose a boy-girl pair from this group?

Given the function y = f(x) with a table of variations as shown. How many distinct real solutions does the equation 3f(x) + 1 = 0 have?

Given the function y = f(x) with a derivative f'(x) = (x^2 - 2)(x - 1) for all x in R. How many extreme points does the given function have?

Given a rectangular box ABCD with a volume of 6. What is the volume of the pyramid ABCD?

In space Oxyz, point M satisfies M = (1; 2; 3). What are the coordinates of M?

In space Oxyz, the plane 2x + 3y + 6z = 0 intersects the y-axis at the point with what y-coordinate?

Assuming F(x) and G(x) are two antiderivatives of f(x) on R such that F(1) - G(1) = 2. What is the value of G(5) - F(5)?

Let a and b be any positive real numbers and α be a real number. Which of the following statements is false?

Given the function y = f(x) with the table of variations as shown. On which interval is the function invertible?

Given a cone with a base circumference of 10π and a height of 12. What is the volume of the cone?

How many integers are contained in the solution set of the inequality x^2 < 16?

The graph shown is of one of the following functions. Which function is it?

What is the domain of the function y = 2 log(x + 1) + log(3 - x)?

What is the modulus of the complex number z = 2 - 3i?

Given the arithmetic sequence u(n) with u(3) = 5 and u(6) = 26. What is the common difference of the sequence?

How many local maxima does the function y = f(x) have based on the derivative sign table?

What is the horizontal asymptote of the function y = (x^2 + 1)/(x + 1)?

What is the minimum value of the function y = f(x) + 1 on the interval [−1; 3]?

Given that a and b are positive real numbers satisfying a^3 + b^2 = 3, what is the value of the expression log(a/b)?

Given the complex numbers z1 = 3 - 2i and z2 = 1 - 3i, what is the imaginary part of z = z1 + z2?

What is the imaginary part of the complex number 2z = -3 + 2i and 1z = -2 + 3i?

For the pyramid SABCD with a square base of side 3 and SA perpendicular to ABCD, what is the angle between SCD and ABCD?

In the space Oxyz, for triangle ABC with A(-1, 2, 0), B(-2, 1, 3), C(0, 1, 1), what is the equation of the median AM?

Given that ∫ from -1 to 2 of f(x) dx = 7 and ∫ from 0 to 1 of f(x) dx = 2, what is the value of ∫ from 0 to 2 of f(x) dx?

For the complex number z = 3 - 4i, which statement is false regarding the points M, N, and P representing z, -z, and z?

There is a row of 6 chairs arranged in a line. If 6 people consisting of 3 male students, 2 female students, and 1 teacher sit randomly, what is the probability that the teacher sits next to and in between the 2 female students?

For the function y = f(x) with derivative f'(x) = 3(x - 1)(x - 3), how many extreme points does the function g(x) = 3f(x) - 4 have?

What are the antiderivatives of the function f(x) = 2 + sin^2(x)?

Assume a and b are positive real numbers different from 1, the graphs of the functions log_a(y) = x and log_b(y) = x as shown. The line 3x = intersects the x-axis, the graphs of log_a(y) = x and log_b(y) = at points H and B respectively. Knowing that HA = 2, which of the following statements is correct?

In space Oxyz, given the line 1: x + y - z = 1 and two points A(1; 5; 2) and B(3; 1; 2). Point M belongs to the line such that triangle MAB is right-angled at M. The x-coordinate of M is:

Given that the equation 2z^2 + 4z + 7 = 0 has two complex roots z1 and z2. In the Oxy plane, A(2;1) and M, N are the points representing z1 and z2 respectively. The area of triangle AMN is:

Let f(x) = ax^3 + bx^2 + cx + d where a, b, c, d are real numbers. The sign table of f'(x) is as follows: How many of the coefficients a, b, c, d are positive, knowing that f(-1) = 0?

Given a prism ABC A'B'C' with all edges equal to a. The distance between the lines AC' and BB' is:

In space Oxyz, given two points A(2;1;1) and B(0;3;1) and the plane P: 2x + y + z - 4 = 0. Let I be the intersection point of AB and P. The y-coordinate of I is:

Given a sphere S with diameter AB having a volume of 8. The sphere centered at A with radius AB has a volume of:

Given a cubic function y = f(x) with a graph as shown. The function g(x) = f(x - 1) - 2 is increasing in which of the following intervals?

The function g(x) = f(x - 2) is increasing on which interval?

The volume of the box formed by using 6 square metal sheets, each with an area of 16 cm² is?

The value of the integral ∫ from 0 to 4 of f(x) dx is?

How many integer values of m make the equation 2z² - 7z + m - 6 = 0 have two distinct roots z1 and z2 satisfying z1 + z2 = 1?

The volume of the pyramid SABCD with a trapezoidal base is?

How many integer values of m make the inequality (log(22) + m)x + log(22) - 1 ≥ 0 have a solution, with each m having no more than 24 integer solutions?

Calculate the volume of gasoline in a cylindrical tank with a height of 8.5m and a diameter of 2.4m, given the distance from the top of the tank to the gasoline surface is 0.6m.

What is the maximum value of the expression P = 5z^3 - z^2 + w - 1, given the complex numbers z and w satisfy the equations?

How many planes are parallel to both lines and tangent to the sphere defined by the equation?

How many integer values of the parameter m exist such that the function g(x) = f(x) - 2x^2 + 4x + m has exactly 2 critical points in the interval (0; 24)?

How many integers a exist such that for each a, there is exactly one positive integer b satisfying the equation?

What is the volume of the stone mortar closest to the given value?

In space Oxyz, given three points (5;1;10), M(9;15;6), A(3;9;6)B and the plane (2x^2 + 2y - 27z = 0), what is the maximum length of the segment MC?

Given that F(x) = 1/2 x^2 is an antiderivative of the function f(x) = sin(2x) + x and g(x) = cos(2x) + x on the area bounded by the curves y = f(x), y = g(x), and x = 2π, what is the area of the region?

For the function f(x) = 2x^3 - 9x^2 + mx, how many integer values of m are there such that max(f(x)) = f(3) for x in [0; 3]?

Given a cylinder with a base area of 1 and a height of 5, what is the volume of the cylinder?

A group of students consists of 5 boys and 6 girls. How many ways can a boy-girl pair be selected from this group?

For the function y = f(x) with a given variation table, how many distinct real solutions does the equation 3f(x) + 1 = 0 have?

Given the function y = f(x) with a derivative f'(x) = 2x - 12, how many critical points does the function have?

Given a rectangular box ABCD with a volume of 6, what is the volume of the pyramid ABCD?

In the space Oxyz, given point M satisfies M = (1, 2, 3), what are the coordinates of M?

In the space Oxyz, the plane 2x + 3y + 6z = 0 intersects the y-axis at which point?

Assuming F(x) and G(x) are two antiderivatives of f(x) such that F(1) - G(1) = 2, what is the value of G(5) - F(5)?

With a and b being any positive real numbers and a being a real number, which of the following statements is false?

Given the function y = f(x) with a change table as shown, on which interval is the function decreasing?

Given the function y = f(x) with the variable table as shown. On which interval is the function decreasing?

A cone has a base circumference of 10π and a height of 12. What is the volume of the cone?

How many integers are contained in the solution set of the inequality 2x^2 < 16?

The graph shown is of one of the following functions. Which function is it?

What is the domain of the function y = log(x + 1) + log(3 - x)?

What is the modulus of the complex number z = 2 - 3i?

For the arithmetic sequence u_n with u_1 = 3.5 and u_2 = 6, what is the common difference?

Given an arithmetic progression with a common difference of 3.5, what is the common difference of the given arithmetic progression?

How many local maxima does the function have based on the derivative sign change?

What is the horizontal asymptote of the graph of the function y = (1 + x)/(x - 1)?

What is the minimum value of the function y = f(x) + 1 on the interval [1, 3]?

Given that a and b are positive real numbers satisfying ab = 32, what is the value of the expression log_a(b)?

What is the imaginary part of the complex number z1 + z2 where z1 = 32 - 3i and z2 = 13 - i?

What is the angle between the line segment SC and the plane ABCD in the pyramid SABCD where the base is a square with side length a?

In space Oxyz, for triangle ABC with A(-1; 2; 0), B(2; 1; 3), C(0; 1; 1). What is the equation of the median AM of triangle ABC?

Given that ∫ from 0 to 1 of f(x) dx = 2 and ∫ from 1 to 2 of f(x) dx = 7, what is the value of ∫ from 0 to 2 of f(x) dx?

For the complex number z = -3 + 4i, which of the following statements is false?

There is a row of 6 chairs arranged in a line. If 6 people consisting of 3 male students, 2 female students, and 1 teacher are randomly seated, what is the probability that the teacher sits next to and in between the 2 female students?

What is the probability of the event A where the teacher sits next to and between 2 female students?

How many extreme points does the function g(x) = 3(x - 4) - 5 have?

What are the antiderivatives of the function f(x) = 2 + sin^2(x)?

Given the lines log_a(y) and log_b(y), what is the correct statement if the line 3x intersects the x-axis at points HA and HB?

In the space Oxyz, given the line D: 111xyz+- and two points A(1; 5; 2) and B(3; 1; 2). Point M belongs to D such that triangle MAB is right-angled at M. The x-coordinate of M is:

Given the equation 2z^2 + 4z + 7 = 0 has two complex roots z1 and z2. In the Oxy plane, A(2; 1) and MN are the points representing z1 and z2 respectively. The area of triangle AMN is:

For the function f(x) = ax^3 + bx^2 + cx + d, how many of the coefficients a, b, c, d are positive, given that f(1) = 0?

For the prism ABC, where all edges are of length a, what is the distance between the lines AC' and BB'?

In the space Oxyz, given two points A(2;1;1) and B(0;3;1) and the plane P: 24x + 0y + z = 0. Let I be the intersection point of AB and P. What is the y-coordinate of I?

Given a sphere S with diameter AB having a volume of 8. What is the volume of the sphere centered at A with radius AB?

For the cubic function y = f(x) with the graph shown, on which interval is the function g(x) = f(x) - (1 + 2) decreasing?

Using 6 square metal sheets, each with an area of 16 cm², to make the faces of a cube. What is the volume of the cube formed?

Given the continuous function y = f(x), what is the value of the integral from 0 to 4 of f(x) dx?

On the complex number set, consider the equation 2m^2 - 7m + 6 = 0 with m as a parameter. How many integer values of m make the equation have two distinct roots z1 and z2 satisfying z1 + z2 = 1?

For the pyramid SABCD with a trapezoidal base, where AD = 2, AB = AC = CD, and the angle between SC and SAD is 30 degrees, what is the volume of the pyramid?

How many integer values of the parameter m are there such that the following inequality has solutions, and at the same time, for each m, the solution set contains no more than 24 integers?

A gasoline tank has the shape of a cylinder with a length of 8.5 m and a base diameter of 2.4 m. The distance from the top edge of the tank to the horizontal gasoline surface is 0.6 m. Calculate the volume of gasoline contained in the tank (ignore the thickness of the tank walls, round the result to the nearest thousandth).

Consider the complex numbers z, w satisfying 12zi + zi = -1 and w = 1. What is the maximum value of the expression 53Pzizw = -?

In space Oxyz, how many planes are parallel to both lines 113: 213xyz + D = 0 and 331: 451xyz - D = 0 and simultaneously tangent to the sphere Sxyz = 12?

Given the function y = f(x) with a derivative satisfying 2(1 + 3)x^2 = f'(x), how many integer values of the parameter m exist such that for each m, the function g(x) = f(x) - 2x^2 + 4x - 1 has exactly 2 critical points in the interval (0; 24)?