Geometry

We are tasked with assembling a square shape using a set of pieces, where one of the pieces must have a single round hole near the center, and pieces cannot overlap. Let's analyze the options.

Step 1: Identify the requirement.
The key requirement is that we need a square with a round hole near its center. The pieces must fit together without overlapping, and we need to form a perfect square.
Step 2: Examine the options.
- Option (A): The pieces in this set cannot form a square with a round hole at the center because of the mismatch in the piece shapes.
- Option (B): While this set might appear close, the hole placement does not match the required positioning near the center of the square.
- Option (C): This set fits the requirement perfectly. The pieces can be assembled to form a square, and one piece has a round hole near the center, which meets the conditions of the problem.
- Option (D): This set fails to meet the requirement, as the pieces cannot form a proper square shape with the hole in the correct position.
Step 3: Conclusion.
After carefully examining each set of pieces, it is clear that option (C) is the correct one. It allows us to form a square with a round hole near the center. Thus, the correct answer is (C).

The given frequency distribution shows the number of students who scored different marks in an exam. We are asked to identify the correct relationship between the mean, mode, and median. In a symmetric distribution, the mean, mode, and median are equal. From the given frequency distribution, we can observe that the distribution appears fairly symmetric with the highest frequency at the middle marks (5 and 6 marks), and it does not show extreme skewness. Therefore, for this distribution, the mean, median, and mode will be approximately equal.
Thus, the correct answer is (B) mean = mode = median.

In the given diagram, we are working with areas of squares inscribed in a circle. The points and lines are defined such that:
- The area of the square is the area enclosed by the tangent line and the radial line from the center .
- Similarly, the areas of the other squares and are determined by the distances defined by the lines and the tangents.
By analyzing the geometric relationships and using the fact that the squares are inscribed, the correct relation between the areas of these squares is: This is derived from the fact that the areas of the squares depend on the lengths of the sides, and the side lengths are related in such a way that this equation holds. Therefore, the correct answer is (B).

To solve this problem, we need to calculate the area between the two lines in the interval on the x-axis. Step 1: Equation of the lines.
- Line 1 (through (0, 0) and (1, 3)): The slope of the line is: The equation of the line is: - Line 2 (through (0, 0) and (1, 2)): The slope of the line is: The equation of the line is: Step 2: Calculate the area between the lines.
The area between the lines is given by the integral of the difference in the y-values of the two lines over the interval : The integral is: Thus, the area is . Therefore, the correct answer is (A).