Learn Mathematics with Free Lessons & Tips

To find the radius of the circle formed by bending a wire into an arc subtending an angle of 60∘, we can use the formula for the length of an arc of a circle:

The circumference of a circle is given by the formula:

To find the area of the shaded region, we need to subtract the area of the square OABC from the area of the quadrant OABQ.

To find the area of the shaded region, we need to subtract the area of the semi-circle from the area of the quadrant.

To solve this problem, we first need to find the areas of the two circles with diameters 10 cm and 24 cm. Then, we'll add these areas together to find the total area. Finally, we'll find the diameter of the larger circle whose area is equal to the sum of the areas of the two smaller circles.

To find the probability of getting a number less than 3 when a die is thrown once, we need to determine the total number of favorable outcomes (rolling a number less than 3) and divide it by the total number of possible outcomes (rolling any number on the die).

Given:

• The die has 6 faces numbered 1 through 6.

1. Total number of possible outcomes: There are 6 possible outcomes when rolling a die.

2. Number of favorable outcomes (rolling a number less than 3): There are 2 favorable outcomes: rolling a 1 or rolling a 2.

Now, let's calculate the probability of getting a number less than 3:

When a standard six-sided die is thrown once, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6.

To find the probability of getting a number greater than 5, we need to determine the number of favorable outcomes (rolling a number greater than 5) and divide it by the total number of possible outcomes (rolling any number on the die).

Given:

• Total number of possible outcomes: 6 (since there are 6 faces on the die).

1. Number of favorable outcomes (rolling a number greater than 5): There is only 1 favorable outcome: rolling a 6.

Now, let's calculate the probability:

The mean of the first n natural numbers can be calculated using the formula for the arithmetic mean:

To show that the mode of the combined series obtained by combining the two series S1 and S2 is different from that of S1 and S2 taken separately, we first need to find the mode of each series individually and then compare it with the mode of the combined series.

To find the median for a frequency distribution, we need to use the relationship between the mean, median, and mode in a symmetrical distribution.

In a symmetrical distribution:

• The mean, median, and mode are all equal.

Given that the mode is 7.88 and the mean is 8.32, we can see that they are not equal. This suggests that the distribution is not perfectly symmetrical.

However, we can still estimate the median based on the mode and mean using the following relationship: