Learn Mathematics with Free Lessons & Tips

To find the area swept by the minute hand in 5 minutes, we need to consider the area covered by the sector formed by the minute hand during that time.

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 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:

To find the probability of drawing a card with an even number, we need to determine the total number of even-numbered cards in the bag and divide it by the total number of cards in the bag.

Given:

• Numbers on the cards range from 3 to 20.

To find the probability that two friends born in the year 2000 have the same birthday, we need to consider the total number of days in the year 2000 and the number of ways they could share a birthday.

Given that they were born in the same year:

1. Total number of days in the year 2000: Since the year 2000 was a leap year, it had 366 days.

2. Number of ways they could share a birthday: Each friend's birthday can fall on any of the 366 days in the year.
   

If each observation in a continuous frequency distribution is increased by 5, the median of the new data set will also increase by 5. This is because the median is the middle value of the data set, and adding the same constant to every observation simply shifts the entire data set by that constant.

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: