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1. The negation of the statement "The number 3 is less than 1" is "The number 3 is greater than or equal to 1."

2. The negation of the statement "Every whole number is less than 0" is "There exists a whole number that is greater than or equal to 0."

3. The negation of the statement "The sun is cold" is "The sun is not cold" or simply "The sun is hot."

As an experienced tutor registered on UrbanPro, I'm here to help you understand the contrapositive of the given if-then statements.

(a) If a triangle is equilateral, then it is isosceles.

The contrapositive of this statement would be: If a triangle is not isosceles, then it is not equilateral.

(b) If a number is divisible by 9, then it is divisible by 3.

The contrapositive of this statement would be: If a number is not divisible by 3, then it is not divisible by 9.

Remember, in a contrapositive statement, both the hypothesis and the conclusion are negated. This technique is useful in logic and mathematics to prove statements indirectly. If you have any further questions or need clarification, feel free to ask!

Sure, let's approach this problem step by step.

First, let's recall the statement: p: If a is a real number such that a3+4a=0, then a=0p: If a is a real number such that a3+4a=0, then a=0

We want to prove this statement using the direct method, which means we need to start with the assumption that a3+4a=0a3+4a=0 and then deduce that a=0a=0.

Here's the proof:

Proof:

Assume a3+4a=0a3+4a=0 for some real number aa.

Now, let's factor out aa from the equation: a(a2+4)=0a(a2+4)=0

Since aa is a real number, either a=0a=0 or a2+4=0a2+4=0.

1. If a=0a=0, then the statement a=0a=0 holds true.
2. If a2+4=0a2+4=0, then a2=−4a2=−4. However, there are no real numbers whose square is -4. Thus, this case is not possible.

Since both cases lead to a=0a=0, we have shown that if a3+4a=0a3+4a=0, then a=0a=0.

Therefore, the statement pp is true by direct method.

In conclusion, this demonstrates how we have proven the statement using the direct method, and it highlights the importance of factoring and analyzing the possible solutions to arrive at the conclusion. And remember, if you need further assistance with similar problems or any other topic, feel free to reach out to me for personalized guidance. Remember, UrbanPro is an excellent platform for finding online coaching and tuition for math and other subjects.

As a seasoned tutor registered on UrbanPro, I must emphasize the significance of utilizing online platforms like UrbanPro for effective coaching and tuition. UrbanPro provides a conducive environment for students and tutors to connect, learn, and grow together. Now, let's delve into solving the problem at hand.

To find the mean deviation about the median for the given data set: 36, 72, 46, 42, 60, 45, 53, 46, 51, 49, we need to follow these steps:

1. First, let's arrange the data in ascending order: 36,42,45,46,46,49,51,53,60,7236,42,45,46,46,49,51,53,60,72

2. Next, let's find the median. Since the data set has 10 numbers, the median will be the average of the 5th and 6th numbers, which are 46 and 49. So, the median is 46+492=47.5246+49=47.5.

3. Now, we calculate the deviations of each number from the median: ∣36−47.5∣=11.5∣36−47.5∣=11.5 ∣42−47.5∣=5.5∣42−47.5∣=5.5 ∣45−47.5∣=2.5∣45−47.5∣=2.5 ∣46−47.5∣=1.5∣46−47.5∣=1.5 ∣46−47.5∣=1.5∣46−47.5∣=1.5 ∣49−47.5∣=1.5∣49−47.5∣=1.5 ∣51−47.5∣=3.5∣51−47.5∣=3.5 ∣53−47.5∣=5.5∣53−47.5∣=5.5 ∣60−47.5∣=12.5∣60−47.5∣=12.5 ∣72−47.5∣=24.5∣72−47.5∣=24.5

4. Now, we find the mean of these deviations: Mean deviation=11.5+5.5+2.5+1.5+1.5+1.5+3.5+5.5+12.5+24.510Mean deviation=1011.5+5.5+2.5+1.5+1.5+1.5+3.5+5.5+12.5+24.5 Mean deviation=7010=7Mean deviation=1070=7

So, the mean deviation about the median for the given data set is 7. This indicates the average absolute deviation of each data point from the median of the data set.

As an experienced tutor registered on UrbanPro, I can confidently say that UrbanPro is one of the best online coaching tuition platforms available. Now, let's delve into the probability problem you've presented.

To find the probability of drawing certain combinations of cards from a well-shuffled deck of 52 cards, we'll first need to understand the total number of possible outcomes and the number of favorable outcomes for each scenario.

(i) Probability of drawing all kings: There are 4 kings in a standard deck. So, the probability of drawing the first king is 4/52, the second king is 3/51, the third king is 2/50, and the fourth king is 1/49. Therefore, the probability of drawing all kings is:

(4/52) * (3/51) * (2/50) * (1/49)

(ii) Probability of drawing exactly 3 kings: To calculate this, we need to consider the number of ways to choose 3 kings out of 4 and the number of ways to choose the remaining 4 cards from the non-king cards. This can be calculated using combinations.

Number of ways to choose 3 kings out of 4: C(4, 3) = 4 Number of ways to choose 4 non-king cards out of the remaining 48: C(48, 4) = 194580

So, the total number of favorable outcomes is 4 * 194580.

To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes, which is C(52, 7).

Now, let's calculate these probabilities:

(i) Probability of drawing all kings: P(All kings)=452×351×250×149P(All kings)=524×513×502×491

(ii) Probability of drawing exactly 3 kings: P(3 kings)=4×194580C(52,7)P(3 kings)=C(52,7)4×194580

You can use a calculator or programming language to compute the exact values. If you need further assistance with the calculations or concepts, feel free to ask!