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3.2: Problem or Exercise?


The main activity of mathematics is solving problems. An exercise is different from a problem.

In a problem, you probably don’t know at first how to approach solving it. You don’t know what mathematical ideas might be used in the solution. Part of solving a problem is understanding what is being asked, and knowing what a solution should look like. Problems often involve false starts, making mistakes, and lots of scratch paper!

In an exercise, you are often practicing a skill. You may have seen a teacher demonstrate a technique, or you may have read a worked example in the book. You then practice on very similar assignments, with the goal of mastering that skill.

Note

What is a problem for some people may be an exercise for other people who have more background knowledge! For a young student just learning addition, this might be a problem:

[ extit{Fill in the blank to make a true statement} : \_\_\_ + 4 = 7 ldotp]

But for you, that is an exercise!

Both problems and exercises are important in mathematics learning. But we should never forget that the ultimate goal is to develop more and better skills (through exercises) so that we can solve harder and more interesting problems.

Learning math is a bit like learning to play a sport. You can practice a lot of skills:

  • hitting hundreds of forehands in tennis so that you can place them in a particular spot in the court,
  • breaking down strokes into the component pieces in swimming so that each part of the stroke is more efficient,
  • keeping control of the ball while making quick turns in soccer,
  • shooting free throws in basketball,
  • catching high fly balls in baseball,

and so on.

But the point of the sport is to play the game. You practice the skills so that you are better at playing the game. In mathematics, solving problems is playing the game!

On Your Own

For each question below, decide if it is a problem or an exercise. (You do not need to solve the problems! Just decide which category it fits for you.) After you have labeled each one, compare your answers with a partner.

  1. This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers.(Note: Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15. )


Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).

  1. A soccer coach began the year with a $500 budget. By the end of December, the coach spent $450. How much money in the budget was not spent?
  2. What is the product of 4,500 and 27?
  3. Arrange the digits 1–6 into a “difference triangle” where each number in the row below is the difference of the two numbers above it.
  4. Simplify the following expression: $$frac{2 + 2(5^{3} - 4^{2})^{5} - 2^{2}}{2(5^{3} - 4^{2})} ldotp$$
  5. What is the sum of (frac{5}{2}) and (frac{3}{13})?
  6. You have eight coins and a balance scale. The coins look alike, but one of them is a counterfeit. The counterfeit coin is lighter than the others. You may only use the balance scale two times. How can you find the counterfeit coin?

  1. How many squares, of any possible size, are on a standard 8 × 8 chess board?
  2. What number is 3 more than half of 20?
  3. Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.

Exercise 3.2: Application of Integration in Economics and Commerce

1. The cost of over haul of an engine is ₹10,000 The operating cost per hour is at the rate of 2x − 240 where the engine has run x km. Find out the total cost if the engine run for 300 hours after overhaul.


2. Elasticity of a function Ey/ Ex is given by Ey/ Ex = −7x / (1 − 2x )( 2 + 3x ). Find the function when x = 2, y = 3/8 .



3. The elasticity of demand with respect to price for a commodity is given by ( 4 − x) / x , where p is the price when demand is x. Find the demand function when price is 4 and the demand is 2. Also find the revenue function.


4. A company receives a shipment of 500 scooters every 30 days. From experience it is known that the inventory on hand is related to the number of days x. Since the shipment, I ( x) = 500 − 0.03x 2 , the daily holding cost per scooter is ₹ 0.3. Determine the total cost for maintaining inventory for 30 days.


5. An account fetches interest at the rate of 5% per annum compounded continuously An individual deposits ₹1,000 each year in his account. How much will be in the account after 5 years. (e 0.25 = 1.284) .


6. The marginal cost function of a product is given by dC/dx = 100 −10x + 0.1x 2 where x is the output. Obtain the total and the average cost function of the firm under the assumption, that its fixed cost is ₹ 500.


7. The marginal cost function is MC = 300 x 2/5 and fixed cost is zero. Find out the total cost and average cost functions.


8. If the marginal cost function of x units of output is a/ √ [ax + b] and if the cost of output is zero. Find the total cost as a function of x.


9. Determine the cost of producing 200 air conditioners if the marginal cost (is per unit) is C ′ ( x) = x 2 /200 + 4 .


10. The marginal revenue (in thousands of Rupees) functions for a particular commodity is 5 + 3 e −0 .03 x where x denotes the number of units sold. Determine the total revenue from the sale of 100 units. (Given e −3 = 0.05 approximately)


11. If the marginal revenue function for a commodity is MR = 9 − 4x 2 . Find the demand function.


12. Given the marginal revenue function 4/( 2x + 3) 2 − 1 , show that the average revenue function is P = 4/[6x + 9] −1.


13. A firm’s marginal revenue function is MR = 20e −x 10 (1− x/10). Find the corresponding demand function.


14. The marginal cost of production of a firm is given by C ′ (x) = 5 + 0.13x , the marginal revenue is given by R ′ (x) = 18 and the fixed cost is ₹ 120. Find the profit function.


15. If the marginal revenue function is R ′ ( x ) = 1500 − 4x − 3x 2 . Find the revenue function and average revenue function.


16. Find the revenue function and the demand function if the marginal revenue for x units is MR= 10 + 3xx 2 .


17. The marginal cost function of a commodity is given by MC = 14000/ √ [7x + 4] and the fixed cost is ₹18,000.Find the total cost and average cost.


18. If the marginal cost (MC) of a production of the company is directly proportional to the number of units (x) produced, then find the total cost function, when the fixed cost is ₹ 5,000 and the cost of producing 50 units is ₹ 5,625.


19. If MR = 20 − 5x + 3x 2 , find total revenue function.


20. If MR = 14 − 6x + 9x 2 , find the demand function.


Step-By-Step Solution

3. 2 Consider a diffusion area that has the dimensions and the abrupt junction depth is 32 . Its n-type impurity doping level is and the surrounding p-type substrate doping level is . Determine the capacitance when the diffusion area is biased at 1.2V and substrate is biased at 0V. In this problem, assume that there is no channel-stop implant.


Why we should exercise - and why we don't

If the benefits of physical activity are legion, so are the reasons for avoiding it. We've got suggestions for adding some to your day.

You already know that exercise is good for you. What you may not know is just how good — or exactly what qualifies as exercise. That's what this issue of the Health Letter is all about. The notion that physical activity helps keep us healthy is very old news indeed. Hippocrates wrote about the dangers of too little activity (and too much food). Tai chi, an exercise system of graceful movements that originated in China, dates from the 12th century B.C. Yoga's roots in India go back much further.

But old ideas aren't necessarily good ones, or have much evidence to back them up. This isn't a problem for exercise — or physical activity, the term many researchers prefer because it's more of a catchall. A deluge of studies have documented its health benefits. Many are observational, which always pose the problem of showing associations (people who exercise happen to be healthy) not proof of cause and effect (it's the exercise that makes those people healthy). But after statistical adjustments, these studies suggest that the connection between exercise and health is more than just an association. Besides, results from randomized clinical trials, which are usually seen as making the case for causality, also point to exercise making people healthier.

What's impressive about this research, aside from the sheer volume, is the number of conditions exercise seems to prevent, ameliorate, or delay.

We're used to hearing about exercise fending off heart attacks. The American Heart Association promulgated the country's first set of exercise guidelines in 1972. And it's not hard to envision why exercise helps the heart. If you're physically active, your heart gets trained to beat slower and stronger, so it needs less oxygen to function well your arteries get springier, so they push your blood along better and your levels of "good" HDL cholesterol go up.

It's also not much of a surprise that physical activity helps prevent diabetes. Muscles that are used to working stay more receptive to insulin, the hormone that ushers blood sugar into cells, so in fit individuals blood sugar levels aren't as likely to creep up.

But exercise as a soldier in the war against cancer? It seems to be, and on several fronts: breast, colon, endometrial, perhaps ovarian. The effect of physical activity on breast cancer prevention may be stronger after menopause than before, although some research suggests that it takes quite a lot to make a difference: four to seven hours of moderate to vigorous activity a week. Three studies have found that if you've had colon cancer or breast cancer, physical activity reduces the chances of it coming back.

To top things off, moving the body seems to help the brain. Several studies have found that exercise can reduce the symptoms of depression, and it changes the brain in ways similar to antidepressant medications. In old age, physical activity may delay the slide of cognitive decline into dementia, and even once that process has started, exercise can improve certain aspects of thinking.

Easy to avoid

We have to eat, so following nutritional advice is a matter of making choices. Swap out saturated fats for healthy oils. Eat whole grains instead of refined carbohydrates.

But in this day and age, many (perhaps most) people don't need to be physically active unless they choose to be. And most evidence suggests that the choice of the kind of activity is far less important than whether to be active at all. About half of adult Americans don't meet one of the most oft-cited guidelines, which calls for at least 30 minutes of moderate-intensity activity (a fast walking pace) most days of the week — and you can accumulate that total in bouts of 10 to 15 minutes. About a quarter of American adults say they devote none of their free time to active pursuits.

Clearly some of us are less athletic than others — and some unathletic individuals were simply born that way. Twin studies suggest that about half of the difference in physical activity among people is probably inherited. And researchers are making headway in identifying particular genes that may influence how we respond to physical exertion. For example, they've identified some of the genes responsible for variation in the beta-agonist receptors in the lungs. How your lungs and heart react to strenuous exercise depends, in part, on those receptors.

But genetic explanations for behaviors like exercising only go so far. Many other influences come into play: family, neighborhood, cultural attitudes, historical circumstances. Research has shown, not surprisingly, that active children are more likely to have parents who encouraged them to be that way. Perceptions of how active parents are also seem to matter. The safety and layout of neighborhoods is a factor, particularly for children. In a dangerous place, having children stay home and watch television instead of going to the park to play might be the healthier choice simply because it's safer.

The trip of a thousand miles begins.

In addition to getting at least 30 minutes of moderate-intensity exercise most days of the week we should also resistance training to build up muscle strength twice a week. But some exercise, even if it is pretty minimal, is better than none, particularly among people who are very sedentary.

So in that spirit, we've made 27 suggestions for ways to become a little bit more physically active.

1. Take the far away spot. Walking from the farthest corner of the parking lot will burn a few calories. If it's a parking garage, head for the roof and use the stairs.

2. Walk to the next stop. If you take a bus or train, don't wait at the nearest stop. Walk to the next one. Or, at the end of your journey, get off a stop early and finish up on foot.

3. Hang loose. During your bus or train trip, stand and don't hold on too tightly. You'll improve your sense of balance and build up your "core" back and abdominal muscles.

4. Get into the swing of it. Swinging your arms when you walk will help you reach the brisk pace of 3 to 4 miles per hour that is the most healthful.

5. Walk and talk. If you are a member of a book group, propose 15 to 20 minutes of peripatetic discussion of the book before you sit down and chat.

6. Walk while you watch. Soccer moms, dads, and grandparents can circle the field several times during a game and not miss a single play.

7. Walk tall. Maintaining good posture — chest out, shoulders square but relaxed, stomach in — will help keep your back and abdominal muscles in shape. Besides, you'll just look a whole lot healthier if you don't slouch (mom was right).

8. Adopt someone as your walking, jogging, or biking buddy. Adding a social element to exercise helps many people stick with it.

9. That buddy might have four legs. Several studies have shown that dog owners get more exercise than the canine-less.

10. Be part of the fun. Adults shouldn't miss a chance to jump into the fray if kids are playing on a playground or splashing around in the water. Climbing on the jungle gym (be careful!) and swinging on a swing will strengthen muscles and bones and set a good example.

11. Dine al fresco. Tired of eating at home? Skip the restaurant meal, which tends to be heavy on the calories. Pack a picnic. You'll burn calories looking for the best spot and carrying the picnic basket.

12. Put on your dancing shoes. Exercise doesn't have to be done in a straight line. Dancing can get your heart going and helps with balance. Dance classes tend to have lower dropout rates than gyms. Or just turn up the volume at home and boogie.

13. Wash and dry the dishes by hand. The drying alone is a mini-workout for the arms.

14. Don't use an electric can opener. It's good for your hand, wrist, and arm muscles to use a traditional opener. For the same reason, peel and chop your own vegetables and avoid the precut versions.

15. Clean house. Even if you have a cleaning service, you can take responsibility for vacuuming a couple of rooms yourself. Fifteen minutes burns around 80 calories. Wash some windows and do some dusting and you've got a pretty decent workout — and a cleaner house.

16. Hide that remote. Channel surfing can add hours to screen time. If you have to get up to change the channel, you are more likely to turn it off and maybe do something else that's less sedentary.

17. Go swimmingly somewhere. Swimming is great exercise if you have arthritis because the water supports your weight, taking the load off of joints. The humid air around a pool sometimes makes breathing more comfortable for people with lung problems.

18. Take a walk on the waterside. Even people who can't, or don't like to, swim can get a good workout by walking through the water. Try walking fast, and you'll get cardiovascular benefits. Walking in water is a great way to rehabilitate if you're recovering from an injury and certain types of surgery because the water acts as a spotter, holding you up.

19. Don't e-mail. In the office, get out of your chair, walk down the hallway, and talk to the person. At home, write an old-fashioned letter and walk to a mailbox — and not the nearest one — to mail it.

20. Stand up when you're on the phone. Breaking up long periods of sitting has metabolic benefits. Even standing for a minute or two can help.

21. Grow a garden. No matter how green the thumb, the digging, the planting, the weeding, and the picking will ramp up your activity level and exercise sundry muscles.

22. Use a push mower. Even if you have a large lawn, pick a small part of it to mow in the old-fashioned way. You get a nice workout, you're not burning any gas, and it's usually quieter. The same reasoning favors the rake over the leaf blower.

23. Think small. Small bouts of activity are better than knocking yourself out with a workout that will be hard to replicate.

24. Be a stair master. Take the stairs instead of the elevator or escalator whenever you can. It's good for your legs and knees, and your cardiovascular health will benefit from the little bit of huffing and puffing. Don't overdo. One flight at a time.

25. Stairs tip #2. You'll give the gluteal muscles a nice little workout if you can climb up two stairs at a time.

26. Stairs tip #3. You can give your calf muscles a nice little stretch by putting the ball of the foot on the stair and lowering your heel.


3.2: Problems on Conditional Probability

  • Contributed by Paul Pfeiffer
  • Professor emeritus (Computational and Applied Mathematics) at Rice University

(P(A) = 0.55), (P(AB) = 0.30), (P(BC) = 0.20), (P(A^c cup BC) = 0.55), (P(A^c BC^c) = 0.15)

Determine, if possible, the conditional probability (P(A^c|B) = P(A^cB)/P(B)).

In Exercise 11 from "Problems on Minterm Analysis," we have the following data: A survey of a represenative group of students yields the following information:

  • 52 percent are male
  • 85 percent live on campus
  • 78 percent are male or are active in intramural sports (or both)
  • 30 percent live on campus but are not active in sports
  • 32 percent are male, live on campus, and are active in sports
  • 8 percent are male and live off campus
  • 17 percent are male students inactive in sports

Let A = male, B = on campus, C = active in sports.

  1. A student is selected at random. He is male and lives on campus. What is the (conditional) probability that he is active in sports?
  2. A student selected is active in sports. What is the(conditional) probability that she is a female who lives on campus?

In a certain population, the probability a woman lives to at least seventy years is 0.70 and is 0.55 that she will live to at least eighty years. If a woman is seventy years old, what is the conditional probability she will survive to eighty years? Note that if (A subset B) then (P(AB) = P(A)).

Let (A=) event she lives to seventy and (B=) event she lives to eighty. Since (B subset A), (P(B|A) = P(AB)/P(A) = P(B)/P(A) = 55/70).

From 100 cards numbered 00, 01, 02, (cdotcdotcdot), 99, one card is drawn. Suppose Ai is the event the sum of the two digits on a card is (i), (0 le i le 18), and (B_j) is the event the product of the two digits is (j). Determine (P(A_i|B_0)) for each possible (i).

(B_0) is the event one of the first ten is draw. (A_i B_0) is the event that the card with numbers (0i) is drawn. (P(a_i|B_0) = (1/100)/(1/10) = 1/10) for each (i), 0 through 9.

  1. What is the (conditional) probability that one turns up two spots, given they show different numbers?
  2. What is the (conditional) probability that the first turns up six, given that the sum is (k), for each (k) from two through 12?
  3. What is the (conditional) probability that at least one turns up six, given that the sum is (k), for each (k) from two through 12?

a. There are (6 imes 5) ways to choose all different. There are (2 imes 5) ways that they are different and one turns up two spots. The conditional probability is 2/6.

b. Let (A_6) = event first is a six and (S_k = ) event the sum is (k). Now (A_6S_k = emptyset) for (k le 6). A table of sums shows (P(A_6S_k) = 1/36) and (P(S_k) = 6/36, 5/36, 4/36, 3/36, 2/36, 1/36) for (k = 7) through 12, respectively. Hence (P(A_6|S_k) = 1/6, 1/5. 1/4, 1/3. 1/2, 1), respectively.

c. If (AB_6) is the event at least one is a six, then (AB_6S_k) = 2/36) for (k = 7) through 11 and (P(AB_6S_12) = 1/36). Thus, the conditional probabilities are 2/6, 2/5, 2/4, 2/3, 1, 1, respectively.

Four persons are to be selected from a group of 12 people, 7 of whom are women.

  1. What is the probability that the first and third selected are women?
  2. What is the probability that three of those selected are women?
  3. What is the (conditional) probability that the first and third selected are women, given that three of those selected are women?

(P(W_1W_3) = P(W_1W_2W_3) + P(W_1W_2^c W_3) = dfrac<7> <12>cdot dfrac<6> <11>cdot dfrac<5> <10>+ dfrac<7> <12>cdot dfrac<5> <11>cdot dfrac<6> <10>= dfrac<7><22>)

Twenty percent of the paintings in a gallery are not originals. A collector buys a painting. He has probability 0.10 of buying a fake for an original but never rejects an original as a fake, What is the (conditional) probability the painting he purchases is an original?

Let (B=) the event the collector buys, and (G=) the event the painting is original. Assume (P(B|G) = 1) and (P(B|G^c) = 0.1). If (P(G) = 0.8), then

Five percent of the units of a certain type of equipment brought in for service have a common defect. Experience shows that 93 percent of the units with this defect exhibit a certain behavioral characteristic, while only two percent of the units which do not have this defect exhibit that characteristic. A unit is examined and found to have the characteristic symptom. What is the conditional probability that the unit has the defect, given this behavior?

Let (D=) the event the unit is defective and (C=) the event it has the characteristic. Then (P(D) = 0.05), (P(C|D) = 0.93), and (P(C|D^c) = 0.02).

A shipment of 1000 electronic units is received. There is an equally likely probability that there are 0, 1, 2, or 3 defective units in the lot. If one is selected at random and found to be good, what is the probability of no defective units in the lot?

Let (D_k =) the event of (k) defective and (G) be the event a good one is chosen.

Data on incomes and salary ranges for a certain population are analyzed as follows. (S_1)= event annual income is less than $25,000 (S_2)= event annual income is between $25,000 and $100,000 (S_3)= event annual income is greater than $100,000. (E_1)= event did not complete college education (E_2)= event of completion of bachelor's degree (E_3)= event of completion of graduate or professional degree program. Data may be tabulated as follows: (P(E_1) = 0.65) , (P(E_2) = 0.30) and (P(E_3) = 0.05) .

  1. Determine (P(E_3 S_3)) .
  2. Suppose a person has a university education (no graduate study). What is the (conditional) probability that he or she will make $25,000 or more?
  3. Find the total probability that a person's income category is at least as high as his or her educational level.

a. (P(E_3S_3) = P(S_3|E_3)P(E_3) = 0.45 cdot 0.05 = 0.0225)

b. (P(S_2 vee S_3|E_2) = 0.80 + 0.10 = 0.90)

c. (p = (0.85 + 0.10 + 0.05) cdot 0.65 + (0.80 + 0.10) cdot 0.30 + 0.45 cdot 0.05 = 0.9425)

In a survey, 85 percent of the employees say they favor a certain company policy. Previous experience indicates that 20 percent of those who do not favor the policy say that they do, out of fear of reprisal. What is the probability that an employee picked at random really does favor the company policy? It is reasonable to assume that all who favor say so.

(P(S) = 0.85), (P(S|F^c) = 0.20). Also, reasonable to assume (P(S|F) = 1).

A quality control group is designing an automatic test procedure for compact disk players coming from a production line. Experience shows that one percent of the units produced are defective. The automatic test procedure has probability 0.05 of giving a false positive indication and probability 0.02 of giving a false negative. That is, if (D) is the event a unit tested is defective, and (T) is the event that it tests satisfactory, then (P(T|D) = 0.05) and (P(T^c|D^c) = 0.02) . Determine the probability (P(D^c|T)) that a unit which tests good is, in fact, free of defects.

Five boxes of random access memory chips have 100 units per box. They have respectively one, two, three, four, and five defective units. A box is selected at random, on an equally likely basis, and a unit is selected at random therefrom. It is defective. What are the (conditional) probabilities the unit was selected from each of the boxes?

(H_i =) the event from box (i). (P(H_i) = 1/5) and (P(D|H_i) = i/100).

Two percent of the units received at a warehouse are defective. A nondestructive test procedure gives two percent false positive indications and five percent false negative. Units which fail to pass the inspection are sold to a salvage firm. This firm applies a corrective procedure which does not affect any good unit and which corrects 90 percent of the defective units. A customer buys a unit from the salvage firm. It is good. What is the (conditional) probability the unit was originally defective?

Let (T) = event test indicates defective, (D) = event initially defective, and (G =) event unit purchased is good. Data are

(P(G) = P(GT) = P(GDT) + P(GD^c T) = P(D) P(T|D) P(G|TD) + P(D^c) P(T|D^c) P(G|TD^c))

At a certain stage in a trial, the judge feels the odds are two to one the defendent is guilty. It is determined that the defendent is left handed. An investigator convinces the judge this is six times more likely if the defendent is guilty than if he were not. What is the likelihood, given this evidence, that the defendent is guilty?

Let (G) = event the defendent is guilty, (L) = the event the defendent is left handed. Prior odds: (P(G)/P(G^c) = 2). Result of testimony: (P(L|G)/P(L|G^c) = 6).

Show that if (P(A|C) > P(B|C)) and (P(A|C^c) > P(B|C^c)), then (P(A) > P(B)). Is the converse true? Prove or give a counterexample.

(P(A) = P(A|C) P(C) + P(A|C^c) P(C^c) > P(B|C) P(C) + P(B|C^c) P(C^c) = P(B)).

The converse is not true. Consider (P(C) = P(C^c) = 0.5), (P(A|C) = 1/4).

(P(A|C^c) = 3/4), (P(B|C) = 1/2), and (P(B|C^c) = 1/4). Then

(1/2 = P(A) = dfrac<1> <2>(1/4 + 3/4) > dfrac<1> <2>(1/2 + 1/4) = P(B) = 3/8)

Since (P(cdot |B)) is a probability measure for a given (B), we must have (P(A|B) + P(A^c|B) = 1). Construct an example to show that in general (P(A|B) + P(A|B^c) e 1).

Suppose (A subset B) with (P(A) < P(B)). Then (P(A|B) = P(A)/P(B) < 1) and (P(A|B^c) = 0) so the sum is less than one.

a. (P(A|B) > P(A)) iff (P(AB) > P(A) P(B)) iff (P(AB^c) < P(A) P(B^c)) iff (P(A|B^c) < P(A))

b. (P(A^c|B) > P(A^c)) iff (P(A^c B) > P(A^c) P(B)) iff (P(AB) < P(A) P(B)) iff (P(A|B) < P(A))

c. (P(A|B) > P(A)) iff (P(AB) > P(A) P(B)) iff (P(A^c B^c) > P(A^c) P(B^c)) iff (P(A^c|B^c) > P(A^c))

Show that (P(A|B) ge (P(A) + P(B) - 1)/P(B)).

(1 ge P(A cup B) = P(A) + P(B) - P(AB) = P(A) + P(B) - P(A|B) P(B)). Simple algebra gives the desired result.

Show that (P(A|B) = P(A|BC) P(C|B) + P(A|BC^c) P(C^c|B)).

An individual is to select from among (n) alternatives in an attempt to obtain a particular one. This might be selection from answers on a multiple choice question, when only one is correct. Let (A) be the event he makes a correct selection, and (B) be the event he knows which is correct before making the selection. We suppose (P(B) = p) and (P(A|B^c) = 1/n) . Determine (P(B|A)) show that (P(B|A) ge P(B)) and (P(B|A)) increases with (n) for fixed (p).

(dfrac = dfrac) increases from 1 to (1/p) as (n o infty)

Polya's urn scheme for a contagious disease. An urn contains initially (b) black balls and (r) red balls ((r + b = n)). A ball is drawn on an equally likely basis from among those in the urn, then replaced along with (c) additional balls of the same color. The process is repeated. There are (n) balls on the first choice, (n + c) balls on the second choice, etc. Let (B_k) be the event of a black ball on the (k)th draw and (R_k) be the event of a red ball on the (k)th draw. Determine

b. (P(B_1B_2) = P(B_2) P(B_2|B_1) = dfrac cdot dfrac)

c. (P(R_2) P(R_2|R_1) P(R_1) + P(R_2|B_1) P(B_1))

d. (P(B_1|R_2) = dfrac) with (P(R_2|B_1) P(B_1) = dfrac cdot dfrac). Using (c), we have


  • Learn the signs of overtraining, such as ongoing fatigue, muscle soreness or joint aches and pains.
  • Vary the types of exercises, as well as their intensity and the order in which you do them.
  • Increase the length and intensity of your workouts gradually.
  • Build light workouts and rest days into your schedule.
  • Be sure to get enough sleep and eat a healthy diet.


NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.2

NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.2 are part of NCERT Solutions for Class 10 Maths. Here we have given NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.2

Ex 3.2 Class 10 Maths Question 1.
Form the pair of linear equations of the following problems and find their solutions graphically:
(i) 10 students of class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
(ii) 5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of one pencil and that of one pen.
Solution:



Worksheets for Class 10 Maths

Ex 3.2 Class 10 Maths Question 2.
On comparing the ratios (frac < < a >_ < 1 >>< < a >_ < 2 >>), (frac < < b >_ < 1 >>< < b >_ < 2 >>)
and (frac < < c >_ < 1 >>< < c >_ < 2 >>) , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
(i) 5x – 4y + 8 = 0, 7x + 6y – 9 = 0
(ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0
(iii) 6x – 3y + 10 = 0, 2x -y + 9 = 0
Solution:


Ex 3.2 Class 10 Maths Question 3.
On comparing the ratios (frac < < a >_ < 1 >>< < a >_ < 2 >>), (frac < < b >_ < 1 >>< < b >_ < 2 >>)
and (frac < < c >_ < 1 >>< < c >_ < 2 >>), find out whether the following pairs of linear equations are consistent, or inconsistent:


Solution:


Ex 3.2 Class 10 Maths Question 4.
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically.
(i) x + y = 5, 2x + 2y = 10
(ii) x-y – 8, 3x – 3y = 16
(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
Solution:




Ex 3.2 Class 10 Maths Question 5.
Half the perimeter of a rectangular garden, whose length is 4 m more than its width is 36 m. Find the dimensions of the garden graphically.
Solution:


Ex 3.2 Class 10 Maths Question 6.
Given the linear equation 2x + 3y – 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i) intersecting lines
(ii) parallel lines
(iii) coincident lines
Solution:



Ex 3.2 Class 10 Maths Question 7.
Draw the, graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.
Solution:


NCERT Solutions for Class 10 Maths Chapter 3 Pairs of Linear Equations in Two Variables (Hindi Medium) Ex 3.2






















NCERT Solutions for Class 10 Maths

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Forecasting: Principles and Practice (3rd ed)

Consider the GDP information in global_economy . Plot the GDP per capita for each country over time. Which country has the highest GDP per capita? How has this changed over time?

For each of the following series, make a graph of the data. If transforming seems appropriate, do so and describe the effect.

  • United States GDP from global_economy
  • Slaughter of Victorian “Bulls, bullocks and steers” in aus_livestock
  • Victorian Electricity Demand from vic_elec .
  • Gas production from aus_production

Why is a Box-Cox transformation unhelpful for the canadian_gas data?

What Box-Cox transformation would you select for your retail data (from Exercise 8 in Section 2.10)?

For the following series, find an appropriate Box-Cox transformation in order to stabilise the variance. Tobacco from aus_production , Economy class passengers between Melbourne and Sydney from ansett , and Pedestrian counts at Southern Cross Station from pedestrian .

Show that a (3 imes5) MA is equivalent to a 7-term weighted moving average with weights of 0.067, 0.133, 0.200, 0.200, 0.200, 0.133, and 0.067.

Consider the last five years of the Gas data from aus_production .

  1. Plot the time series. Can you identify seasonal fluctuations and/or a trend-cycle?
  2. Use classical_decomposition with type=multiplicative to calculate the trend-cycle and seasonal indices.
  3. Do the results support the graphical interpretation from part a?
  4. Compute and plot the seasonally adjusted data.
  5. Change one observation to be an outlier (e.g., add 300 to one observation), and recompute the seasonally adjusted data. What is the effect of the outlier?
  6. Does it make any difference if the outlier is near the end rather than in the middle of the time series?

Recall your retail time series data (from Exercise 8 in Section 2.10). Decompose the series using X-11. Does it reveal any outliers, or unusual features that you had not noticed previously?

Figures 3.19 and 3.20 show the result of decomposing the number of persons in the civilian labour force in Australia each month from February 1978 to August 1995.

Figure 3.19: Decomposition of the number of persons in the civilian labour force in Australia each month from February 1978 to August 1995.

Figure 3.20: Seasonal component from the decomposition shown in the previous figure.

  1. Write about 3–5 sentences describing the results of the decomposition. Pay particular attention to the scales of the graphs in making your interpretation.
  2. Is the recession of 1991/1992 visible in the estimated components?

This exercise uses the canadian_gas data (monthly Canadian gas production in billions of cubic metres, January 1960 – February 2005).

  1. Plot the data using autoplot() , gg_subseries() and gg_season() to look at the effect of the changing seasonality over time. 1
  2. Do an STL decomposition of the data. You will need to choose a seasonal window to allow for the changing shape of the seasonal component.
  3. How does the seasonal shape change over time? [Hint: Try plotting the seasonal component using gg_season() .]
  4. Can you produce a plausible seasonally adjusted series?
  5. Compare the results with those obtained using SEATS and X-11. How are they different?

The evolving seasonal pattern is possibly due to changes in the regulation of gas prices — thanks to Lewis Kirvan for pointing this out.↩︎


The 5 Whys in daily life

Although the 5 Whys is most widely used for manufacturing/development use, I’ve found that it is also quite applicable to daily life in any situation where one might seek deeper understanding—of a problem, a challenge or even a motivation behind an action.

This quick graphic from Start of Happiness provides a great example:

Ever since learning about the 5 Whys, I find myself asking “why?” a lot more often.


Watch the video: Class - 10th, Ex -, Q6 Maths Pair of Linear Equations in Two Variables NCERT CBSE (October 2021).