if n is an odd integer then it is the difference of two perfect squares can you show

ring. a) Calculate the work done to compress the spring. (2 marks) b) What happens to the work done on the spring ? (1 mark) c) If the spring is released, what happens to the energy of the spring? (1 mark) d) Calculate the total mechanical energy of the ball at the instant it leaves the spring. (2 marks) e) What will be the speed of the ball at the instant it leaves the spring? (2 marks) f) If the ball is fired up into the air by the spring, how much gravitational potential energy will it gain? (1 mark) g) What will be the maximum height of the ball? (2 marks

2,3,4,7). If it lands tails, a fair six-sided die is thrown (with values 3,4,5,6,7,9). Regardless of which die is used, Alice eats n grains of rice, where n is the largest prime factor of the die result (for example, the largest prime factor of 9 is 3). (a) What is the conditional probability that the coin lands heads, given that Alice eats three grains of rice? (b) Suppose that the entire experiment is conducted twice on the following day (starting with a new coin toss on the second run-through). What is the conditional probability that the coin lands heads on both run-throughs, given that Alice eats a total of five grains of rice during the two run-throughs? (Do not count the two grains from part (a) in part (b); we assume two brand new experiments, each with a new coin toss. Start your solution by defining a suitable partition of the sample space. Please use an appropriate notation and/or justification in words, for each value that you give as part of your solution.) Exercise 5) Alice and Bob throw an unfair coin repeatedly, with probability 2/5 of landing heads. Alice starts with £2 and Bob starts with £3 . Each time the unfair coin lands heads, Alice gives Bob £1 . Each time the unfair coin lands tails, Bob gives Alice £1 . The game ends when one player has £5 . (a) Draw a labelled Markov chain describing the problem, and write down a transition matrix P. Write down the communication classes, and classify them as either recurrent or transient. (b) Using the transition matrix, calculate the probability that Alice loses all of her money in exactly four tosses of the unfair coin. (c) Calculate the (total) probability that Alice loses all of her money (before Bob loses all of his). (d) Calculate the expected (mean) number of tosses of the unfair coin, for the game to end.

a prize with probability p independently of the other curtains. The player chooses one curtain (a) The game host tells the contestant there's a prize hiding behind at least one of the other curtains, and offers the player a chance to change his choice. Should he? (b) The game host tells the player there's a prize hiding behind exactly one of the other curtains. Should the player change his choice? How does your answer depend on p (c) How would your answer to section (b) change if there are n curtains in total? (d) Bonus (4 points): How would your answer to section (a) change if there are 4 curtains in total?

Recursive Step: if (n,m,r) ∈ A then (n+1,m+2,m+r+2) ∈ A Prove, using structural induction, that for every .( n , m , r ) ∈ A , 0 ≤ n < m and n + m ≤ r .

Since he became the president, did President Trump act with the transparency and the integrity that you expect from a president?” 675 voters responded the poll and 351 responded “YES.” Assume that 40% of the U.S. population supports Trump. a. Define a binary random variable, Y, for supporting Trump (Y=1) vs. not (Y=0). Calculate the population mean (????????) and variance (???????? 2 ) for supporting Trump. b. Calculate the sample mean ????̅ and the sample standard deviation of ????̅ (????????̅ ) for the poll. c. Calculate the standard error of ????̅ and construct a 95% confidence interval from the poll using ????̅ and its sample standard error. d. Conduct a two-sided hypothesis test at 5% significance level to determine whether 40% of the U.S. population supports Trump. State the null and the alternative hypotheses, calculate the test statistics and the associated p-value, and conclude. Is the Fox News survey reliable? Why? Why Not? e. Suppose that you wanted to design a survey that had a margin of error of at most 1%. That is: the difference between the upper bound and the lower bound of the confidence interval should be a maximum of 2 percentage points. For example, for ????̅ = 0.52 you are aiming for the 95 % CI to be [0.51 0.53]. How large should n be if the survey uses simple random sampling?

34.0 N when it’s not in the water. When it’s submerged in water (the density of water is 1.00 x 103 kg/m3) the scale now reads 27.0 N. (a) What is the density of the block? (b) If you suspended another object from the block that has a density of 3.20 x 103 kg/m3, with both objects submerged, what would the object's mass need to be for the scale to once again read 34.0 N? Note: Part (a) is worth 7 points, and part (b) is worth 8 points.

f the students make their choices independently and each is as likely to pick one integer as any other, what is the probability that 8 or more will select 4,5 or 6? b Having observed eight students who selected 4, 5, or 6, what conclusion do you draw based on your answer to part (a)?A missile protection system consists of n radar sets operating independently, each with a probability of .9 of detecting a missile entering a zone that is covered by all of the units. a If n = 5 and a missile enters the zone, what is the probability that exactly four sets detect the missile? At least one set? b How large must n be if we require that the probability of detecting a missile that enters the zone be .999?

ng n distinct integers in just his thoughts. He plays turns on the list. In each turn he does the following– He takes a number (in its index’s order ) and swap it with any number in the list including itself i.e. if it swap it with itself it doesn’t move at all (The selection of the number is completely random). He does the same for all the elements in their index’s order in that turn. If initially the list was unsorted, such that, no element was in sorted position, then find the probability that the list is sorted after m such turns. Note : Take the assumption that if an element is not in its sorted position then it can be in any other n − 1 positions equally likely.

ng the snow. The first child exerts a force of F1 = 11 N at an angle θ1 = 45° counterclockwise from the positive x direction. The second child exerts a force of F2 = 6 N at an angle θ2 = 30° clockwise from the positive x direction. Find the magnitude (in N) and direction of the friction force acting on the sled if it moves with constant velocity. magnitude direction (counterclockwise from the +x-axis) What is the coefficient of kinetic friction between the sled and the ground? What is the magnitude of the acceleration (in m/s2) of the sled if F1 is doubled and F2 is halved in magnitude?

el tabletop. A child attempting to use the spring to pull the block directly horizontally from rest extends the spring by 30.0 cm a) Does the block slip? Explicitly state how you know. b) Calculate the magnitude of the acceleration if the block accelerates or, if it does not, calculate the magnitude of the static frictional force acting on the block.

hey have the same remainder after division by n. Prove that a ≡ b (mod n) if and only if n|(a − b).

hey have the same remainder after division by n. Prove that a ≡ b (mod n) if and only if n|(a − b).

te data object as present inside each element of the array children . The output will be series of lines as follows: "data": {...} "data": {...} The contents between the curly braces will vary from one data object to another, and the subfields can be objects themselves (i.e., the subfields may have their values surrounded by curly braces). At the time of this writing, 26 such objects printed. It is critical to keep in mind that the order of the fields within the data objects is not important. The closest this can be described in C is as follows: struct data { char c; long n; }; The application code mustn't rely on the fact that field c was defined before the field n . In other words, a well written C application would work perfectly, without any code changes, even if the structure was redefined as follows: struct data { long n; char c; };