Math Answers

Express 1/(cosθ − i sinθ) in the form of a + ib and hence prove that

cos θ + i sinθ

cos θ − i sinθ = cos 2θ + i sin2θ.

1. Suppose that three items are selected at random from a manufacturing process. Each item is inspected and classified defective, D, or nondetective, N. To list the elements of the sample space providing the most information, construct the tree diagram. Now, the various paths along the branches of the tree give the distinct sample points. Starting with the first path, we get the sample point DDD, indicating the possibility that all four items inspected are defective.

a. What is the sample space? b. Make tree diagram of (a)

1) dx/dy+4y=12 for y(0)=1 (general nad particular solution)

A farmer can plant upto to 8 acres of land with wheat and barley. He can earn $5000 for every acre he plants with wheat and $3000. His use of necessary pesticides is limited by federal regulations of 10 gallons for his entire 8 acres. Wheat requires 2 gallons of acre for each planted and barley one gallon per acre. What is the maximum profit he can make

Among 200 persons interviewed as part of an urban mass transportation study, some live more than 5 miles from the center of the city (X), some now regularly drive their own car to work (Y), some would gladly switch to public mass transportation if it were available (Z). Use the information given in the below figure to find the probabilities the person

a) Lives more than 5 miles from the center of the city

b) Lives more than 5 miles from the center of the city given that he or she would gladly switch to public mass transportation

c) Regularly drives his or her car to work given that he or she lives more than 5 miles from the center of the city

d) Would not want to switch to public mass transportation given that he or she does not regularly drive his or her car to work

The breakdown voltage of a randomly chosen diode of a certain type is known to be normally distributed with mean value 39 V and standard

deviation 1.1 V.

a) What is the probability that the voltage of a single diode is between 37.98 and 41.24?

b) What value c is such that only 1% of all diodes have voltages exceeding that value?

c) If 6 diodes are independently selected, what is the probability that at least 2 has a voltage exceeding 41.24?

Let A be {1, 2, 3, 4}. Then the sequences 124, 421, 341 and 243 are same  permutations of A taken 3 at a time. The sequences 12, 43, 31, 24, and 21 are examples of different permutations of A taken two at a time.

As for those three guys, John, Bill and Bob, I will only vote for one. Translate into proposition logic.

1.     The following formulas have been abbreviated based on the common abbreviation rules. Follow the steps below and translate the formulas into good English.

·       Step 1: Re-add the omitted brackets.

·       Step 2: If necessary, convert them into some other logically equivalent formula

so as to make it more readable. Write out the rule(s) you use for conversion.

·       Step 3: Translate the formulas into `good' English. Try to make your translation as brief/understandable as possible. (For instance, `John and Bill are coming' is better than `John is coming and Bill is coming.')

p: John wants to come to the class.

q: John will come to the class today.

r: John audits the class.

s: John is enrolled in the class.

Hint:

`No matter whether John is going or not, I'm going.' is the translation for (j à i) ^ (⌐j à i),

in which j = John is going, i = I'm going.)

Janny opened a savings account and deposited $300.00 as principal. The account earns 6% interest, compounded annually. What is the balance after 6 years?