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1. Let l_1 and l_2 be lines through the origin in R^2 that intersect in an angle pi/n, and let r_i be the reflection about l_i. Prove that r_1 and r_2 generate a dihedral group Dn.
2. The symmetric group S_3 operates on two sets U and V of order 3. Decompose the product set U x V into orbits for the "diagonal action" g(u, v) = (gu, gv), when
A) the operations on U and V are transitive
B) the operation on U is transitive, the orbits for the operation on V are {v_1} and {v_2, v_3}
3. Let G be the group of symmetries of a cube, including the orientation-reversing symmetries. Describe the elements of G geometrically.
Probability density function sample question, Let E be the event that the number which comes up when a single die is tossed is divisible by 3. Let X be the number of times that event E occurs in 3 tosses of the die
What is the probability of the following card hands from a 52 card deck. In poker, aces are either high or low. A bridge hand is made up of thirteen cards.
given a finite set of integers such that all of them are less than or equal to 3, how could you solve in polynomial time that a subset of integers exists whose sum is zero?
Is there a vector r such that the directional derivative of f at (1,1,0) in the direction of r equals 1? If there is, find one such vector. If not, explain why not.
A fire detection sensor is tested for its failure rate and found to have a mean failure rate of 3.6E-2 per hour of exposure with a standard deviation of
Show that the equation AX = B represents a linear system of two equations in two unknowns Solve the system and substitute into the matrix equation to check your results
The surface area of the right squre pyramid is given by S=b square rt. b^2 + 4h^2. If the pyramid has height of 10 feet and surface area of 100 square feet, what is the length of a side b of its base?
Drawing at least one black card when you draw a card from a standard deck 6 times, replacing the card each time you draw so that there are always 52 cards in the deck.
What is the relationship between exponents and logarithms? How would you distinguish between the two, using both a graph and a sequence? Explain by showing examples of each.
Perform the appropriate analysis and state the conclusions
How many passengers are required in the sample to reduce in half the sampling error within which we need to estimate the proportion of passengers who had purchased tickets for more than $400 over a year's time?
Evaluate the given expression and Perform the indicated operation - Evaluate -2x 4 - 3x 2 + 5x - 9 for x = 2.
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