Math Answers

Draw the directed graphs representing each of the relations on {1, 2, 3, 4}

a) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}

b) {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)}

c) {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}

d) {(2, 4), (3, 1), (3, 2), (3, 4)}

Represent each of these relations on {1, 2, 3} with a matrix (with the elements

of this set listed in increasing order).

a) {(1, 1), (1, 2), (1, 3)}

b) {(1, 2), (2, 1), (2, 2), (3, 3)}

c) {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)}

d) {(1, 3), (3, 1)}

Show that the relation R = ∅ on the empty set S = ∅ is

reflexive, symmetric, and transitive.

Arithmetic mean of a certain number of observations was 60.8. Five new observations 85, 

60, 73, 81, 90 were added to the group. Now the arithmetic mean of all the observations 

becomes 65.05. Find how many observations were there originally.

Show that the relation R = ∅ on a nonempty set S is sym-

metric and transitive, but not reflexive.

Pebbles are poured out of a tube at one cubic meter per second. It forms a pile which has the shape of a cone. The height of the cone is equal to the radius of circular base. How fast is the pile of pebbles rising when it is 2 meters high

The cumulative distribution for a random variable x is

F(x) = 1/2 sqrt x-1; 1≤x≤5

Calculate the probabilities

Pr(1≤x≤4.1)

A random variable x has the cumulative distribution function given as

f(x) = 8>x<7

0; for x<1

X^2 -2x+2

2; for 1≤x2

1; for x≥2

Calculate the variance of x.

1. Convert the polar coordinates (-8, 2π/3) into rectangular coordinates.
2. Convert the rectangular coordinates (3, -3) into polar coordinates with r > 0 and 0 ≤ θ < 2π.
3. Convert the rectangular equation x2 + y2 = 100 into a polar equation that expresses r in terms of θ.
4. Convert the polar equation 4r cos θ + r sin θ = 8 into a rectangular equation that expresses y in terms of x.

1. The mean of a random sample of size 50 drawn from a population with mean 112 and standard deviation 40.

a. Find the sample mean.

b. Find the standard deviation of sample

c. What is the sample size?

d. What is the population mean?

e. If the population standard deviation is 40, compute for the variance population.