Let P be an external point of a circle. Given two distinct secants PAB and PCD such that AB
and CD are chords of the circle. We know that PA x PB = PC x PD.
(a) Alternatively, if the point P lies on the circle, i.e., P moves from being an external point
to become concurrent with A and C, state why PA x PB = PC x PD is still obtained.
(b) It can be shown that PA x PB = PC x PD even if P is an internal point of a circle. The
power of a point P with respect to a circle is defined as ?2 − ?2 where d is the distance
from P to the centre of the circle and R is the radius of the circle. Using the results above,
determine the three possible locations of P when its power is zero, positive and negative,
respectively.

why is understanding quantities and units
important

The temperature T at a point (x,y,z) in space is inversely proportional to the square of the distance from (x,y,z) to the origin. It is known that T(0,0,1) = 500. a. [2] Compute T(2,0,0). b. [3] Find the rate of change of T at the point (2,3,3) in the direction of the point (3,1,1). c. [3] What is the maximal rate of change of T at the point (2,3,3)?

1a. An ATM requires a four-digit PIN, using the digits 0-9. How many PINs have no repeated digits?

1b. How many ways can president and vice president be determined in a club with twelve members?

1c. A security team visits 12 offices each night. How many different ways can the team order its visits?

1d. In a certain lottery you select seven distinct numbers from 1 through 39, where order makes no difference. How many different ways can you make your selection?

1e. First, second, and third prizes are to be awarded to three different people. If there are ten eligible candidates, how many outcomes are possible?

1f. Three identical “Outstanding Teacher” awards are to awarded to three different people. If there are ten eligible candidates, how many outcomes are possible?

Exercise 4.9.29: Solve the following systems of congruences, or state that there is no solution. Be sure to state if there are multiple solutions.

a. {6 = 13a + b(mod 26), 13 = 4a + b(mod 26)

b. {14 = 17a + b(mod 26), 8 = 7a + b(mod 26)

c. {1 = 15a + b(mod 26), 10 = 9a + b(mod 26)

Goofus and Gallant try out for the quiz team. Goofus, Gallant, and seven of their classmates are trying out for the quiz bowl team. There are four positions available on the team.

(a) If each position is different, how many ways are there that the team can be made?
(b) If each position is the same, how many ways are there that the team can be made?
(c) Suppose each position is the same. How many teams include both Goofus and Gallant?
(d) Suppose each position is the same. How many teams include Goofus or Gallant? ("Or" means "at least one of.")

Solve listed initial value problems by using the Laplace Transform:

6.       y^ll − 3y^l − 4y = 4t − 5,     y(0) = 2, y^l(0) = 4

Find at least the first six nonzero terms in the power series expansion about x=0 for a general solution to the differential equation:

y'' + (sinx)y = 0

Given that x =0 is a regular singular point of the given differential equation, show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x = 0.

xy"+(1-x)y'-y=0

what is social contract and stakeholder theories of Corporate Social Responsibility (CSR) andexpalin use the social demandingness, social activist and stakeholder theories of corporate social responsibility in order to discuss the case study below.

Solve the differential equation

y''(x)-2xy'(x)+2ny(x)=0

using the Hermite Polynomials

Find two distinct subgroups of order 2 of the group D3 of symmetries of an equilateral triangle. Explain why this fact alone shows that D3 is not a cynic group.