Straight Line Equation

Let us study EQUATION OF A STRAIGHT LINE

In Mathematics, equations such as

2x+y=6

are designated as linear equations, and their graphs are shown to be straight lines. The purpose of this discussion is to study the relationship of slope to the equation of a straight line.

POINT-SLOPE FORM

Suppose that we want to find the equation of a straight line that passes through a known point and has a known slope. Let (x,y) represent the coordinates of any point on the line, and let (x,,y,) represent the coordinates of the known point. The slope is represented by m.

Recalling the formula defining slope in terms of the coordinates of two points, we have

STRAIGHT-LINE AND COMPASS CONSTRUCTIONS

Let us study about straight line and compass constructions,
You can draw straight lines, and set the compasses to any radius you like (obviously you can't measure the radius, because you've no markings on your ruler); in particular, if you've managed to mark two points on the (infinite) plane that you're drawing on (normally points are defined when arcs and lines intersect) then you can set the compass to exactly the distance between those points.

example : Draw an equilateral triangle.

Solution :
1. Draw a line, A.
2. Mark two arbitrary points on A, b and c, not coincident. ( The distance bc will be the length of each side of the triangle.)
3. Draw a circle, centre b, radius bc. ( This is two steps rolled into one: 1. Set the compasses to radius bc 2. Draw a circle centred on b )
4. Draw a circle, centre c, radius bc.
5. The two circles will intersect in two points, d and d', one either side of A.
6. Draw the line segments bd and cd. bcd is the equilateral triangle. ( Since lengths bd = bc = cd )

I hope the above explanation was useful.

Determining the Eigenvalues of a Matrix

Since every linear operator is given by left multiplication by some square matrix, finding the eigenvalues and eigenvectors of a linear operator is equivalent to finding the eigenvalues and eigenvectors of the associated square matrix; this is the terminology that will be followed. Furthermore, since eigenvalues and eigenvectors make sense only for square matrices, throughout this section all matrices are assumed to be square.


Given a square matrix A, the condition that characterizes an eigenvalue, λ, is the existence of a nonzero vector x such that A x = λ x; this equation can be rewritten as follows:
This final form of the equation makes it clear that x is the solution of a square, homogeneous system. If nonzero solutions are desired, then the determinant of the coefficient matrix—which in this case is A − λ I—must be zero; if not, then the system possesses only the trivial solution x = 0. Since eigenvectors are, by definition, nonzero, in order for x to be an eigenvector of a matrix A, λ must be chosen so that
I hope the above explanation was useful.

Sampling Distributions

Let us study about Sampling Distributions,
Sampling variability is the tendency of the same statistic computed from a number of random samples drawn from the same population to differ.

Suppose that the first sample of 100 magazine subscribers was “returned” to the population (made available to be selected again), another sample of 100 subscribers selected at random, and the mean income of the new sample computed. If this process were repeated ten times, it might yield the following sample means:

27,500 27,192 28,736 26,454 28,527

28,407 27,592 27,684 28,827 27,809

These ten values are part of a sampling distribution. The sampling distribution of a statistic—in this case, of a mean—is the distribution obtained by computing the statistic for a large number of samples drawn from the same population.
Hope the above explanation helped you.

Multiplying Polynomials

In this session i will explain how to Multiply Polynomials,

The following are rules regarding the multiplying of variable expressions.

Rule 1: To multiply monomials with the same base, keep the base and add the powers:
Rule 2: To find the power of a base, keep the base and multiply the powers.
Rule 3: To find a power of a product, raise each factor in the product to the power.
I hope the above explanation was useful, now let me explain about remainder theorem.

Law of Sines

Let us study about Law of Sines,
You can use the Law of Cosines discussed in the last section to solve general triangles, but only under certain conditions. The formulas that will be developed in this section provide more flexibility in solving these general triangles.

The following discussion centers around Figure 1 .
Figure 1

Reference triangles for Law of Sines.

Line segment CD is the altitude in each figure. Therefore Δ ACD and Δ BCD are right triangles. Thus,


In Figure 1 (b), ∠CBD has the same measure as the reference angle for β Thus,
It follows that


Similarly, if an altitude is drawn from A,


Combining the preceding two results yields what is known as the Law of Sines.
In other words, in any given triangle, the ratio of the length of a side and the sine of the angle opposite that side is a constant. The Law of Sines is valid for obtuse triangles as well as acute and right triangles, because the value of the sine is positive in both the first and second quadrant—that is, for angles less than 180°. You can use this relationship to solve triangles given the length of a side and the measure of two angles, or given the lengths of two sides and one opposite angle. (Remember that the Law of Cosines is used to solve triangles given other configurations of known sides and angles).

First, consider using the Law of Sines to solve a triangle given two angles and one side.
Hope the above explanation was helpful.

Permutations

Permutations:

Introduction:

Let us understand the meaning of the term Permutation with the help of few examples.In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging in an ordered fashion) objects or values.Informally, a permutation of a set of values is an arrangement of those values into a particular order. Thus there are six permutations of the set {1,2,3}, namely [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], and [3,2,1].
The different arrangements that can be made with a given number of things taking some or all of them at a time are called permutations.

The symbol nPr or P(n,r) is used to denote the number of permutations of n things taken r at a time.

Examples:


1. 2 and 3 are two digits and with these digits, the numbers 32 and 23 are formed. Although both numbers viz., 32 and 23 consist of the digits 2 and 3, the order of digits is different. Each of the above arrangements is called a 'permutation'. Thus, the number of arrangements or permutations of two distinct digits 2 and 3 is 2.
2. The permutation of the three letters a, b, c taken two at a time are ab,ba,ac,bc,cb.(6 in number)

Question: Find the number of permutations of the elements in the set {1, 3, 5, 7, 9, 11}.

A ) 6
B ) 21
C ) 36
D ) 720

Steps to derive

1 Number of elements in the set = 6

2 Number of permutations of 6 distinct elements of the set is 6P6.
[nPn = n!.]
3 = 6!

4 = 6 · 5 · 4 · 3 · 2 · 1

5 = 720

Hope you like the above example of Permutations.Please leave your comments, if you have any doubts.