Euclid’s Division Lemma

Euclid’s Division Lemma:
Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.
Euclid’s division algorithm is based on this lemma.

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
The word algorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi. In fact, even the word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabrw’al muqabala. A lemma is a proven statement used for proving another statement.

Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers.

Equiangular Triangles

Two triangles are similar, if

(i) their corresponding angles are equal and
(ii) their corresponding sides are in the same ratio (or proportion). Note that if corresponding angles of two triangles are equal, then they are known as equiangular triangles.

A famous Greek mathematician Thales gave an important truth relating to two equiangular triangles which is as follows:
The ratio of any two corresponding sides in two equiangular triangles is always the same. It is believed that he had used a result called the Basic Proportionality Theorem (now known as the Thales Theorem) for the same.

Arithmatic Progressions

Arithmetic Progressions:
Arithmetic Progressions was known ever three thousand years ago.
You must have observed that in nature, many things follow a certain pattern, such as
the petals of a sunflower, the holes of a honeycomb, the grains on a maize cob, the
spirals on a pineapple and on a pine cone etc.
We now look for some patterns which occur in our day-to-day life. Some such
examples are :

(i) Jia applied for a job and got selected. She
has been offered a job with a starting monthly
salary of Rs 8000, with an annual increment of
Rs 500 in her salary. Her salary (in Rs) for the
1st, 2nd, 3rd, . . . years will be, respectively
8000, 8500, 9000, . . . .

(ii) The lengths of the rungs of a ladder decrease
uniformly by 2 cm from bottom to top The bottom rung is 45 cm in
length. The lengths (in cm) of the 1st, 2nd,
3rd, . . ., 8th rung from the bottom to the top
are, respectively
45, 43, 41, 39, 37, 35, 33, 31

(iii) In a savings scheme, the amount becomes 5/4 times of itself after every 3 years.
The maturity amount (in Rs) of an investment of Rs 8000 after 3, 6, 9 and 12
years will be, respectively :10000, 12500, 15625, 19531.25

The number of unit squares in squares with side 1, 2, 3, . . . units (see below Fig)
are, respectively
12, 22, 32, . . . .

Circles

Circles:
The different situations that can arise when a circle and a line
are given in a plane.
So, let us consider a circle and a line PQ. There can be three possibilities given
in Fig. below:

The line PQ and the circle have no common point. In this case,
PQ is called a non-intersecting line with respect to the circle. In above Fig. (ii), there
are two common points A and B that the line PQ and the circle have. In this case, we
call the line PQ a secant of the circle. In Fig. 10.1 (iii), there is only one point A which
is common to the line PQ and the circle. In this case, the line is called a tangent to the
circle.
The tangent to a circle is a special case of the secant, when the two end
points of its corresponding chord coincide.

Matrices

The knowledge of matrices is necessary in various branches of mathematics.

Matrices are one of the most powerful tools in mathematics.
This mathematical tool simplifies our work to a great extent when compared with other straight forward methods. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrices are not only used as a representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use. Matrix notation and operations are used in electronic spreadsheet
programs for personal computer, which in turn is used in different areas of business
and science like budgeting, sales projection, cost estimation, analyzing the results of an
experiment etc. Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices. Matrices
are also used in cryptography. This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
management.

Example 1:
Consider the following information regarding the number of men and women
workers in three factories I, II and III
Men workers Women workers
I 30 25
II 25 31
III 27 26
Represent the above information in the form of a 3 × 2 matrix. What does the entry
in the third row and second column represent?

MATHEMATICS
Solution The information is represented in the form of a 3 × 2 matrix as follows:



30 25
A=25 31
21 26


The entry in the third row and second column represents the number of women
workers in factory III.

arithmetic reasoning help problems

Arithmetic reasoning help is for students who are preparing for test or competitive exams.We will have 4 options for each question ,in which we have to choose one correct answer.Let's see one of these kind of examples.

Question:-

A doctor can treat 4 cancer patients per hour; however stroke patients need 3 times as much of the doc's time. If the doctor treats patients 6 hours a day and has already treated 10 cancer patients and 3 stroke patients today, how many more stroke patients can the doctor treat?

A.1 B.2 C.3 D. 5

Answer:-

Let's see what is an expert online tutor explain us.

5,5' is the average of 5' and 6': (5 + 6)/2 = 5,5

One would expect that the height of the average is the average of the heights:Height converter will help us with this.

(110 + 170)/2 = 140



HEIGHT WEIGHT
5' 110
6' 170

?

I could just multiply 5x60 to find the LBS of a 5' person, why is this given? I tell you, that's "military math"...

Anyway, let's talk about that doctor:

"4 cancer patients per hour" => one quarter of an hour per cancer patient.

"3 times as much" for stroke patients => three quarters of an hour per stroke patient.

"has already treated 10 cancer patients and 3 stroke patients today" => has already consumed 10 quarters plus 3x3=9 quarters of his time. Makes 19 quarters altogether.

"the doctor treats patients 6 hours a day" => the doctors has 6x4=24 quarters of an hour at his disposal per day.

But he already spent 19 quarters => he has 24-19=5 quarters of an hour still available today.

How many stroke patients fit in 5 quarters of an hour? You already know from above that each stroke patient consumes 3 quarters.

So the answer is "c".

How to find the equation of a line which is passing through two points

Question:-

Find the equation of line passing through the points(-1,7)and (0,15)

Answer:-

Step 1:-

finding the slope of line = m

                      y2-y1     15-7         8
Formula to find slope ------- = ------- = ------ = 8
                      x2-x1     0-(-1)      0+1

so ,slope of a line is 8

step 2:-

The point slope form of equation (y-y1)=m(x-x1)

We know that (x1,y1) is (-1,7)

and m=8

So (y-7)=m-(x-(-1))

(y-7)=8(x+1)

y-7=8x+8
+7   +7
---------
y=8x+15

step 3:-

The equation of the line becomes

             y=8x+15

For more help on this,you can reply me.