Learn Matrices Online

Introduction about learn matrices online:

Matrix  is a rectangular array of numbers, arranged in rows and columns.

The number of rows (say m) and the number of columns (say n) find the order of the matrix.

It can be written as m x n (to be read as m by n).

Learn matrices online:

The plural  of matrix is matrices. Matrix is usually denoted by capital letters such as A, B, C,…

The Matrices can be defined as the set of m × n numbers arranged in a rectangular form. here m is row of the matrix and  n is column of the matrix.

for eg: A =  `[[a,b],[c,d]]` is a matrix of order 2 × 2.

Matrices Online is  used to verify the consistency of the linear equation system. Also, used to find the solution for the linear equation system with two and three variables by matrix inversion. Please express your views of this topic Matrices Calculator by commenting on blog.



Example Problems to learn solving online matrices:


Learn online solving  matrices Example 1:

If  `[[2,x],[x,1]]` = `[[5,2],[4,1]]`   ,then find x?

Solution:

Since the given matrices are equal,their corresponding elements must be equal.

Equating the corresponding elements, implies

2 - x2 = 5 - 8

x2 = 5 then

x = `+- sqrt(5)` .

Learn solving online matrices Example 2:

Evaluate determinant   = `[[1,2,2],[2,2,1],[1,2,1]]`

Solution:

= 1(2-2) - 2(2-1) + 2(4-2)

= 0 - 2(1) + 2(2)

= -2 + 4

= 2.

Learn solving online matrices Example 3:

If  `[[x+3,z+4,2y-7],[-6,a-1,0],[b-3,-21,0]]` = `[[0,6,3y-2],[-6,-3,2c+2],[2b+4,-21,0]]`   Find the values of a, b, c, x, y and z.

Solution:

Since the given matrices are equal,their corresponding elements must be equal.

Equating the corresponding elements, implies

x + 3 = 0,

z + 4 = 6,

2y – 7 = 3y – 2,

a – 1 = – 3,

0 = 2c + 2,

b – 3 = 2b + 4,

By Simplification,

Answers:

a = – 2,

b = – 7,

c = – 1,

x = – 3,

y = –5,

z = 2.

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Learn online solving Matrices Example 4:


Find the values of a, b, c, and d from the following matrix equation:

`[[2a+b,a-2d],[5c-d,4c+3d]]` =` [[4,-3],[11,24]]` .

Solution:

since the given matrices are equal,their corresponding elements must be equal.

equating the corresponding elements, implies

2a + b = 4,

5c – d = 11,

a – 2b = – 3,

4c + 3d = 24,

By Simplification,

Answers:

a = 1,

b = 2,

c = 3 and

d = 4.

Learn online Solving matrices Example 5:

If A = `[[sqrt(3) ,1,-1],[2,3,0]]` and B = `[[2,sqrt(5),1],[-2,3,1/2]]` , find A + B.

Solution:

Since, the two matrices is of the same order.

Matrix addition is possible, implies

A+B = `[[sqrt(3)+2,1+sqrt(5),-1+1],[2-2,3+3,0+1/2]]`

= `[[2+sqrt(3),1+sqrt(5),0],[0,6,1/2]]` .

Math Superstars Answers

Introduction to math superstars answers:

The basic concepts of number, measurement, algebra, and geometry. In number basis of simple addition, subtraction, multiplication, and conservation. The Superstars program is to provide the additional or extra challenge that self- motivated to the students. Usually more than one way to solve the problems but student needs the opportunity to discover the shortcut methods. It is best way to learn the math’s and easily understand the concept. Understanding Math Answers Fast is always challenging for me but thanks to all math help websites to help me out.


Math superstars problems 1 to 5::


Math superstar problem 1:  If the 24th day of the month falls on Saturday, on what day did the 8th fall?

Solution:  Students can use a calendar or make a chart with “Su, M, T, W, Th, F, Sa” at the top and begin numbering      backward putting 24 under Saturday. They may also realize that the 17th and 10th fall on Saturdays and count back from the 10th.

Answer: Thursday

Math superstar problem 2:  Together, 12 tigers  and 6 horses weigh 1050 pounds. The horses all weigh the same -- x pounds. Each tiger weighs 55 pounds.

What is the  weigh of one horse?

Solution: Students will probably solve this by first finding the total weight of the 12 tigers: 12 × 55 = 660 pounds. Then   they will compute 1050 - 660 = 390 pounds, the weight of the 6 horses. Then 390 ÷ 6 = 65 pounds per horse.:

Answer: 65pounds

Math superstar problem 3: The sum of 3 consecutive numbers is 405. What are the numbers?

Solution:   Students may use the guess-check-revise method. Some students might know that the numbers they seek are about 1/3 of the total, and approximate the numbers by dividing 405 by 3. This gives 135, which is the middle number.

Answer: 134, 135, 136

Math superstar problem 4: . How many corners are on 3 squares?

Solution: 1 square = 4 corners

3 square = 4 × 3 = 12 corners

Answer: 12 corners

Math superstar problem 5:  How many sides are on 2 squares?

Solution: 1 Square = 4 sides

2 Square = 4 × 2 = 8 sides

Answer: 8 sides

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Math superatars problem 6 to 10::


Math superstar problem 6:  I saw 1 cat and 2 dogs outside. How many legs did i see?

Solution: 1 cat = 4 legs

2 dogs= 8 legs

8 + 4 = 12 legs

Answer: 12 legs

Math superstar problem 7:  I have 6 balls, half of them are red. How many are Red?

Solution: Total balls = 6

Half of ball = 6/2 = 3

Answer: 3 Red balls

Math superstar problem 8:  I have a nickel and 2 pennies, how much money do I have all together?

Solution: 1 nickel = 5 pennies

So, Total = 5 + 2 = 7 pennies

Answer: 7 pennies

Math superstar problem 9: There are 3 cars, 2 bicycles, and 4 tricycles in the neighbor's garage. How many wheels are there in all? Forget about any "spare tires"!

Solution:  There would be 12 wheels on the 3 cars, 4 on the 2 bicycles, and 12 on the 4 tricycles

Answer: 28wheels

Math superstar problem 10: .I saw three triangles, how many sides are there all together on two triangles?

Solution: 1 triangle has 3 sides

3 triangle has 3 × 3 = 9 sides

Answer: 9 sides

Algebra 2. Quadratic Functions

General Form of quadratic function:

In general the quadratic function is of the form f(x) = ax^2 + bx + c,

Where a, b, and c are numbers and a not equal to zero.

The graph of a quadratic function is a curve and it is called as a parabola. Parabolas may open upward or downward and different length in "width" or "steepness", but they all have the same basic "U" shape. For example, y = 2x^2is a quadratic function since we have the x-squared term. y = x^2+1/x would not be a quadratic function because the 1/x term is x-1 which does not valid the form.


Sketching Parabola


To sketch the graph of the quadratic equation, following steps are used  :

(i)Verify if a > 0 or a < 0 to decide if it is opened upwards or downwards respectively.

(ii)If a>0, the quadratic function has a minimum value, and if a<0 a="" br="" function="" has="" maximum="" quadratic="" the="" value.="">
(iii)The x-coordinate of the minimum or maximum point is given by

X=-b/2a

We need to substitute x-value into our quadratic function Then we will have the (x, y) coordinates of the minimum or maximum point. This is called the vertex of the parabola.

(iv)To find the he coordinates of the y-intercept, substitute x = 0. This is always simple to find!

(v)We substitute y = 0 to find the values of y-intercepts in the quadratic equation.


Example for Quadratic functions


Sketch the graph of the function y = 2x^2 ? 8x + 6

Solution:

We first identify that a = 2, b = -8 and c = 6.

Step (i) Since a = 2, a > 0, it is a parabola with a minimum point and opens upwards

Step (ii) The x co-ordinate of the minimum point is

X= -b /2a  = -8/  2(2)= 2

y value of the minimum point is

y = 2(2)2 - 8(2) + 6 = -2

Hence the minimum point is (2, -2)

Step (iii) the y-intercept is found by substituting x = 0

y = 2(0)2 - 8(0) + 6 = 6

Hence (0, 6) is the y-intercept.

Step (iv) The x-intercepts are found by setting y = 0

2(x^2 - 4x + 3) = 0

2(x - 1)(x - 3) = 0

Hence x = 1, or x = 3.

Precalculus with Limits

Introduction:

In general, the limit is an extension up to which something can go. In mathematics, a limit is an intended height of a function. In other words, a limit is defined as the boundary of a specific area. The function limit is most often used in calculus problems.

Generally it is written as

`lim_(x->c)`  f (x) = K

Where    c - real number (when f(x) is a real valid function).


Properties:

Let the given function be f(x) and x approaches to h.
State all the possibilities.
Simplify the given function to apply the limits and solve.
Replace x for h.
Simplify the function.
The limits of a function could be obtained.

Example problems:

Problem 1: Find the limit of the function f (x) = 3x as x approaches 6.

Solution: Given f (x) = 3x

Substituting the value of x the equation becomes

f (6) = 3 (6) = 18.

So, the limit of f (x) = 3x as x approaches 6 is 18.

Problem 2: Find the limit of the function 9x^2 + 2x – 5 as x approaches 2.

Solution:

Given 9x^2 + 2x – 5

Substituting the values of x

= (9) (2)2 + 2 (2) – 5

= (36 + 4 – 5)

= 35

Hence the limit of 9x^2 + 2x – 5 as x approaches 2 is 35.

Problem 3: Solve (|x^2 - 5x + 6| / (x – 2))  for x approaching to 1

Solution:

Given (|x^2 - 5x + 6| / (x – 2))

Factorize the numerator |x^2 - 5x + 6|

x^2 - 5x + 6 = 0

(x – 2) (x – 3) = 0

The roots are x = 2 and x = 3.

Substitute the roots in the given function

((x – 2) (x – 3) / (x – 2))

Applying the limit

= ((1 - 2) (1 – 3) / (1 – 2))

= (-1) (-2) / (-1)

= 2 / (-1)

= -2

Hence the solution of (|x^2 - 5x + 6| / (x – 2)) is -2

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Practice Problems:

Determine the limit of the function f(y) = -2y as y approaches to 0.5
Answer: -1
Find the limit of f(x) = sin x as x approaches to 0
Answer: 0
Find the limit: f(x) = x^2 - 5x + 6 when x tends to 2
Answer: 0
Solve the following: `lim_(x->3)` (x^2 + 8x + 3) / (x3 + 2x + 1)
Answer: 18 / 17

Language Proof and Logic

Introduction to Language Proof and Logic

Statement is logic. Logic is a set of sentences in a perfect language. Theorems are individual sentences. Equivalence is the two declarations are logically equivalent if they have the identical truth values for all grouping of truth value of their variables. Language proof done in logic. Now we will see the examples for language proof and logic. I like to share this Population Correlation Coefficient with you all through my article.

Basic Operations- Language Proof and Logic

Negation (~p)    : If the condition is true then change into false and vice versa.

Disjunction (pvq)  : All declarations answer will be true. Only the answer will be false if two declarations are false.

Conjunction (p^q)  : If two statements are true then the result will be true otherwise false.

Conditional (p?q) : Truth of the statement ‘p’ is sufficient to truth of statement ‘q’.

Bi-conditional (p?q):  p?q to be true, both p and q must have the equal accuracy values. Otherwise it is false.

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Examples-language Proof and Logic

Example 1

What is the conjunction form for the following statements?

P: Michel reading newspaper.

Q: Peter reading story books.

Solution:

Given statement is as follows,

P: Michel reading newspaper.

Q: Peter reading story books.

Conjunction is a ‘and’ declaration. The representation of ‘and’ condition is ^.

We can write the above sentence in the following way.

‘Michel reading newspaper and Peter reading story books’.

So logical form = P v Q.

Example 2

Solve the logical term for the following statement.

‘If you go to the theater early then you can buy the ticket easily’.

Solution:

Let P: If you go to the theater early.

Q: Then you can buy the ticket easily.

The given state is ‘if-then’. The sign of ‘or’ is ?.

So logical form=P?Q.

These example problems are used to understand about the language proof and logic.

Calculate Inches to Cubic Feet

Introduction to calculate inches to cubic feet:

The area measurement is done with various units of area measurements. There are various type of units for the measurement of the area like the metric system, SI units etc. The inches and feet are the basic units of the area measurement commonly applied for the small areas. In this article we will see more about the measurement of area using the inches and square feet.

More on Calculate Inches to Cubic Feet:

The area measurements of the small areas are measured with the basic units like the square inch and the square feet. The relation between the square inch and the square feet helps in the conversion of the area measured using the square inch to square feet. The relation between the square inches and the square feet is given by the relation,

1 square feet = 144 square inches

The relation can also be written as the form for the inches as,

1 square inch = `1/144` square feet

Example Problems on Calculate Inches to Cubic Feet

1. Convert the given area of 152 square inches into its equivalent square feet.

Solution:

1 square feet = 144 square inches

1 square inch = 1/144 square feet

152 square inch = `152/144` square feet

152 square inch = 1.05 square feet

2. Convert the area of 245 square inches into square feet.

Solution:

1 square feet = 144 square inches

1 square inch = 1/144 square feet

245 square inch = `245/144` square feet

245 square inch = 1.70 square feet

3. Convert the area of 683 square inches into its equivalent square feet.

Solution:

1 square feet = 144 square inches

1 square inch = 1/144 square feet

683 square inch =` 683 /144` square feet

683 square inch = 4.74 square feet

Practice problems on calculate inches to cubic feet

1. Convert the given area of 1231 square inches into its equivalent square feet.

Answer: 8.54 square feet.

2. Convert the area of 510 square inches into square feet.

Answer: 3.54 square feet.

3. Convert the given area of 195 square inches into square feet measurement.

Answer: 1.35 square feet.

Greater than or less than Calculator

Introduction to greater than or less than calculator:

In this article we discuss about the greater than or less than calculator. The greater than or less than function deals with algebra mathematics. The greater than or less than calculator operates the four basic functions such as addition, subtraction, multiplication and division. In greater or less than calculator we are using many mathematical symbols. Each one of the symbol represents the particular operation. The some of the mathematical symbol is given below,

< It denotes the less than symbol

<= It denotes the less than or equal to symbol

> It denotes the greater than symbol

>= It denotes the greater than or equal to symbol

!= It denotes the not equal to symbol

= It denotes the equal to symbol

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Explanations of Greater than or less than Calculator:

Less Than(<)

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Less Than or equal to(<=)

If a<=5, the variable ‘a’ is less than or equal to value 5, the variable ‘a’ may be equal to 5 or may be any one of the following value 4, 3, 2, ……… so on.

Greater Than(>)

If a>5, the variable ‘a’ is greater than value 5, the variable ‘a’ may be any one of the following value 5, 6, 7,……… so on.

Greater Than or equal to(>=)

If a>=5, the variable ‘a’ is greater than or equal to value 5, the variable ‘a’ may be 5 or may be any one of the following value 6, 7, 8, ……… so on. Please express your views of this topic syllabus of class 9 cbse by commenting on blog.

Examples for Greater than or less than Calculator:

Addition and subtraction of greater than or less than calculator:

4x+6<-2x br="">
4x+6-6<-2x br="">
4x<-2x br="">
4x+2x<-2x br="" x="">
7x<2 br="">
`x>2/7`