Broad Differentiation

In differentiation we use different formulas. Differentiation is the process of finding the derivative of the given function. The differentiation of the function is denoted as f'(x). The function can differentiated by different variables. Broad differentiation is the process of finding the derivative value one or more time of the given function. Broad differentiation includes third derivative and fourth derivative values. Now in this article we learn about broad differentiation and their example problems.

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Example problems for broad differentiation

Broad differentiation example problem 1:
        Find the third derivative value of the given function f(x) = 5x⁴ - 7x² + 61x - 9
Solution:
  Given function is f(x) = 5x⁴ - 7x² + 61x - 9
Differentiate the given function with respect to x, we get
                                f'(x) = 20x³ - 14x + 61
Again differentiate the given function for finding the second derivative, we get
                               f''(x) = 60x² - 14
For finding the third derivative, again differentiate the given function
                                f'''(x) = 120x
Answer:
      The final answer is 120x
Broad differentiation example problem 2:
        Find the third derivative value of the given function f(x) = 52x⁴ - 17x² + 33x
Solution:
  Given function is f(x) = 52x⁴ - 17x² + 33x
Differentiate the given function with respect to x, we get
                           f'(x) = 208x³ - 34x + 33
Again differentiate the given function for finding the second derivative, we get
                          f''(x) = 624x² - 34
For finding the third derivative, again differentiate the given function
                          f'''(x) = 1248x
Answer:
The final answer is 1248x

Broad differentiation example problem 3:
Find the fourth derivative value of the given function f(x) = 10x⁴ + 4x³ + 14x² - x
Solution:
  Given function is f(x) = 10x⁴ + 4x³ + 14x² - x
Differentiate the given function with respect to x, we get
                                f'(x) = 40x³ + 12x² + 28x - 1
Again differentiate the given function for finding the second derivative, we get
                                 f''(x) = 120x² + 24x + 28
For finding the third derivative, again differentiate the given function
                                 f'''(x) = 240x + 24
For fourth derivative, we get
                                f''''(x) = 240
Answer:
      The final answer is 240

Mean Frequency Table

In Statistics, mean is a basic term used frequently. A frequency table is one in which the mean and other terms can be determined. Basically, the frequency table consists of mean, median and mode. In this article, we shall discuss about the terms used in frequency table. Also we shall solve some sample problems based on mean of a frequency table.

Terms in a frequency Table:

Mean:
Mean is the method of the addition of values divided by the number of values in the given set of data.
Mean = addition of the given values / Entire number of values

Median:
Median is actually a middle number of a table. Some cases may have even values. In such cases, there will be 2 middle values. The median for such cases will be the average of the two vales.

Mode:
The most commonly repeated value in the given set of data is called as the mode for the given table..
Range:
The Variation between the greatest and the least values in the given set is said to be the range of the particular set.

Example problems for Mean Frequency Table:

Example 1:
Determine the mean, median, mode, and the range of the set of data given below.
13  17   15  16  22  25  22  27 14
Solution:
Given set of data can be arranged in the order.
13  14   15  16  17  22  22  25  27

Mean:
                Mean = ((13 + 14 + 15+ 16 + 17+ 22+ 22+ 25 + 27)/9)
                          = 171/9
                          = 19
              Median:
                  The middle number is 17. Hence the median is 17
              Mode:
                  Commonly repeated value is 22. Therefore mode is 22
              Range:
                  27 - 13 = 14.
The results are
                Mean = 19
              Median = 17
                Mode = 22
               Range = 14.

Example 2:
Consider the frequency distribution shown in scores of 20 students in a science test.
The mean of these marks = total marks /number of students
Frequency table:

Marks(x)	Frequency(f)
40	1
50	2
60	4
70	3
80	5
90	2
100	3
Total	20

                             
Then,  x = (40*1)+ (50*2)+ (60*4)+ (70*3)+ (80*5)+ (90*2)+ (100*3)
             =  40  + 100 + 240 + 210 + 400 + 180 + 300
             =  1470 / 20
             = 551 / 7
              = 73.5 marks

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    It will be very easy to discover the products of the scores and the frequencies by adding an extra column to the frequency table and the work as below:

Marks(x)	Frequency(f)	fx
40	1	40
50	2	50
60	4	240
70	3	210
80	5	400
90	2	180
100	3	300
Total	20	1470

                                                                
   Mean = sum of the xf column/ sum of the f column  
            = 1470/20

   Mean = 73.5 marks

Saxon Math Calculus

• Saxon math calculus deals with solving basic calculus problems for test preparation. Most of the Saxon’s math book deals with solving test problems for exam preparation. Saxon’s Calculus mainly deals with the differentiation of functions with one or more variables. Calculus can be divided into two parts namely differential calculus and integral calculus. Mostly Saxon’s calculus test involves in solving differential problems. The following are the test problems in math calculus solved by Saxon.

  
  

  Example problems on saxon math calculus:

  The following are the Saxon’s test problems in calculus with detailed solution.
  

  Ex:1 Solve the given function by differentiation f(u) = u ³ – 48u + 10

  Sol: The given function f(u) is the set of all real numbers. The first derivative f' (u)is given as
  f '(u) = 3 u ² - 48
  f '(u) is defined for all real numbers. Let us now solve f '(u) = 0
  3 u ² - 48 = 0
  Add 48 on both sides,
   3 u ² – 48 + 48 = 48
  3 u ² = 48
  u ² = 16
  u = 4 or u = -4
  Since u = 4 and u = -4 are in the domain of f so these both are  critical numbers.
  
  Ex:2 Solve the given function by differentiation. f(u) = | u - 8 |
  Sol:
  The domain of f(u) is the set of all real numbers. Let us use the fact sqrt (s ²) = | s | to modify function f as follows
  f(u) = sqrt (s ²) , with s = u - 8
  Using the chain rule, f '(u) is given by
  f '(u) = (1/2) 2 s s'(u) / | s |
  Since s '(u) = 1, f '(u) simplifies to
  f '(u) = (u - 8) / | u - 8 |
  f ' is undefined at u = 8 and 8 is in the domain of f. u = 8 is a critical number of function f given above.
  
  Q:3 Determine the critical number(s) of the absolute value function f given by f(x) = | x - 8 |
  Sol: The domain of f is the set of all real numbers. Let us use the fact sqrt (u ²) = | u | to modify function f as follows
  f(x) = sqrt (u ²) , with u = x - 8
  Using the chain rule, f '(x) is given by
  f '(x) = (1/2) 2 u u'(x) / | u |
  Since u '(x) = 1, f '(x) simplifies to
  f '(x) = (x - 8) / | x - 8 |
  f ' is undefined at x = 8 and 8 is in the domain of f. x = 8 is a critical number of function f given above.
  
  Practice problems on saxon math calculus:
  
  The Saxon math test problems in calculus are given below for practice.
  Q:1 Solve the given function by differentiation. f(x) = u ⁴ - 108u + 100
  Ansr:  u = 3 or u = -3
  
  Q:2 Solve the given function by differentiation  f(u) = | u - 7 |
  Ans:  f ' is undefined at u = 7 and 7 is in the domain of f. u = 7 is a critical number of function f given above. 

Math Names of Polygons

In this article we see about some names of each polygon in math, three sided polygon is triangle, some 4 sided polygons are square, rectangle, rhombus, parallelogram, trapezoid and quadrilateral. Each polygons have several properties. In math a closed geometric figure with three or more sides are also called polygon. Polygons have regular and irregular polygons shapes.

Math names of polygons:

Three sided polygons in math:
In 3-sided polygon the total angle of a triangle is 180 degrees.   
                                   
The triangle consists of many names in math:
• Right triangle
• Acute triangle
• Obtuse triangle
• Equilateral triangle
• Isosceles triangle
• Scalene triangle
Names of special polygons in math with 4 sides:
                                          
• In math 4-sided polygons, the total angle of a quadrilateral is 360 degrees.
• In math 4-sided polygon having all right angles. The total angle of a rectangle is 360 degrees.
• In math 4-sided polygon having equal length for all sides meeting at right angles. The total angle of a square is 360 degrees.

• In math 4-sided polygon have two pairs of parallel sides. The total angle of a parallelogram is 360 degrees.
• In math 4-sided polygon having all four sides of equal length. The total angle of a rhombus is 360 degrees.
• In math 4-sided polygon having accurately one pair of parallel sides. The two sides that are parallel in trapezoid are called the bases. The total angle of a trapezoid is 360 degrees.
Some 3d polygons in math are
Cube, pyramid, cone etc
                                              

Math names of other polygons with many sides:

Regular and Irregular polygons with many sides:
• Regular polygon – all angles are equal and all sides are the same length. Regular polygons have both equiangular and equilateral.
• Equiangular – all the angles are equal
• Equilateral – all the sides are same length
• Irregular polygon –A polygon whose all sides are not the same length or whose interior angles do not all have the same measure.
• Convex polygon – you can draw a straight line through a convex polygon crosses at most two sides. Every interior angle is less than 180 degree.
• Concave polygon –You can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angles is more than 180 degree.


Other polygons:
If a figure contains 5 sides, then the name of the polygon is Pentagon
If a figure contains 6 sides, then the name of the polygon is Hexagon
If a figure contains 7 sides, then the name of the polygon is Heptagon
If a figure contains 8 sides, then the name of the polygon is Octagon
If a figure contains 9 sides, then the name of the polygon is Nonagon
If a figure contains 10 sides, then the name of the polygon is Decagon
If a figure contains 12 sides, then the name of the polygon is Dodecagon

Quadrilaterals 4 Types

Quadrilaterals are four sided polygons. They are classified by their sides and angles. an important distinction between quadrilaterals is whether or not one or more pairs of sides are parallel. One of the more familiar quadrilaterals is a parallelogram. You will see that a square, a rectangle, and a rhombus are all different types of a parallelogram. The quadrilaterals 4 types basically but there are some other types that satisfy the properties of quadrilaterals. Let us see the 4 types of quadrilaterals.

quadrilaterals 4 types:

Trapezoid:
A trapezoid is a quadrilateral that has one pair of parallel sides.

                                      
Parallelogram:
A parallelogram is a quadrilateral of  two pairs of parallel sides.

                                     
Additional properties:

•  Opposite sides parallel

•  Opposite sides equal in measure

•  Opposite angles equal in measure

Rectangle:
A rectangle is a parallelogram with four right angles.

                                       
Additional properties:

•  Opposite sides parallel

• Opposite sides equal in measure

•  All angles measure 90°

•  Diagonals equal in length

Square:
A square is a rectangle with all sides equal.

                                
Additional properties:

•  Opposite sides parallel

• All sides equal in measure

•  All angles measure 90°

•  Diagonals equal in length

Rhombus:
A rhombus is a parallelogram with all sides equal.

                                        
Additional properties:

•  Opposite sides parallel

•  All sides equal in measure

•  Opposite angles equal in measure

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Isosceles Trapezoid:
An isosceles trapezoid is a quadrilateral.
                             
Additional properties:

• One pair of parallel sides

• Nonparallel sides are equal in length

quadrilaterals 4 types:

Example 1:
          Find the base of a parallelogram if its area is 512 cm² and altitude is 14 cm.
Solution:
                   Area = base × height.
                      512 = base × 14.
                         b = 512 / 14
                            = 512 cm.
                  Base = 36.5 cm.
Example 2:
Find the perimeter of square whose sides are 11 cm.
Solution:
                given the side if square is 11cm
                Perimeter of the square, P = 4a
                                                                 = 4 × 11 cm
                                                                 = 44 cm
               Hence the perimeter of square is 44 cm.

Meaning of the Triangle

Geometric figures are congruent if they have the same shape and the same size. We shall learn some properties of geometric figures that are of the same shape but not necessarily of the same size. Such figures are said to be similar. It is obvious that the congruent figures are similar but the converse is not necessarily true.
         
A triangle is a three-sided polygon. In fact, it is the polygon with the least number of sides. We write D ABC instead of writing “Triangle ABC”.

Meaning of the triangle - Properties:

We know already two important properties of a triangle,

            (i) The sum of the angles of a triangle is 180 degree
            (ii) The sum of any two sides of a triangle is greater than the third side.
Observe that one of these statements is about the angles of a triangle, while the other is about the sides of a triangle.

1. Classify the following triangles on the basis of the sides into,
Scalene triangle:
A triangle in which all the sides are of different lengths and no two sides are equal, the triangle is called a scalene triangle.


Isosceles triangle:
A triangle in which two sides are of equal lengths is called an isosceles triangle.



 Equilateral triangle
A triangle in which all the three sides are of equal lengths is called an equilateral triangle.



2. Classify the following triangle on the basis of the angles into,

Acute angled triangle
A triangle whose all angles are acute is called an acute angled triangle or simply an acute triangle
    
 Right angled triangle:
A triangle whose one of the angles is a right angle is called a right angled triangle, or simply a right triangle.
    
Obtuse angled triangle:
A triangle one of whose angles is obtuse is called an obtuse angled triangle or simply an obtuse triangle.

Meaning of the triangle - Example problems:

Meaning of the triangle problem 1:
Two angles of a triangle measure 55 degree and 85 find the measure of the third angle.



Solution:
            Let the measure of the third angle be x degree.
            We know that the sum of the angles of a triangle is 180 degree
                        55 + 85 + x = 180
                        140 + x = 180
                                    X = 180 – 140
                                        = 40


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Meaning of the triangle problem 2:
            The angles of a triangle are in the ratio 4: 2:3, Find the angles of a triangle.
Solution:
            Let the angles of the given triangle be 4x, (2x), (3x).
            The sum of the angles of a triangle is 180 degree
                        4x + 2x + 3x = 180
                                    9x = 180
                                    X = 180 / 9
                                    X = 20
            4x = 4 * 20 = 80
            2x = 2 * 20 = 40
And     3x = 3 * 20 = 60

The angles of the triangle are 80, 60, 40

Parallelogram Activity

Parallelogram is a special case of a quadrilateral where the opposite sides are parallel and equal. The sum of the angles in the quadrilateral is equal to 360 degree.We will learn about parallelograms in this lesson

Brief Explanation about the activities of Parallelogram

Meaning of the word Parallelogram:

There are 4 sides in the Parallelogram. Here the opposite sides are parallel. It means there is no intersection point between these two sides they are not met. In this Parallelogram we have two pair of sides so it is named us Parallelogram.

They are

Activity 1: The sum of angles in the Parallelogram is equal to 360 degree.

Activity 2: The opposite angles are equal in measure.

Activity 3: The sum of the adjacent angles is equal to the measure of 180 degree.

Activity 4: The opposite sides are equal in measure.

Activity 5: The height of the Parallelogram is the distance between the parallel sides.

Activity 6: The base of the Parallelogram is b and then the height of the Parallelogram is defined by h then the area of the Parallelogram is b x h square units.

Activity 7: The perimeter of the Parallelogram is the sum of the four sides of the Parallelogram. That is 2(b + h) units.

Activity 8: The two diagonals of the Parallelogram bisect at an angle 90 degree.

Example Problems

Example 1:

The angles p, 3p, p, 3p are the angles of the Parallelogram. Find the measure of p and identify the adjacent and opposite angles of the Parallelogram

Solution:

The sum of angles in the Parallelogram is 360 degree

That is p + 3p + p + 3p = 360

Then 8p = 360

Dividing 8 on both sides we have to get,

p=45 degree

Therefore 3p = 3(45) = 135 degree

Then the opposite angles are 45 degree and then the adjacent angles are 135 degree.

Example 2:

The base of the Parallelogram is 6 cm and the height of the Parallelogram is 16 cm. Calculate the value of the area of the Parallelogram.

Solution:

The area of the Parallelogram = base x height

That is 6 x 16

= 96 square cm.

Standard Form of Ellipse

Standard form of Ellipse:
             Ellipse is the two dimensional closed geometric figure formed by the intersection of the circular cone and the plane cutting through the circular cone completely. The distance between any point on the ellipse and the foci is always constant one. The addition of distance between any two fixed points is constant

standard formula

The standard form of the ellipse is
                       ` (x^2/m^2)` +`(y^2/n^2)` = 1 where m and n are greater than 0
                       Here
                                  Origin is (0, 0)
                                  The length of the major axis is 2m
                                  The length of the minor axis is 2n

Model problems for standard form of ellipse

1.The equation of the ellipse is
                         4x²+9y²=36
a.Find the x intercept and y intercept.
b.Find the points of the foci
c.Find the length of the major and minor axis
d.Draw the graph
Solution:
          Here the standard equation is
                   The standard form of the ellipse is
                       ` (x^2/m^2)` +`(y^2/n^2)` = 1 where m and n are greater than 0
                       Here
                                  Origin is (0, 0)
                                  The length of the major axis is 2m
                                  The length of the minor axis is 2n
Rewrite the equation in the standard form so, divide the entire equation by 36
              ` ((4x^2)/36)` + `((9y^2)/36)` = `(36/36)`
                        `(x^2/9)` +`(y^2/4)` = 1
                    Here m²= 9, m=3
                            n² =4,n=2             m>n>0
^{a.Find the x intercept and y intercept:}
   To find the x intercept put y=0 in the given  equation
                                 ` (x^2/9)` +0=1
                                         x²=1*9
                                         x²= 9
                                            x= + or – 3
   To find the y intercept put x=0 in the given equation
                                0 +`(y^2/4)` =1
                                            y²=1*4
                                             y²=4
                                               y= + or – 2


b.Find the point of foci:
                     We need to find the p
                                           p²= 9-4
                                           p²=5
                                              p = + or – 2.23
       The foci points are (2.23, 0) and (-2.23, 0)
c.The length of the major axis and minor axis
           The length of the major axis is 2m = 2*3= 6
           The length of the minor axis is 2n = 2*2=4
d.We need to draw the graph

                                                  

Glossary of Algebraic Symbols

Algebra is a branch of mathematics. Algebra plays an important role in our day to day life. The glossary of algebraic symbols involved in the four basic operations such as addition, subtraction, multiplication and division. The most important terms with the glossary of algebraic symbols are variables, constant, coefficients, exponents, terms and expressions. In Algebra, besides numerals we use symbols and alphabets in place of unknown numbers to make a statement. Hence, glossary of algebraic symbols may be regarded as an extension of Arithmetic.  

Glossary of algebraic symbols:

Most important terms for glossary of algebraic symbols:

Expressions

          An algebraic Expression is the combination of variables, constant, coefficients, exponents, terms which are combined by the following arithmetic operations Addition, subtraction, multiplication and division. The example of an algebraic expression is given below

                           2y + 5
Term

          Terms of the algebraic expression is concatenated to form the algebraic expression by the arithmetic operations such as addition, subtraction, multiplication and division. In the following example 3n² + 2n the terms 3n², 2n are combined to form the algebraic expression 3n² + 2n by the addition operation ( + )

Coefficient

          The coefficient of an algebraic expression is the value is present just before the terms. From the following example, 3n² + 2n the coefficient of 3n² is 3 and 2n is 2

Equations

          An algebraic equation equals the numbers or expressions. Most probably algebraic equation is used for the value of the variable. The example of the equation is given below

                          2y + 5

Examples for glossary of algebraic symbols:

Example 1:

8x - 3 = 2x

Solution:

8x - 3 = 2x

8x – 3 + 3 =2x + 3 (Add 3 on both sides)

8x =2x +3

8x – 2x =2x -2x + 3 (Add -2x on both sides)

6x = 3

6x /6 = 3 / 6  (both sides by divided 6)

X = 1/2

Example 2:

Solve the equation |-3x + 3| -8 = -6

Solution:

|-3x + 3| -8 = -6

|-3x + 3| -8 + 8 = -6 + 8 (Add 8 on both sides)

|-3x + 3| = 2

Case (i)

+ (-3x+3) = 2

-3x + 3 = 2 

Subtract 3 on both sides,

-3x + 3 -3 = 2-3

-3x=-1

x =1/3

Case (ii)

- (-3x+3) = 2

3x-3 =2

Add 3 on both sides

3x-3+3=2+3

3x=5

x=5/3

What is Absolute Value in Math

Absolute value in math is nothing but if we represent a number as absolute value the result will be positive. If the number is positive or negative the result of the absolute value is math is positive. For example |-2| = +2 and |+ 2| = +2. Here we are going to learn what absolute value in math is and operations using the absolute values in math. It is help us to understand what the absolute value in math is.

Examples for what is absolute value in math:

Example 1:
           What is the absolute value of |-5|?
Solution:
            We know in math absolute value of any value is its positive  value. Here we won’t consider the sign.
             So |-5| = +5

Example 2:
             What is the absolute value of |10|?
Solution:
            We know in math absolute value of any value is its positive value. Here we won’t consider the sign.
             So |10| = +10

Example 3:
             What is the absolute value of |-9|?
Solution:
             We know in math absolute value of any value is its positive value. Here we won’t consider the sign.
               So |-9| = +9

Operations using absolute value in math:

 Addition operation:
            Perform the following operation. |-3| + |-2|
Solution:
             Given numbers are |-3| + |-2|
              Absolute value of |-3| = +3
              Absolute value of |-2| = +2
              So |-3| + |-2| = +3 + 2 = +5

Subtraction operation:
               Perform the following operation. |-8| - |-2|
Solution:
               Given numbers are |-8| - |-2|
                Absolute value of |-8| = +8
                Absolute value of |-2| = +2
                So |-8| - |-2| = 8 – 2 = +6


Multiplication operation:
               Perform the following operation. |6| `xx` |-5|
Solution:
            Given numbers are |6| `xx` |-5|
            Absolute value of |6| = +6
            Absolute value of |-5| = +5
            So |6| `xx` |-5| = 6 `xx` 5 = 30

       These are some of the examples for absolute value in math. It is better to understand the absolute values.

Singular Matrices

The term determinant was first introduced by Gauss in 1801 while discussing quadratic forms. He used the term because the determinant determines the properties of the quadratic forms. We know that the area of a triangle by means of vertices (x1, y1) (x2, y2) and (x3, y3) is

1/2 [x1(y2-y3) + x2 (y3-y1) + (y1-y2) ]

Similarity the condition for a second degree equation in x and y to represent a pair of straight lines is

abc + 2fgh – af2 – bg2 –ch2 =0.

Definition of determinant:

To every square matrix A of order n with entries as real or complex numbers, we can associate a number called determinant of matrix A and it is denoted by | A | or det (A) or Δ.

Thus determinant formed by the elements of A is said to be the determinant of matrix A.

If A = `[[a_(11),a_(12)],[a_(21),a_(22)]]` then its | A | =

Minors:

Let | A | = |[aij]| be a determinant of order n. The minor of an arbitrary element aij is the determinant obtained by deleting the ithjth column in which the element aij stands. The minor of aij is denoted by Mij. row and

Cofactors:

The cofactor is a signed minor. The cofactor of aij is denoted by Aij and is defined as

Aij = (− 1)i + j Mij.

Singular and non-singular matrices:

A square matrix A is said to be singular if | A | = 0 (i.e) determinant zero

A square matrix A is said to be non-singular matrix, if | A | ≠ 0. (i.e) determinant is not zero


Examples of determinant zero:

Let us see some examples of determinant zero:

Example 1:

A = `[[1,2, 3],[4,5,6],[7,8,9]]`

Solution:



= 1(45 - 48) - 2(36 -42) +(32 -35)

= -3 +12-9 = 0

|A| = 0

So, A is singular  matrix.


Example 2:

Solve for x if

Solution:

`rArr` (x2 - 35) + (1-2) = 0

`rArr` x2 -35 -1 = 0

`rArr` x2 - 36 =0

`rArr` x2 = 36

`rArr` x = `+-` 6.

These are examples of determinant zero.

Isosceles Acute Triangle

 An Isosceles triangles has

• All angles add up to 180 degrees
• Has two equal sides.
• only one unequal side called the base.
• Base angles of isosceles are all equal.        

An acute triangle is a triangle which has all the angles less than 90°. We know all the internal angles in triangle is equal to 180°. In acute triangle the sum of all the internal angles is equal to 180° but not even the one angle is equal to or greater than 90°.  acute triangle drawn below  L, M and N are all acute angles.

    

Example Problems for Isosceles acute triangle:

Example 1:
   Find the area of the Isosceles triangle of b = 13 cm, h = 7 cm..
Solution:
   Given side of the Isosceles triangle, b = 13 cm, h = 7 cm.
        Area of the Isosceles triangle   = `1/2` * b * h
                                                      = `1/2 ` * 13 * 7
                                                      = 0.5 * 91
                                                      = 45.5 cm²
  Area of  Isosceles triangle   = 45.5 cm²
Example 2:
Find the area of the Isosceles triangle of b = 15 cm, h = 8 cm..
Solution:
Given side of the Isosceles triangle, b = 15 cm, h = 8 cm.
 Area of the Isosceles triangle   = `1/2` * b * h
                                                      = `1/2 ` * 15 * 8
                                                      = 0.5 * 120
                                                      = 60 cm²
  Area of  Isosceles triangle   = 60 cm²
Example 3:
Find the perimeter of Isosceles triangle that has side S₁ =16 cm,S₂ = 16,S₃ = 10 cm.
Solution:
 Given, S₁ =16 cm,S₂ = 16,S₃ = 10 cm.
  perimeter of Isosceles triangle   = S₁+S₂+S₃
                                                         = 16+16+10
  perimeter of Isosceles  triangle  =  42 cm

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Some other examples regarding Isosceles acute triangle:

Example 1:

A triangle has angle 43º, 64º and 73º. What type of triangle is this?

Solution:

  Here all the given angles are less than 90°, hence the given triangle angles are an acute triangle.

Example 2:

A triangle has angle 39º, 65º and 76º. What type of triangle is this?

Solution:

  Here all the given angles are less than 90°, hence the given triangle angles are an acute triangle.

Example 3:

A triangle has angle 45º, 56º and 79º. What type of triangle is this?

Solution:

  Here all the given angles are less than 90°, hence the given triangle angles are an acute triangle.

Associative Property of Addition

Associatively is the property of addition is nothing but in an expression containing two or more occurrences in a row of the addition operator, the order of operations that are performed will not be a matter it can be performed in any order of operation, and rearranging the brackets so that the values will not be changed. Here we are going to see about the associative property of addition

Associative property of Addition is

(a + b) + c = a + (b + c)

Examples for associative property of addition

Example problem 1 for associative property of addition:

Prove that (1 + 2) + 3 = 1+ (2 + 3) by using associative property of addition.

Solution:

Given (1 + 2) + 3 = 1+ (2 + 3) it is in the form (a + b) + c = a + (b + c)

Left hand side:   (1 + 2) + 3 = 3 + 3

= 6

Right hand side: 1 + (2 + 3) = 1 + 5

= 6

Left hand side is equal to right hand side. Therefore it is proved using associative property of addition.
Example problem 2 for associative property of addition:

Prove that (3 + 6) + 9 = 3+ (6 + 9) by using associative property of addition.

Solution:

Given (3 + 6) + 9 = 3+ (6 + 9) it is in the form (a + b) + c = a + (b + c)

Left hand side:   (3 + 6) + 9 = 9 + 9

= 18

Right hand side: 3 + (6 + 9) = 3 + 15

= 18

Left hand side is equal to right hand side. Therefore it is proved using associative property of addition.


Example problem 3 for associative property of addition:

Prove that (2 + 5) + 8 = 2+ (5 + 8) by using associative property of addition.

Solution:

Given (2 + 5) + 8 = 2+ (5 + 8) it is in the form (a + b) + c = a + (b + c)

Left hand side:   (2 + 5) + 8 = 7 + 8

= 15

Right hand side: 2 + (5 + 8) = 2 + 13

= 15

Left hand side is equal to right hand side. Therefore it is proved using associative property of addition.

Nonlinear Equation Algorithms

Nonlinear equation is the form of the multi variable equations and functions. In the equation there will be having more number of terms available. In the nonlinear equations the variables are not dependent to each other in the equation. Nonlinear equation has the more number of different orders of degrees. The graph of the nonlinear equation is not a straight line. It includes the quadratic function of equation and cubic function of equation. Here we are showing about the nonlinear equation algorithms and example problems in it .




Step by step algorithm for solving nonlinear equation:

Nonlinear equation algorithms:

Find whether the given equation is linear or nonlinear.

If the equation is equal to y = m x + b then the equation is linear equation
If the equation is not equal to y = mx + b then the given equation is nonlinear equation.

Find the equations of the order equal to any one of their variable and substitute the value in another equation.
Calculate the value for that equation and find two values.
Substitute the two values of the first equation in the second equation
Now we get each variable having two values.


Nonlinear equation algorithms - Example Problems:

Nonlinear equation algorithms - Problem 1:

Solve the nonlinear equations and find the value of x and y.

x2 - 8y = - 32

- x + y = 4

Solution:

Given equations

x2 - 8y = - 32   ------> Equation 1

- x + y = 4         ------> Equation 2

From the equation 2 rearrange and equal to y

y = 4 + x

Substitute the y value in the equation 1

x2 - 8(4 + x) = - 32

x2 - 32 -8x = -32

x2 - 8x -32+32 = 0

x2 -8x = 0

Solve the above equation by using factorization, we get

x(x - 8) = 0

Therefore the x value will be,

x=0            x - 8 = 0

x = 0 and x = 8

Substitute the x values in equation 2, we get

For x = 0,

y = 4 -x

y = 4 - 0

y = 4

For x = 8

y = 4 - 8

y = - 4

The value of y is 4, - 4


Nonlinear equation algorithms - Practice Problems:

1, Solve the nonlinear equations and find the value of x and y.

x2 - 7y = - 31

- x + y = 3

Answer:

x = 2 , 5 and y = 5, 8

2.

Find the nonlinear equations and find the value of x and y.

x = 2y - 3

y2 + 3x = - 2

Answer:

x = - 1, - 17 and y = 1, - 7

Learn Points and Lines

To learn about points and lines, A point used to represent a place in a plane with a help of pencil, a point is nothing but the dot , it has no dimension or no width, it’s only a simple black dot. In geometry co ordinates of a point which shows the particular place in a segment for representation.Line has two end points is called segment. Line segment is denoted with a connected piece of line.line segments names  has two endpoints and it is named by its endpoints.


learn about points and lines:

To learn about the geometric points and lines we have to know the classification of a points and lines.points and lines classification are as follows.
Collinear points:
When three or more points lies on the same line is said to be collinear points.

Midpoint:
A halfway point where line segment divides into two equal parts are called midpoint.

Equidistant point:
A point which is said to be equidistant in a line segment where point is equal length from other points which are in congruent then the point is equidistant point.

Parallel line segment:
Two lines which does not touch each other are called parallel lines.

Perpendicular line segment:
Two line segment  that form a L shape are called perpendicular lines.


learn problems in points and lines:

Example 1:
Find the distance between the points A(5,2) and B (7,3).


Solution:
Let assume "d" be the distance between A and B.           (x1,y1)= (5,2), (x2,y2)= (7,3).

Then d (A, B) =`sqrt((x2-x1)^2+(y2-y1)^2)`

= `sqrt((7-5)^2 +(3-2))^2)`

= `sqrt(2^2+1^2)`

= `sqrt(4+1)`

=`sqrt5`

Example 2:
Find co-ordinate of the mid point of the line segment joining given points A(-1,1) and B(3,4)

Solution:
The required mid point is
Formul a   `((x_1+x_2)/2 ,(y_1+y_2)/2)` here,  (x1, y1) = (-1,1),(x2, y2) = (3,4)

=  `((-1+3)/(2))``((1 +4)/(2)) `

= `(2/2) ` ,  ` (5/2)`

=`(1,5/2)`

Example 3:
Find the slope of the lines given (2,-1) and (1,3)

Solution:
(x1,y1)= (2,-1), (x2,y2)= (1,3).
We know to find slope of line,m=` (y_2-y_1) /(x_2-x_1)`

=`(3+1)/(1-2)`

m =`4/-1` = -4
Example 4:

Find the equation of the line having slope `1/2` and y-intercept −3.
Solution:
Applying the equation of the line is y = mx + c
Given,       m = `1/2` ,c = −3
y = `1/2` x + (−3)

or  2y = x − 6
or  x− 2y − 6 = 0.

Forms of Quadrilaterals

Forms of Quadrilateral

What is Quadrilateral?

A quadrilateral is a 2-dimensional closed shape with four straight sides.  E.g. The shape ABCD shown here is a quadrilateral.

A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral.  AC is a diagonal of quadrilateral ABCD, as is BD.

Properties of a quadrilateral:

Four sides (or edges)
Four vertices (or corners).

The interior angles add up to 360 degrees

Forms of Quadrilateral:

Parallelograms

Quadrilaterals are called parallelograms if both pairs of opposite sides are equal and parallel to each other.  Different parallelograms and their properties are described below.

Opposite sides of a parallelogram are parallel and equal in length.
Opposite angles are equal in size.

Rectangle

Opposite sides of a rectangle are parallel and equal in length.
All angles are equal to 90°



Square

A square has equal sides and every angle is a right angle (90°)
Also opposite sides are parallel.
A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

Rhombus

All sides of a rhombus are equal in length
Opposite sides are parallel.
Opposite angles of a rhombus are equal.

The diagonals of a rhombus bisect each other at right angles

Trapezium

A trapezium has one pair of opposite sides parallel

A regular trapezium has non-parallel sides equal and its base angles are equal, as shown in the diagram

Kite

Two pairs of adjacent sides of a kite are equal in length
One pair of opposite angles (the ones that are between the sides of unequal length) are equal in size.
One diagonal bisects the other.
Diagonals intersect at right angles.

Pyramid of Biomass

Pyramid of Biomass is the graphic representation of biomass of the successive trophic levels of an Eco system. Biomass refers to the total weight of dry matter present in the ecosystem at anyone time. The pyramid of biomass indicates the gradual reduction in biomass at each trophic level from base to top. The total biomass of producers is much more than the total biomass of herbivores. Like wise the total biomass of the third level will be less than the herbivores and so on. The pyramid of biomass are generally upright, but in case of parasites, the pyramid of biomass is inverted. Similarly, if organisms of lower levels are much smaller than those of higher levels, biomass pyramid may be inverted.                                                            

Examples of Pyramid of Biomass.1

Pyramid of Biomass in a Marine ecosystem

In the marine Eco system the Phytoplankton occupy the basic trophic level and the next trophic level is occupied by the Zoo plankton, the third and fourth are occupied by carnivores such as worms, and molluscs and carnivores fishes respectively. Here too the biomass falls in a graded manner from the basic trophic level to higher trophic level.

Examples of Pyramid of Biomass.2

Pyramid of Biomass in  a  grass land 

In a grass land the biomass of grasses is the maximum and it gradually decreases towards the consumer level. That is  the biomass is decrease in this order. Grasses have higher biomass than that of snake  and snake have higher biomass than that of Hawk.

                                                 Grass ------------> Snake -------------> Hawk

How to Rewrite Radicals

Radicals are nothing but a root which we also called square root. Square root is indicated with the symbol (sqrt) and sqrt (). Normally, radicals are rewrite like, sqrt of (a) is rewrite as a ^(1/2). Likewise radicals are expressed in various forms. The expression sqrt (8) is read as “radical eight”, or “the square root of eight”. Thus, we are going to see how to rewrite radicals in different ways.

Formula for how to rewrite radicals:

Formula for how to rewrite radicals:
General expression with exponent and radical:
                                          `( ^nsqrt (a) ) ^m`   = `( ^nsqrt (a) ) ^m`   = `(a1/n) ^m ` = `am/n`                         
Multiplication property for radical expression:  (  `^nsqrt (ab)`   ) = ( ` ^nsqrt (a)` )  ( ` ^nsqrt (b)` )
Division property for radical expression:  ( `^nsqrt (a/b)` ) = (  `^nsqrt (a)` ) / (  `^nsqrt (b)`   )
Different forms of radicals is,
                    A square (second) root is written as (  ` sqrt(x)` )

     A cube (third) root is written as (  `^3sqrt(x)`  ),

     A fourth root is written as (  `^4sqrt(x)`   ),

     A fifth root is written as:  (  ` ^5sqrt(x)` ).

Example for how to rewrite radicals:

Example for how to rewrite radicals: Rewrite radical(`sqrt (1225)` )
Given: (`sqrt (1225)` )
Solution: Given question says, radical (1225),
      When, we take radical for 1225, we obtain 25*49.
Because,                 (  `^nsqrt (ab)`   ) = ( ` ^nsqrt (a)` )  ( ` ^nsqrt (b)` )
                                       `sqrt (1225)` = `sqrt (25)` `sqrt (49)`
                                                        = `sqrt(5)`  * `sqrt(7)` 
                                       `sqrt (1225)` = `35`
Thus, we can do how to rewrite radicals in the prefered way.

Example for how to rewrite radicals: Rewrite radical( `^3sqrt (512)` )
Given: (  `^3sqrt (512)` )
Solution: Given question says, cubic root of (512),
      When, we take cubic root for 512, we obtain 8.
Because,                           `8*8*8 = 512`
                                          `^3sqrt(512)`` =` `8^3`
  Therefore, cubic root for (512) = 8^3
Thus,we can do how to rewrite radicals in the prefered way

Example for how to rewrite radicals: Rewrite radical `^3sqrt (729)`
Given: (`^3sqrt (729)` )
Solution: Given question says, cubic root of (729),
      When, we take cubic root for 729, we obtain 9.
   Because,                         `9*9*9 = 729`
                                         `^3sqrt(729) = 9^3`
     Therefore, cubic root of (729) = 9^3


Example for how to rewrite radicals: Rewrite radical `^4sqrt (1296)`
Given: (`^4sqrt (1296)` )
Solution: Given question says,  fourth root of (1296),
      When, we take fourth radical for 1296, we obtain 6.
Because,                             `6*6*6*6 = 1296`
                                             `^4sqrt(1296 )= 6^4`
     Therefore,   fourth root of (1296) = 6^4

Example for how to rewrite radicals: Rewrite radical `^5sqrt (3125)`
 Given: (`^5sqrt (3125)` )
Solution: Given question says, fifth root of (3125),
      When, we take Fifth radical for 3125, we obtain 8.
Because,                       `5*5*5*5*5 = 3125`
                                             `^5sqrt(3125) = 5^5`
     Therefore,    fifth root of (3125) = 5

           

Math Scientific Expressions:

In this article we shall discuss about math scientific expressions. Scientific notation expressions are meant by transform very big numbers into simplest form of a number. That is called as scientific notation expressions to integer. Scientific notation expressions is nothing but exponential of the base integer 10 have expressed in the number scientific notation expressions. Example for scientific notation expressions is 785000000 this can be note down as 7.85 x 10⁸. Here, 8 is the power of 10.   

Example problems based on math scientific expressions:

The example problems based on math scientific expressions to integer is given below that,

• Example 1:

How do you write 67000 in math scientific expressions?
Solution:
Step 1:

The given number is 67000.

Step 2:

The given number is not a big number, but it is simple to convert scientific notation.

First rounding 6.7 x 10⁴ = 67000 [10⁴ = 10000]  

Step 3:

The concluding answer for math scientific expressions is 6.7 x 10⁴.

• Example 2:  

How do you write 2800000 in math scientific expressions?
Solution:
Step 1:

The given number is 9100000.

Step 2:

The given number is a big number, but it is simple to convert scientific notation.

First rounding 9.1 this is not equivalent to the given number.

Step 3:

Here, 6 digits subsequent the decimal point.

So, 9.1 × (1000000)

Therefore the rounding will be 10⁶.

Step 4:

The concluding answer for math scientific expressions is 9.1 x 10⁶

• Example 3:

How do you write 843000 in math scientific expressions?
Solution:
Step 1:

The given number is 843000.

Step 2:

The given number is a big number, but it is simple to convert scientific notation.

First rounding 8.43 this is not equivalent to the given number.

Step 3:

Here, 5 digits following the number are there so we can write 10⁵.

Step 4:

The concluding answer for math scientific expressions is 8.43 x 10⁵.

.

Practice problems based on math scientific expressions:

The practice problems based on math scientific expressions is given below that,

• Problem 1:

How do you write 9510000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 9.51 x 10⁶.

• Problem 2:

How do you write 852000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 8.52 x 10⁵.

• Problem 3:

How do you write 75000 in math scientific expressions?
Answer: The concluding answer for math scientific expressions is 7.5 x 10⁴.

Just Math Answer

Some of the main branches of mathematics are algebra, geometry and calculus. Various number of problems exists in these branches.
In this article of just math answer, several problems related to various branches of mathematics are given with their respective answer. In addition, several practice problems with answer key are given for better understanding.


Math Example problems with answers:

Example 1:

Find the Mean of the given data set: { 2, 6, 19, 29 }

Solution:

Mean   =  ( 2 + 6 + 19 + 29) / 4

=  56 / 4

= 14

Answer: Mean  = 14

Example 2:

Find the volume of cylinder given the radius is 13 cm and 32 cm.

Solution:

Volume of cylinder = `pi` r2 h cubic units.

= (3.14) * 132 * 32

=  3.14 * 169 * 32

= 16981.12 cm3

Answer: Volume of cylinder = 16981.12 cm3

Example 3:

Find the area of the square with the side length of 9 cm.

Solution:

Area of square  =  ( a ) 2

= ( 9 )2

= 9 * 9

= 81 cm2

Answer: Area of square =  81 cm2

Example 4:

Find the area of a triangle through base of 15 m and a height of 8 m.

Solution:

Area of a triangle = ½ b h

= ½ ( 15 ) ( 8 )

= 0.5 * 15 * 8

= 60 m2

Answer: Area of triangle = 60 m2

Example 5:

Solve :  6 ( x + 3 ) = 72

Solution:

6( x + 6)  =  72

6x + ( 6 x 6 )  =  72

6x + 36  =  72

6x  =  36

x  =  6

Answer:  x =  6

Example 6:

Find lateral surface area of cone with radius 5 cm and slant height 16 cm.

Solution:

Given:  Radius r = 5 cm

Slant height l =16 cm

Lateral surface area  = `pi` r l   square units.

= 3.14 x 5 x 16

= 251.2 cm2

Answer: Lateral surface area = 251.2 cm2



Example 7:

Find the range of the data set { 25, 42, 36, 21, 48 }

Solution:

Here, The maximum value is 48

The minimum value is 21

Range  =  Maximum value – Minimum value

=  48 – 21

= 27

Answer: Range  = 27


Math Practice problems with answer:

1) Find the range of the data set { 15, 43, 29, 21, 14 }

Answer: 29

2) Find the area of the square with the side length of 11 cm.

Answer: 121 cm2

3) Find the Mean of the given data set: { 8, 16, 20, 26 }

Answer: 17.5

4) Find lateral surface area of cone with radius 8 cm and slant height 18 cm.

Answer: 452.16 cm2

Practice Divide Monomials

A polynomial is an algebraic expression with literal. Mostly we use x, y literal in polynomials. Literals are also called as variables. In polynomial, the variables have only positive integral exponents.

For example,  15 + 2x + x2 , 7x3 + 5xy2 + 12 y3 .

The first example is polynomials in one variable x.

The second example is polynomials in two variables x and y.

Polynomials having only one term are known as monomials. Monomial is one of the type of polynomials.  For example,

x3 , x2  x , y5, 5x5, 6y3 .

Let us practice division of a monomial problems.



Divide monomial problems for Practice:

There are two rules for dividing a monomial by a monomial. They are following,

Rule 1: The coefficient of the quotient of two monomials is equal to the quotient of the coefficients of the monomials in question.

Rule 2: In the quotient of two monomials, the variable part is equal to the quotient of the variable parts in the monomials in question.

Problem 1:

Divide by x5 by x3 monomials.

Solution:

x5 ÷ x3 = `x^5/x^3 `

= x2 .

Problem 2:

Divide by 15x5 by 5x4 monomials.

Solution:

15x5 ÷ 5x4 = `(15x^5)/(5x^4) `

= 5x .

Problem 3:

Divide by -20x4 by 10x monomials.

Solution:

- 20x4 ÷ 10x = `(-20x^4)/(10x) `

= `(-20/10)`` (x^4/x)`

= - 2 x3 .

Problem 4:

Divide by 3y3 by `sqrt(3)` y monomials.

Solution:

3y3 ÷ `sqrt(3)` y = `(3y^3)/(sqrt(3)y) `

= `(3/sqrt(3))`` (y^3/y)`

= `sqrt(3)` y2.

Problem 5:

Divide by 2x2 by 2x monomials.

Solution:

2x2 ÷ 2x      = `(2x^2)/(2x) `

= `(2/2)`` (x^2/x)`

=  x .

Problem 6:

Divide by -3x3 by x2 monomials.

Solution:

- 3x3 ÷ x2 = `(-3x^3)/(x^2) `

= `(-3/1)`` (x^3/x^2)`

= - 3 x .

Practice Problems on monomials:

Practice Problem 1:

Divide by `2/3` x2 by x monomials.

Answer:

`2/3` x .

Practice Problem 2:

Divide by `sqrt(5)` x4 by 5x3 monomials.

Answer:

`x/sqrt(5)` .

Practice Problem 3:

Divide by `sqrt(3)` a3 by 2a monomials.

Answer:

`(sqrt(3)a^2)/2` .

Practice Problem 4:

Divide by 4a4 by - 2 `sqrt(2)` a2 monomials.

Answer:

`-sqrt(2)a^2` .

Multipling and Dividing

The result of multiplying is the total number (product) that would be obtained by combining several (multipling) groups of similar size (multiplicand). Division is clear as a mathematics connotation which is the opposite procedure of multiplication. From the method of division the amount or relation of 2 numbers can be calculate. Sign of multiplying is ‘*’.The same result can be obtain by recurring multiplying. Otherwise the process of find how many period of one number is including in an additional one. Sign of division is ‘/’ or ‘÷’.If we are combining 6 groups with 3 objects in each group, we could arrive at the same answer by addition. For example, 6+6+6=18 is equivalent to the multiplication equation 6*3=18.

Descriptions about multipling and dividing:-

Steps for multiplying and dividing problems,
Step1:-
Multiplying and division of two digits by the similar signs will be positive sign
• (+ sign) ÷ (+ sign) = (+ sign).
• (- sign) ÷ (- sign) = (- sign).
• (+ sign) * (+ sign) = (+ sign).
• (- sign) * (- sign) = (+ sign).
Step2:-
Multiplying and division of two digits by the dissimilar signs will be negative
• (+ sign) ÷ (- sign) = (- sign).
• (- sign) ÷ (+ sign) = (- sign).
• (+ sign) * (- sign) = (- sign).
• (- sign) * (+ sign) = (- sign).

Example problems for multiplying and dividing:-

Problem1:-
Solve 24 dividing by 8
Solution:-
24 ÷ 8
= 3.

Problem2:-
Solve (-12) dividing by (-2)
Solution:-
(-12) ÷ (-2)
= 6

Problem3:-
Solve 4 dividing by (-2)
Solution:-
4 ÷ (-2)
= -2

Problem4:-
Solve (-9) dividing by 9
Solution:-
(-9) ÷ 9 = -1

Problem5:-
Solve 12 multipling by 3
Solution:-
12*3
=36.

Problem6:-
Solve (-10) multiplying by (-2)
Solution:-
(-10)*(-2)
=20.



Problem7:-
Solve 14 multiplying by (-4)
Solution:-
(14)*(-4)
=-56.

Problem8:-
Solve (-12) multiplying by 3
Solution:-
(-12)*(3)
=-36.