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. 

Frequency Distribution Data

A Frequency Distribution shows us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. It is a way of showing unorganized data e.g. to show results of an election, income of people for a certain region, sales of a product within a certain period, student loan amounts of graduates, etc. Some of the graphs that can be used with frequency distributions are histograms, line graphs, bar charts and pie charts. Frequency distributions are used for both qualitative and quantitative data.

(Source: Wikipedia)

Types of Frequency Distribution data:

Different types of frequency distribution data are,

• Univariate frequency tables
• Joint frequency distribution

Univariate distribution tables:

                It is a list of values that can be ordered by the quantity. It can show the values for each value appear for number of times.

Joint frequency distribution:

                 It is used as two-way tables. It is also called as bivariate joint frequency distribution.

Example problem for Univariate frequency distribution data:

Example 1:
Construct the univariate frequency distribution table for the given data. For the following students in a class have marks in the exam.
Students scored marks in between 31-40 are 5
                                                        41-50 are 12
                                                        51-60 are 9
                                                        61-70 are 15
                                                        71-80 are 7
                                                        81-90 are 4
                                                        91-100 are 2
Solution:
         Determine the range:
                      100 – 31 = 70
         Determine the intervals:
                   Choose the interval as 10
         Construct the univariate frequency distribution table.

Marks	No of students	Cumulative frequency
31-40	5	5
41-50	12	17
51-60	9	26
61-70	15	41
71-80	7	48
81-90	4	52
91-100	2	54

Joint frequency distribution example problem:

Example 2:
In a school, boys and girls are participated in different sports competition that can be given. Using that set of values construct the joint frequency distribution.
Boys and girls are participated in running, long jump, and volley ball.
In running – 10 boys and 8 girls
In long jump – 8 boys and 7 girls
In volley ball –12 boys and 12 girls are participated.
Solution:
          Construct a joint frequency distribution table for the given set of data.

Joint frequency	Boys	Girls	Total
Running	10	8	18
Long jump	8	7	15
Volley ball	12	12	24
Total	30	27	57

Second Derivative \Help

In calculus (a branch of mathematics) the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity. The process of finding a derivative is called differentiation. Second derivative is higher derivatives of the function. (Source: Wikipedia) 

Example problems for second derivative help

Second derivative help example problem 1:
      Find the second derivative of the given function f (x) = 5x⁴ + 24x³ + 6.
Solution:
 Given function is f (x) = 5x⁴ + 24x³ + 6
Differentiate the given function with respect to x, we get
                           f'(x) = 20x³ + 72x²
Again differentiate the above function with respect to x, we get
                           f''(x) = 60x² + 144x
Answer:
  Second derivative value is f''(x) = 60x² + 144x
Second derivative help example problem 2:
      Find the second derivative of the given function f (x) = 12x² + 3x + 7.
Solution:
 Given function is f (x) = 12x² + 3x + 7
Differentiate the given function with respect to x, we get
                          f'(x) = 24x + 3x
Again differentiate the above function with respect to x, we get
                         f''(x) = 24
Answer:
  Second derivative value is f''(x) = 24
Second derivative help example problem 3:
      Find the second derivative of the given function f (x) = 6x³ + 5x²
Solution:
 Given function is f (x) = 6x³ + 5x²
Differentiate the given function with respect to x, we get
                          f'(x) = 18x² + 10x
Again differentiate the above function with respect to x, we get
                          f''(x) = 36x + 10
Answer:
  Second derivative value is f''(x) = 36x + 10

Practice problems for second derivative help

Second derivative help practice problem 1:
      Find the second derivative of the given function f (x) = 4x⁴ + 14x³
Answer:
  Second derivative value is f''(x) = 48x² + 84x
Second derivative help practice problem 2:
      Find the second derivative of the given function f (x) = 2x² - 56x
Answer:
  Second derivative value is f''(x) = 4
Second derivative help practice problem 3:
      Find the second derivative of the given function f (x) = x³ - 83x²
Answer:
  Second derivative value is f''(x) = 6x - 166 

Probability Compound Events

An event is a one or more possible outcomes from an experiment. An event consisting of one or more simple events is called compound event. An event is called independent event if one event does not affect the other event. An event is called dependent event if one event does affect the other event. Two events cannot happen at the same time, they are called mutually exclusive events.


Probability Compound Events – Formulas

Probability of two independent events:

P(A and B) = P(A) · P(B)

Probability of two dependent events:

P(A and B) = P(A) · P(B following A)

Probability of two mutually exclusive events:

P(A or B) = P(A) + P(B)

Probability Compound Events - Examples

Example 1: Find the probability of rolling a dice and getting a 5 on each one.

Solution:

These events are independent.

P(5) · P(5) = (1/6) · (1/6) or (1/36)

Therefore, probability is 1/36.

Example 2: A box contains pen, pencil, and eraser. Find the probability of choosing first pen and then, without replacing the pen, choosing a pencil.

Solution:

These events are dependent.

The first probability is (1/3).

The probability of choosing a pencil is (1/2) since there are now only pen and eraser are left.

The probability of both is

(1/3) · (1/2) = (1/6)

Example 3: What is the probability of choosing a queen and seven with replacement?

Solution:

P(Queen) = 4/52

P(Seven) = 4/52

P(Queen and Seven) = P(Q) · P(S)

= 4/52 · 4/52

= 16/2704

=1/169

Example 4: A box contains 5 rose, 7 brown, 4 black and 8 white marbles. A marble is chosen at random from the box. After replacing it, a second marble is chosen. What is the probability of choosing a brown and a white marble?

Solution:

P(Brown) = 7/24

P(White) = 8/24 = 1/3

P(Brown and White) = P(Brown) · P(White)

= 7/24 · 1/3 = 7/72



Probability Compound Events - Practice problems

Problem 1: What is the probability of choosing a seven and king of diamond with replacement?

Answer: 1/676

Problem 2: A coin is tossed and a 6-sided die is rolled. Find the probability of tail and 5.

Answer: 1/12