Radical Equations in Grade XII

Hello friends, Previously we have discussed about probability worksheets and in this session we will learn about the radical equations and inequalities problems in grade XII of tamilnadu stateboard syllabus. Radical equations are the type of equations having square root symbol and some of the variables with them. So such equations are the equation of roots. We can perform addition, subtraction, multiplication, and division on such type of equations. To deal with the radical equations we have to perform squaring of the radical expressions and after that we have to solve it in the proper manner as we solve other types of algebraic expressions. For example (√54 + 2x) is a radical equation.
In the radical equations there are the concepts of the nth roots of the expressions in them. The variables are enclosed with root symbol of nth degree where this n could be 1, 2, 3, …, n. The variable which comes in the radical part is called as radicand of the expression. For example in the expression √(5x + 8), (5x + 8) is the radicand part. Radical inequalities are also the part of radical equations. In the similar way radical inequalities are those inequalities which also involve the radical symbol or they are in the square root symbol. Some of the radical equations can be given as:
√(2x + 6), √(7xy) +5, √(x² y²) + x*y, √(7xy > 21) = (x y + 21), √(4x + 4) + 6 <= 12, etc…
Now talking about inequalities, in the grade XII, the inequality level is too complicated. The inequalities are the type of algebraic equations which has some inequality sign with it, and radical inequality is the expressions that have radical symbol with it. In inequality, some facts are used which state that
If x < y that have the meaning that number x is less than that of number y.
If x > y that have the meaning that number a is greater than that of number y.
if a<=x>=b, then x is greater or equals to a and b is less or equals to x.
If x ≠ y have the meaning that x is not equal to the number y. We can perform several operations in the inequalities like addition, subtraction, division, and multiplication as like other algebraic equations.
When we solve the inequality problems in general way, we use some of the standards like after multiplying or dividing them we flip the sign of the inequality. We use the number lines to solve the inequality problems. Here we also learn about the radical inequalities. The examples of the inequalities can be given as:
- (x / 2) < 8, 4 x + 8 < 13, √(4x + 4) + 6 <= 12,  √(x + 4) + 12 = 6, etc.
To solve such type of inequality problems we have to use the pattern of solving for normal radical and inequality.

So this is all about radical equation. I have given a brief Idea of how to deal with radical equation in XII grade and if you want to study about other topics like quartiles and Triangle congruence relationships visit various websites on the internet.

Learn factoring of Two Degree Polynomials

Hello friends today we are going to discuss about factoring of two degree and three degree polynomials which you need to study in grade XII of gujarat secondary education board. In earlier class you must have gone through about the basics of polynomials and For more practice you can use properties worksheets . We are here to study about 2^nd degree and 3^rd degree polynomial. Now let us discuss about the 2^nd degree polynomial.
2^nd degree polynomial:
                                    The degree of polynomial represent the highest degree on non zero term which contain variable.
2^nd degree polynomial:
                                    It means its variable has highest power of 2.
                                   Example -2X²+2y²+9=0
We can do fallowing operations on 2^nd degree polynomial.
Addition
Subtraction
Multiplication To know more about Two Degree Polynomials click here,
Now let us discuss Addition of polynomial with the help of fallowing examples
Example: 2X²+2Y²+2Z² + 3X²-Y²-Z²
                       For solving this type of problem we just need to use the property of associative and distributive law which you have studied in junior classes.
(2X²+2Y²+2Z²)+( 3X²-Y²-Z²)
2X² +3X² +2Y²-Y² +2Z²-Z²
 (2X² +3X²)+ (2Y²-Y²)+(2Z²-Z²)
 5X²+Y²+Z²
  Same things we can apply for subtraction but for more clearance we see one example
Example: Subtract 2X²+4Y²+6Z² from 4X²-6Y²-4Z²
(4X²-6Y²-4Z²)-( 2X²+4Y²+6Z² )
4X²-2X²-6Y^2-4Y²-4Z^2-6Z²
2X²-10Y²-10Z²
 After addition and subtraction we move to multiplication by this process we can change the degree of polynomial. How to multiply we see by example given blow
2X²+2Y²+2Z² multiply by 3X²-Y²-Z²
(2X²+2Y²+2Z²)* (3X²-Y²-Z²)
(2X^2*3X²)+( 2Y² *-Y²)+( 2Z² *-Z²)
6X⁴-2Y⁴-2Z⁴
   As you have seen in the above example that degree of polynomial can be changed by the multiplication in the above problem firstly the degree of polynomial was 2 now it is changed to 4
   Division can be also done by the polynomial that you will be going to study in further class. Now I think we have well understood about 2^nd degree polynomial now let’s move to 3^rd degree of polynomial as it is more important for you .Now you came to know that if polynomial is of  3^rd degree then its highest power will be 3.Like 2^nd degree we can implement same operation on.
Factoring 3^rd degree polynomials:
All algebraic expressions one or more than variables that are called polynomial .it terms are not in negative exponent.
Polynomial in algebraic expression P(x) =a0+a1x+a2x2……..an-1 xn-1+anxn,   where a0, a1,…an.and are real number and n is non negative integer .and in polynomial x real degree of n.
A polynomial of degree is in three term like ax3+bx2+cx+d, a? 0 that is also called cubic polynomial.
Some step for third degree polynomials

1.      In numerical ,plug x=1,-1,-2,2 etc in P(x)

2.      If p (1) =0 then x-1 is factor of p(x) polynomial .

3.      If p(x) is a cubic ,then it divide it by x-1

4.      Now factorize quadratic polynomial.

Now lets the example of three degree

Ex: x3+7x2 +12x

Sol: let P(x) =x3+7x2 +12x

Let’s solve in simple form.

P(x) =x(x2+7x+12)

P(x) =x(x2+4x+3x+12)

P(x) =x(x(x+4) +3(x+4))

P(x) =x(x+4) (x+3)

This value is factor of this polynomial mean x=0, x=-4, x=-3
And this value is satisfying this polynomial...
: Let’s check my answer.
First you plug the value in given polynomial.
P(x) =x3+7x2+12x
Put x= -4
P (-4) = (-4)3+ 7(-4)2+ 12(-4)
P (-4) = -64+ 122 - 48
P (-4) =0
Here -4 values satisfy this polynomial. It means -4 is factor of this polemical.
In this way you can check your all value by this process.

In this section only these parts are important and You can also refer grade XI  blog for further reading on Polynomials in Grade XI.Read more maths topics of different grades such as Angles of triangles and polygons in the next session here.

Linear Equations and Inequalities

Hello friends today we are going to discuss about Linear equations, inequalities which you need to study in grade XII of every education board .As you all know about  solving linear equations and you all studied them in grade X and XI. Linear equation is a equation which have 1,2 ,3 or more than that unknown variables with some known variable.

Linear equation in one variable: This type of equation has only one solution or you can say unique solution.
Example: 3X+5=14
3X=9 or X=3
These equations are very simple to solve you just need to put x one side and other values on other side and u can easily get the desired result.
Linear equation with two variable: any equation is in the form of aX+bY+c=0 is called linear equation with two variable .where a,b,c are real numbers and X,Y are variables .For solving this type of  equations we need to have 2 equations because we are having two unknown variable in the form of  X,Y.
Example: 2X+3Y=15&3X+3Y=18
Now solving this type of equation we need to we need to add or subtract both the equation if sign are same then subtract them and if not same than just add them.
Now in above case we need to subtract on subtraction we will get X=3 and then put this value in above equation to find the value of y .than you will find the value of y is 3.
Note-in above example sign and number before variable are same so the problem solved easily but
if the number before the variable are not same than u need to multiply them by a constant. This constant you have to decide by your own.
Now we will discuss about the inequalities
Inequalities are basically linear equation type problem just the change is that you can change the sign according to inequalities but you can’t change sign in linear equation that is the basic difference between them. For solving inequalities you need to fallow fallowing steps
*Add or subtract both the number same side.
*On switching change the sign of inequalities.
*Multiply and Divide by same positive number
*if you Divide by negative number or multiply by positive number then you need to change the sign of inequalities. It will be more clear when we go through the examples.Know more about inequality here,

Example: Solve this inequality
                       5X+4>-10X+14
Step one  add or subtract same value
5X+4-4>-10X+19-4
5X>-10X+15
5X+10X>15
15X>15
X>1
In this way just fallowing simple steps we can solve inequalities problem easily.
Example: solve this inequality
4X<16
Divide by 4 each side you can easily solve this problem
X<4
Now the example given blow is taken from compound inequality .These are complicated problems .Let us see one  problem .
Example:3X+9>18 and 4X-7<-11
Now for that type of problem you just need to take both the problems one by one and them just combine them.
3X>18-9
3X>9
X>9
And
4X<-11+7
4X<4
X<1
Now answer of the problem will be the combination of both that will be X>9,X<1 by this you can say all the number above 9 and blow 1 are the solution of this equation.
We have discussed about linear equation and inequalities,Inequalities are little difficult to solve  so you need to practice a lot and to know answers of interesting questions like can a rational number be negative wait till next session and You can also refer grade XII  blog for further reading on Formal/informal proofs.Read more maths topics of different grades such as Correlation and causation in the next session here.

Limits at Infinity in Grade XII

Hello friends I think you have understood the last articles today we are going to study about limits and infinity which you need to study in calculus in grade XII.As you are now familiar with functions, range and domain they are very important to understand the concept of math because these are the basic requirements for studying limits. Now let’s see how to solve limits?
Limit of any function can be defined as the behavior of the function near a particular input. Let’s suppose a function f has the output f(x) to every input x and the function has limit L and an input p. If f(x) is close to L when f(x) is close to p. In simple words we can say as f(x) moves closer to L x moves closer to p. and if we talk about infinity we know that it is impossible to reach there but we still try to reach it. For more on limits visit this.
Limits at infinity is used to describe the behavior of the function with respect to its limit and also describes its behavior as the independent variable increase or decrease without any bound. In actual the value we get from the limit is not the exact value but the value we get is very close to real value or we can say tends to that value.
Limit has a wide application in the world of mathematics. (Also see Upper Limits) It can be applied to simplify the function and also used to find the value of the function. General notation of limit is given below

By the above statement we can say that as x approaches to infinity 1/x approaches to 0. Whenever you see limits you just think of approaching to some value. In mathematical language we can say we are not talking about x tending to infinity. We just know that as x gets bigger, value of the function approaches to 0
Let’s see one example of limit approaching to infinity
y=3x
As  x=1  y=3
x=2 y=6
x=3 y=9
x=4 y=12
and so on
x=100  y=3000
Now as x approaches to infinity y approaches to infinity as well. Now we will see one more example
2x² -5x
2x² will always tend towards infinity and -5x always tends towards minus infinity so if x will increase where will the function tends?
It will always depend on the value of if x² will grow more rapidly with respect to x as x increases then the function will surely tend towards the positive infinity.
Now let’s talk about the degree of the function. The degree of the function can be defined as the highest power of variable for example
5x²+6x+7
In this x has highest power 2 so degree of the function will be 2.
By looking at degree of the function we can tell that limit will be positive or negative.
If degree of the function is greater than 0 then limit will always be positive.
If degree of the function is less than 0 then the limit will be 0
This is all about limits at infinity. I hope you have understood all the things in limit at infinity

In upcoming posts we will discuss about Linear Equations and Inequalities and standard deviation of normally distributed random variable. Visit our website for information on 12th biology syllabus Maharashtra board

Patterns and Functions in Grade XII

Hello friends as I believe you have understood the previous topics. Today we are going to discuss about Patterns, relations, and functions which you have to face in grade XII. As you have heard these terms in your previous classes but today we will discuss them in detail. Firstly we will learn about pattern.

You all are familiar with the term algebra pattern, here we are to discuss about the number pattern especially in series. It will be well understood by an example.

Find the value of X in given series

20, 42 ,66, 92, X, 150, 182

For this type of series you just need to check the pattern and if you found the pattern then it is very easy to solve. Now for above problem

42 is greater than 20 by 22

66 is greater than 42 by 24

92 is greater than 66 by 26

Now you can check that the difference between the number is also in a pattern 22,24,26 and so on so next number in difference series will be 28 and X will be 92+28=120, and you solved the problem so you just need to find pattern then it is very easy. Now move to another example and find the advance pattern.

A group of lecturer were at a conference meeting. Every lecturer exchanges his class record with each additional lecturer who were there. If there were 11 lecturers how many class record were exchanged?

To solve this type of problem we just need to focus on number of class note exchanged so class notes will be exchanged=10+9+8+7+6+5+4+3+2+1=55

Number of class notes exchanged=55*2=110

This problem is little hard to understand but I don’t think it’s tough you just need to practice. Now let's move to function and relation.

If we denote relation by R. If R is the relation from set (C,D) and (C,D) belongs to R then we can say C is related to D under the relation of R. There are many types of relations which are as follows

identity Relations

The relation IB’on a set B is called identity relation on B, if IB = (x, y) | x = y

So it will be in identity only when x=y

Universal Relation

If a relation R is A*A than R in universal relation with A

Void Relation

As we know that null set is the sub set of every set so we can say that null set has a void relation with every set.

Reflexive relation

A relation R on a set B is called reflexive if (n, n) belongs to n in R for all n belongs to B

Symmetric relation

if any set (x,y) belong to R than (y,x ) will also belong to R

Transitive relation

If in a group two parameters are common and they are in a relation R then the other 2 parameters are in transtive relation

Now we will talk about functions

There are mainly 3 types of function

1. Injective Function

If we find the value of function at two parameters and value comes the same then it is called injective function

2. Surjective function

A function is said to be surjective only if its range lies within the domain

3. Bijective Function

A function is said to bijective only if it has the properties of injective and surjective. Also see Average Value of a Function.

This is all about patterns, relations, and functions and I hope that this article will help you in understanding this topic.

In upcoming posts we will discuss about Limits at Infinity in Grade XII and Line of best fit - least squares regression. Visit our website for information on 12th biology Maharashtra board syllabus