Exponents in on Grade XII

In previous article, one of the important topics of mathematics that is Quadratic functions was explored and in today's article we are going to move further in grade XII to learn Algebra online syllabus with Exponents and exponential functions. This topic has very short introduction of itself but various situations and applications of it, tends it to cover a wide portion of Algebra unit.
Let us first start with the general description of exponent after that we will move to the main topic of Exponential functions with their growth and decay property. Before coming into 12 standard students learn almost all the aspects of logarithm and in respect of those exponents as well. Whenever an exponent is described it is clear that presence of Euler constant 'e' is surely there in the function.  The main reason to use exponent in any function is that the corresponding constant change in any variable in parallel with other independent variable is measured with the help of graph. For more on exponents visit here.
If any function is described as f(x) = y^p then 'y' is called base and 'p' is its relative exponent. There are some pre-defined properties for exponential function as follows:
Suppose two integer variables are m and n, then
   • yⁿ = y × y × · · · × y with n factors
   • y^1/n means the nth root of a. That is, a1/n is that positive number which satisfies
                                 (y^1/n ) × (y^1/n )) × · · · × (y^1/n ) = y
       Where there are n factors on the left hand side.
   • y^m/n = (y^1/n ) × (y^1/n ) × · · · × (y1/n ) (y^1/n )
   • y⁻ⁿ = 1/ yⁿ
    
The above situations are used when x is an integer but if ‘x’ is a real number then:
Suppose ‘p’ is a rational number
                                
                               p=m/n;          here m, n are integers
                               
Now assume ‘x’ as a real number that cannot be written in a form of rational number. Two common examples of this type or irrational numbers are x = √2 and x = π.
For other possible real numbers that can be easily converted into fraction of rational form use decimal to fraction conversion for example:
                                     x = 3.914712334317 . . .
Then, if we are working to 3 decimal points we would write
                                            x ≈ 3.915
and this number can certainly be expressed as a rational number:                                                
                                       x ≈ 3.915 = 3915/1000                                                  
So, in this case                                                          
                                 y^x = y^3.914712... ≈ y^3.915 = y ^3915/1000
                 
and the final term: y ^3915/1000 can be determined in the usual way by using your calculator.
This is all about Exponents; now let us move towards the next part of this session that includes the description of Exponential functions.
Exponential Functions: - suppose a function is as f(x) = y^x
Now for this function the value of ‘x’ clearly varies according to a fixed base value and its appropriate exponent value.
So the functions (y^x) (as different values are chosen for a) are called exponential functions. Let us explain in more practical manner:
                 If y>z> 0 then    y^x > z^x       only if x>0    
And if x<0 then         y^x < z^x                   
Euler constant ’e’ is the most important and widely used exponential function whose particular base value is 2.7182818 . . . ,
                                          e = 2.7182818 . . .
The number ‘e’ has its own importance in mathematics like number π and, like π it is also an irrational. So ‘e’ also cannot be written as the quotient of two integers. The value of ‘e’ is stored in most of the science calculators to sort out exponential function.
There are various ways of calculating the value of ‘e’. For example, ‘e’ represents the end point of sequence of numbers as follows:
(2/1)¹ . (3/2)² . (4/3)³ . ......  (18/17)¹⁷.....  ..  (65/64)⁶⁴.....
Decimal form of these ( up to 6 decimal points) is as:          2.000000,   2.250000,    2.370370,    ...,   2.637929,     . . . , 2.697345,    ...
This slowly converging unit can represent as single unit as ‘e’s expression as:
e= Lim _{m-> infinity} [( m + 1) /m]^m
There is one more way to calculate the value of ‘e’ and that is infinite series:
e = 1 + 1/1! + ½! + 1/3! + ¼! .........+ 1/m! + .....
Here students know that
                         m! = m × (m − 1) × (m − 2) × . . . (3) × (2) × (1)
According to the standard analysis it can be shown that first eleven terms of this series represents the value of ‘e’ with almost negligible error of 3 × 10⁻⁸.
In general all functions of the form of a^x are called exponential functions but still to be more specific we refer to e^x as
The exponential function.
 Exponential growth:-
If s > 1 then it can be easily represented no matter how large K is:                                  
                                      s^x/x^k   → ∞ as x → ∞
It means if K is a fixed value and can be chosen as large as wanted, still if the value of ‘x’ increases then s^x will also increase and being ahead of it always. It will happen until the value of‘s’ is greater than 1. This phenomenon of s^x growth as x increases is called exponential growth.
 Exponential decay
As explained in previous point that the behavior of ‘e^x’ as x → ∞ is known as exponential growth. Just in similar passion,
 if we explain or characterize the expansion of e^–x then that procedure will called as Exponential decay.
This is all about Exponential functions and their relative properties. As the definition of exponential function defines that in it we observe the relative change in any variable in respect of any other variable. This variation may occur in two different styles as Direct and Indirect.
Let us first talk about the direct variations in exponential function. If two variables are said to be direct variant of each other then it means that if one of them get doubles then other one also changes in same ratio, similarly if the second one is reduced in some amount then first one is also decreased in same ratio. For example: assume a function as y = x + 1
If value of ‘x’ grows up then at the same time value of ‘y’ also gets increased.
Now let’s see “what is Inverse variation?”, this type of variation causes opposite change in any value whenever the related value has increased or decreased. For example, if x and y are two values that are related to each other with Inverse relation then if value if x increases as it doubles then at the same time the value of y gets decreased by half of its value. This type of change gets complicated when more than two independent variables are included in the function. Let us take example of Newton’s kepler theorem:
Formula is as F = Gm/r²
So it represents that whenever mass of the body gets doubled then the strength of force also gets doubled and when distance is increased then the strength of the force gets fall in its fourth.
The thing is that both variations: direct and Inverse, are occurring together so determination of the value gets complicated.
This is all about Exponents, Exponential function with their growth and decay functionality, and types of variations in exponents; all of these grade XII Algebra topics comes as a single chapter in your syllabus because there queries solution requires implementation of their common properties, which are explored in detail in this article.
In the successive article of this one, we will discuss the next included algebra topic in grade XII syllabus and that is co-relation of data.
After reading this article still there is any issue or doubt remains in your mind or you want to solve related math queries of it, and then access the online math tutoring websites where students like you can ask their queries and surely will have immediate assistance by online math tutor. These math education websites provides proper interactive session to students so that they can ask their problems on the regular basis to be clear in every math concept. Students can use some available options like online chat, relevant study material, video seminars, unsolved and solved math worksheets, online math calculators for quick solution of the problem, and many more. All these options increase the ease level of the students’ study that causes their better involvement in studies.
Grade XII math syllabus is comparatively vast than any other standard and the main difficulty in it is that all the topics of it usually have the math application problems whose introductory part was studied by you guys in your previous classes. So always learn every math topic with your pure attention to execute it with ease in the final examination because math is a kind of practical subject that can enrich your aggregate percentage when potentially studied otherwise it also can ruin your overall performance in other subjects.

In upcoming posts we will discuss about Patterns and Functions in Grade XII and Standard distributions. Visit our website for information on 12th physics syllabus Maharashtra board

Quadratic Equations in grade XII

In previous article of Algebra I, we had already explained the fundamentals of Rational expressions and polynomials, today in this article we are going to continue with another topic of algebra 1 that is Quadratic equations and quadratic functions. The introductory part of quadratic equation was learned by students in their previous classes but still to make you guys remember it, let us start with the basics concept of Quadratic equations.
As we all know that every algebraic equation is a polynomial function and so on Quadratic equation is also, but the point is that every quadratic equation is a non-linear polynomial function of second order and it is in the form of monomial because quadratic equation has many derivatives but all of them are of a single variable, Suppose x is an unknown variable then its quadratic equation could be of the form:
Ax² + Bx + C = 0, here A, B, and C are integer constants. One compulsion to have this form as quadratic is that “A is not equal to zero”, if it does then the equation is said to be linear not quadratic.
For example:x² + 2x = 4 is said to be quadratic because it can be written in the form of Ax² + Bx + C = 0,
Let us take another example;
Suppose given equation is as 2x² – 1/x + 2 = 0
Here 1/x can be rewritten as x^-1, and quadratic can’t have x raised to the power other then 1 or 2. So the equation is not quadratic.
Now let us come to the main point and that is “ how to solve any quadratic equation?”, in previous article you guys have learned the factoring procedure and in between that session it was nicely explained that factoring can be used to solve any algebraic equation, so quadratic equation is also an algebraic form.  Let’s see how factoring is being applied to evaluate quadratic equation solutions.
There are 3 simple steps need to be executed to sort out the quadratic equation:
1.    First, write the given equation into standard form and for this you can use distributive property and additional property to combine like terms and put them into one side of ‘=’ sign.
2.    Now factor the obtained equation by using simple factoring technique.
3.    Now evaluate the possible values of variable by putting each factor equals to zero according to Zero product law.

Let us take an example and observe the actual execution of above defined steps:
(6x + 8)x = 14
Apply distributive property
6x² + 8x = 14
Imply addition property
6x² + 8x – 14 = 0 ( in standard form)
Now apply the factoring technique as:
6x² – 14x + 6x -14 = 0
Now take the common factor out
6x( x-1) – 14 ( x-1) = 0
(x - 1) (-6x + 14) = 0
Use the zero product law
 (x - 1) = 0 or (-6x + 14) = 0
x = 1 or x = -14/6 = -7/3

This is the specified way to sort out quadratic equation by using factorization. There is one more way to sort out these types of equations and that is Extracting square roots. After extracting square roots equation automatically comes into the correct form, for example:
( 2x-5 )2 + 5 = 3
Use additional; property to add -5 both sides of the ‘=’ sign
( 2x-5 )2 = -2
Now extract square roots
±√(2x-5)² = ±√(-2) (negative and positive sign both are included when square root of any value is taken)
(2x – 5) = ±i√ 2 ( simplify the radical and apply the definition of ‘I’)
Apply addition property of equality
2x = 5 + ±i√ 2
Now use division property of equality
x = (5 + ±i√ 2)/2
There are two more ways to evaluate any quadratic equation, one of them is “completing the square” and other is “ quadratic formula”. We will discuss both of them but let us start with Completing the square method.
Let’s take an example and in parallel of its solution we will explain the implementation of this method,
Given equation is as: 3x² + 4x – 7 = 0
First step: use addition equality property and isolate the variable terms like x² and x, at one side of the equation.
 3x² + 4x = 7
Second step: use division equality property and divide each term on both sides by integer coefficient of term ‘x²’.
3x²/ 3 + 4x/3 = 7/3
x² + (4/3)x = 7/3
Third step: now take coefficient of term x, square it and add in both sides of equation; here X’s coefficient is 4/3 so its ½ is 2/3, now square of (2/3) is added to both sides of the equation:
x² + (4/3)x + (2/3)² = 7/3 + (2/3)²
x²+ (4/3)x + (4/9) = 7/3 + (4/9)
Multiply 7/3 with 3/3 to have the same denominator on RHS
x² + (4/3)x + (4/9) = 21/9 + (4/9)
x² + (4/3)x + (4/9) = 25/9
Next step: factor the left side by using the standard formula:
( a² + 2ab + b²) = (a + b )²
So , x² + (4/3)x + (4/9) = 25/9
(x + 2/3 )² = 25/9
Next step: use; extract the square root technique for finding the relative roots of the equation
√(x + 2/3 )² =±√( 25/9)
x + 2/3 = ± 5/3
Again use addition property of equality
x = -2/3  ± 5/3
Now two equations can be formed as
x = -2/3  + 5/3  OR x = -2/3  - 5/3
By solving these equations
x = 1 or x = -7/3
This method is used to solve complex problems and for some easy problems prefer the previous method, which is much easier than this one.
Now it’s time to talk about the standard quadratic formula and its use to evaluate Quadratic function. If quadratic equation is in the form of
Ax² + Bx + C = 0 then the standard quadratic formula is represented as:
; here (b² – 4ac) is called determinant of the function, which decides the behaviors of the roots of quadratic equation.
While using this formula, the only task for students is to convert the given equation into standard form, for example:
Given equation is: (y + 3)² = ( y – 2)
Rewrite the above equation as its factors:
( y + 3) ( y + 3) = (y - 2)
Now use the standard distributive property to combine like terms:
y² + 6y + 9 = y – 2
Use addition equality property:
y² + 6y + 9 – y + 2 = 0
y² + 5y + 11 = 0
Now the equation is in standard form so we can implement the quadratic formula, first grab the values of a, b and c from the equation.
a = 1, b = 5 and c = 11.
Put this value in the formula


; by using the ‘i’ definition roots can be easily determined.
This is all about Quadratic equations and functions. In this article we have explored all the possible ways of solving any quadratic equation. These four ways of evaluating quadratic equation covers the topic of quadratic function in grade XII math. Students of this class most of the times fixed in troubles because there are several ways in which a similar problem can be asked and every time there is a bit of changes in the solution process, so it is hard to memorize all the ways and forms of quadratic equation. This task only can be accomplished by doing practice as much they can and to help students in it Online math tutor websites do their role by giving immediate assistance and various worksheets to solve.  If any kind of doubt stays in student's mind related to any topic then the whole fundamentals gets messed up, so it must be required that student asks his queries on regular basis and this will happen only when student does not hesitate in front of tutor which most often happen. To troubleshoot this problem, Online services uses online Chatting option via text or video, to build a better conversation between students and tutors, and because most of us are friendly with Internet platform, the access of this service gets easier.

In grade XII math content Algebra is covering most of the problem region so it gets more essential to learn its each and every problem with all the possible ways of query presentation. For having the detailed descriptions of other topics of Grade XII math syllabus, keep following the successive articles that will surely add a wide range of knowledge.

In upcoming posts we will discuss about Exponents in on Grade XII and Methods of data representation. Visit our website for information on Maharashtra state board

Polynomials in XII Grade

In previous article we had introduced the whole scenario of grade XII^th math and now from here we will start the proper demonstration of each and every topic one by one. Today’s topic is polynomials and their solution by using factoring technique. When the term polynomial arises then its application, rational expression also comes into the account. It is expected from students of this class like you to have the knowledge of polynomials, still for making you guys remind about polynomials, let us start a fast flash back of polynomials:
Polynomials are the mathematics expressions that include variables and constants that are related by using arithmetic operators like addition, subtraction, and multiplication.
Now as the general methodology of math expression, every equation can be of two forms either linear or non-linear. You can also play linear equations worksheets to enhance yours skills, But in both cases polynomials functions are needed to be normalized for simplifying them. Simplification of polynomials every time mean that students need to calculate the values of unknown variables of equation.
Let us start with the representation scenario of any polynomial function after that we will discuss the solution process of it;
F(x)= a_nxⁿ + a_n-1 x^n-1 + .……+ a₂x² + a₁x + a₀

Every polynomial includes three terms as:leading term, constant and integer co-efficient. Here x_n is the leading term and (a_n, a_n-1, ……., a₀) are integer co-efficient.
When any polynomial is said to be as linear then it means all the derivatives of the equation are of same order and similarly when the scenario is opposite means: all the derivatives are not of the same order then it is said to be non-linear polynomial function. visit here for more on polynomials.
For example:
X² + Y² = 8 (here both are of same order so it is Linear equation)
X² + y = 7 (here x and y have different orders, so clearly it is an example of non-linear equation)

There are 4 major elementary properties of any polynomial equation, those are as follows:
1.    Sum of two or more polynomials always results as polynomials.
2.    Product of two polynomials is also a polynomial function.
3.    When two polynomials are combined together then the result is obtained by substituting variable of the first polynomial by the second one.
4.    Suppose a polynomial is a_nxⁿ+ a_n-1x^n-1 + ... + a₂x² + a₁x + a_0,
and  na_nx^n-1 + (n-1)a_n-1x^n-2 + ... + 2a₂x + a₁ is the its derivative thenif the set of co-efficient (an, an-1, ……., a0) does not contain the integer value that time Ka is to be evaluated as k times of a.
Now the basic theme to categorize any polynomial into distinct category is the presence of number of unknown variables in the function. As the name suggests Poly means “many”, so any polynomial function may have various unknown variables. Suppose any equation includes all the derivatives of only one unknown variable so that could be called as Monomial and if two variables are taking part then the formed polynomial is Binomial. Let us learn this one with examples:
-2xy + 3x - 5z ( a trinomial, because of three unknown variables)
-10xy (monomial,)
7xy + z (binomial)

That’s all about polynomial presentation, now it’s time to move on into its solution process. There are various ways of solving any polynomial equation but in this article we are going to elaborate factoring procedure.  As we all know that factoring is a normalization technique and most often used for polynomial evaluation because the only problem while solving any polynomial equation is that it includes number of derivatives of various order, which makes its presentation a bit complex.
So let us first talk about factorization, the problems of factors was studied by you and other grade XII students in their early grades but that time you guys were dealing with linear equations and here the given equation is most of the time of non-linear form.
There are mainly two situations according to which factoring also needs to be implemented in different ways, the situations are:
If the polynomial equation is ax² + bx + c
Then when constant c is positive and other when c is negative,
So let us explore these two situations and will take suitable example for better explanation:
First when the constant c is positive in that case:    polynomial only can be factorized when there are 2 factors of product (ac) that can be added into the absolute value of b.
For example:
6x² + 11x + 3
Here a= 6, b = 11 and c = 3

Now we need to know that are there two factors of (ac= 18) whose sum is 11, the answer is yes, the sum of 2 and 9 is 11 and product is 18.
So by rewriting the equation:
6x² + 11x + 3
6x² + 9x + 2x + 3
Put the common factor out
3( 3x + 1) + 2x( 1 + 3x)
(3 + 2x)(1 + 3x)
Now let us talk about the second situation where c or a  is negative,
-800x² -800x + 600
We can rewrite this equation by taking the common factor out as:
200( -4x² – 4x + 3)
Now if here the product of a and c is taken then the the result is negative.
(ac) = -XII, now the factors are 6 and 2 but 6 should be placed as negative to result the sum as -4.
200( -4x² – 6x + 2x + 3)
200( 3( 1 – 2x) + 2x ( 1-2x))
Now here common factor is (1-2x)
200(3 + 2x)(1- 2x)
This is the way to use the factoring normalization procedure to sort out the complex polynomials. While solving any polynomial related query one thing is to be remembered always that whenever there is any common factor present in all derivatives, then make sure to keep it out. This fundamental is the key statement to sort out the complex rational expressions.
Rational expressions are the fraction form that includes complex polynomials in its numerator and denominator. Now when term polynomial arises students must gets sure that he needs to normalize the expression first by using factorization as done above. Let us take an example to make you better understand this:
(x-2)/ (x+4)  +  (x+1)/(x+6) = (11x + 32)/(x² + 10x + 24)
(x-2)/ (x+4)  +  (x+1)/(x+6) = (11x + 32)/ (x+4)(x+6)
By applying cross-multiplication in the above expression:
(x+4)(x+6) [(x-2)/ (x+4)  +  (x+1)/(x+6) ]  = (11x + 32)
(x+4)(x+6) (x-2)/ (x+4)  + (x-2)/ (x+4)  (x+1)/(x+6) = (11x + 32)
Cancel out the common terms, after that remaining term is as :
(x+6) (x-2) + (x+4) (x+1) = (11x + 32)
Multiply the above terms
X² + 4x – 12 + x² + 5x +4 = 11x +32
2x² + 9x – 8 = 11x + 32
Simplify it now by using the normal arithmetic transitions
2x² -2x -40 = 0
2(x² – x – 20) = 0
Now send 2 into the denominator of RHS.
X² – x -20 =0
(x - 5) (x + 4) = 0
So it gives, x = 5 or -4
This is how any complex rational expression can be sorted out; in this article we have explained the polynomials, factorization and rational expression. All these three terms are related to each other because for solving those similar fundamentals are required to implement like Normalization.
It is quite understanding that when students solve math queries then several doubts occur in their minds and because there is nobody to give them immediate assistance the whole concept of topic is messed up in his mind, that’s why the service of online math tutoring is reaching the success bar because they fulfill the requirement of that immediate assistance. Online math tutor is always virtually present with you through internet platform. Student can ask any kind of mathematical query and surely he will get instant answer in much explained manner. Student can review lessons as many times he want to make himself comfortable in that topic. In present time Online math tutoring is better option than other private tuition classes because private institutes are running as secondary school for students where the rush of students does not allow them to interact in a friendly way with tutor and this is most required term to understand the actual difficulty of the students in particular subject.

When student go with Online math learning then tutors provide some friendly and useful options to student for managing a proper learning session and to compare your analytical skills with other students across the globe, These features are video aids, text chatting option, video conference, 24 x 7 hours availability, Online tests, various worksheets to solve and option of choosing the tutor according to own appropriateness. Every single topic of math is very well categorized in these type of online math websites and you not need to search a lot, just type your query and either the direct solution or the instant assistance by online math tutor is given to you in the mean time.

In upcoming posts we will discuss about Quadratic Equations in grade XII and Correlation and causation. Visit our website for information on 12th state board syllabus Tamilnadu

Syllabus of XII Grade

Introduction:

The twelfth grade mathematics program is something that acquaints a person with the future scenario a student may face in future. The course structure of 12^th grade Mathematics program is precisely concentrated to give an overview of the broader mathematical fields that a student may get into in his or her higher studies. A student interested in pursuing mathematics in 12^th grade has wider prospects in higher studies and a much better career opportunity. A mathematics scholar of 12^th grade can opt from a wider choice of streams such as engineering, accountancy, graduation with mathematics which can further lead to career in research and development in premier organizations which deal with scientific operations.

The course structure of 12^th grade mathematics program is designed to meet the requirements to built a strong base for the future. There are eleven fields of mathematics that are dealt in basic in the 12^th grade math program. These eleven chapters are Algebra i, Algebra ii, Geometry and Measurement, Probability and Statistics, Ap Probability and Statistics, Trigonometry, Pre-calculus, Calculus, Linear Algebra, Mathematical Analysis and Mathematical Models with Applications.

1) Algebra: Let's first see what Algebra is all about. It's the branch of mathematics dealing with the study of the rules of operations and relations. It formats the constructions and concept that comes out, by the help of polynomials, equations and algebraic structures. Algebra is one of the most vital branch of mathematics.

Algebra has been divided into two distinct sub topics in 12^th grade mathematics:

Algebra (I)

Algebra (II)

Algebra (I) constitutes of sub-topics that show computational works on theoretical mathematical calculations and proofs. They are Linear equations/inequalities, Monomials and polynomials, Factoring second and third degree polynomials, Rational expressions/functions, Quadratic equations, Quadratic functions Exponents, factors, variation, exponential growth/decay, Hypothesis, counter examples, Correlation for data Patterns, relations, functions, Changing parameters of given functions, Graphs, matrices, sequences, series, recursive relations, Radical equations/inequalities and Limits and infinity.

Algebra (II) deals with the sub-topics that are related to higher engineering mathematics. These sub topics are Absolute value to solve equations/inequalities, Systems of linear equations/inequalities, Operations on polynomials, Factoring polynomials/trinomials, Operations on complex numbers, Operations on rational expressions, Graphing quadratic equations, Quadratic equations in complex number system, Quadratic functions, Laws and properties of logarithms, Fractional exponents, exponential functions, Conic sections, Ellipse, parabola, hyperbola, Binomial theorem, Types of series, Composition of functions, Continuous and discrete functions, Complex numbers for solutions of quadratic equations, Factoring, exponents, Systems of equations/inequalities, Estimation of solutions, Standard functions, Inverse of a function, Solutions of function equations and Slope, various forms of equation of line.

2) Geometry and Measurement: This topic deals with graphical and geometrical representations algebraic expressions and calculations of phenomena related to geometrical structures.
For this given task Geometry and Measurement has the given constituent subtopics that are, Logical reasoning, Geometric proofs, Euclidean/non-Euclidean geometries, Formal/informal proofs, Conditional statements, Pythagorean Theorem, Intersection of a plane with 3-d figures, Congruence, similarity, triangle inequality theorem, Properties of quadrilaterals, circles, parallel lines cut by a transversal, Angles of triangles and polygons, Basic constructions, Triangle congruence relationships, Similarity properties and transformations, trigonometric ratios, Pythagorean triples, Coordinate geometry, Special right triangles, Properties of inscribed/circumscribed polygons of circles, Rotations, translations, reflections, Planar cross-sections; perpendicular lines/planes, Effect of rigid motions on figures, isosceles triangle theorem; polyhedra, Parabolic functions (vertex, axis of symmetry) and Compound loci in the coordinate plane,

3) Probability and Statistics: Probability and Statistics is the branch of mathematics that measures the likeliness that a random event will occur or not.

Two topics in 12^th grade mathematics deals with the course of probability and statistics :

Probability and Statistics

Ap Probability and Statistics

Probability and Statistics: Probability and Statistics is an introduction to the field of the probability with mathematical concept and has the sub topics such as Types of events, Measures of central tendency/dispersion, quartiles, interquartiles, Permutations and combinations, Methods of data representation, Statistical experiments (more than one variable) and Correlation and causation.

Ap Probability and Statistics: Ap Probability and Statistics is the higher and more complex version of probability. It teaches to apply the mathematical approach to the probability problems. The sub topics under this heading are Probability problems with finite sample spaces, conditional probability worksheet, Discrete/continuous random variables, Mean, variance of discrete random variable , Standard distributions, Mean, standard deviation of normally distributed random variable, Central limit theorem, Line of best fit - least squares regression, Correlation coefficient, Statistics of a distribution, Mean, standard deviation of sampling distribution, Mean, standard deviation of population distribution, Confidence intervals, P-value for a statistics and Chi-square distribution/test.

Trigonometry: In trigonometry, students will study triangles and the relations between their sides and the acquired angles. Trigonometry is all about trigonometric functions showing relations and its applications to cyclical phenomena for example waves. For this topic, several subtopics that a person may deal with are Graphing functions, Trigonometric functions, Formulas for sines and cosiness, Polar/rectangular coordinates, Complex numbers in polar form, DeMoivre's theorem, Inverse, trigonometric functions and Area of triangle (one angle and 2 adjacent sides).

Pre-calculus and Calculus: In these two topics, a 12^th grade student will firstly go through Pre-calculus, get acquainted with several terms and fundamentals that are used in Calculus. And, then he or she shall get into an interesting journey of Calculus that brings differentiations and integrations of expressions.
Pre-calculus has sub topics like Limits of sequences, series, Continuity, end behavior, asymptotes, limits, Regression, Law of Sines/Cosines, area formulas, Parametric/rectangular forms of functions

Vectors, Even/odd functions, significant values, Functions and operations, Conic sections- applications and Sequences and series.

Calculus, after going through pre-calculus, may get easier for a being and he or she shall easily understand calculus topics like Limits- operations on functions, Continuity of a function, Intermediate value, extreme value theorem, Derivatives, differentiability, Derivatives of functions

Chain rule, Rolle's theorem, mean value theorem, and L'Hopital's rule, Maxima, minima, inflection points, intervals, Newton's method (approximating the zeros of a function), Differentiation to solve problems, Definite/indefinite integrals, Riemann sums, Fundamental theorem of calculus, Definite integrals problems, Techniques of integration, Inverse trigonometric functions, Integrals of functions, Simpson's rule, Newton's method, Improper integrals (limits of definite integrals), Tests for convergence/divergence of sequences/series, Power series, Taylor series/polynomials and Elementary differential equations.

Linear Algebra: Linear algebra deals with vector spaces, also called linear spaces, along with linear functions that input one vector and output another. So, this topic deals with Types of linear systems, Gauss-Jordan elimination, Gauss-Jordan method, Vectors and vector addition, Geometrical solution sets of systems of equations, Rectangular matrices to row echelon form, Matrix multiplication, Inverse to a square matrix, Determinants of 2 by 2 and 3 by 3 matrices, Row reduction methods, Cramer's rule and Scalar (dot) product.

Mathematical Analysis: Mathematical Analysis is a branch of mathematics that consist s of the theories of differentiation, integration and measure, limits, infinite series and analytic functions. Various forms of complex numbers, Mathematical induction, Roots and poles of rational functions

Graphing rational functions – asymptotes and Limits of sequence/function are the sub topics that are studied under this topic.

Mathematical Models with Applications: The mathematical model is a study of a system using mathematical concepts and language. This section covers Theoretical/empirical probability , Probability models, Studying patterns and analyzing data, Rates, linear functions, direct/inverse variation, Problems related to personal income, credit, financial planning, Transformations and symmetry.

Best of luck for your foundational step into the world of mathematics towards the brightest future ever!

In upcoming posts we will discuss about Polynomials in XII Grade and interquartiles. Visit our website for information on Tamilnadu education board

Types of Linear Systems in 12th Grade

Mathematics is a kind of practical subject which is never going to be outdated, but when students come in twelfth grade, the math problems become get more complex. To evaluate the way of solving these problems students have to implement all the math principles which they have learned in their previous grades. For students, grade XII is the stage where their actual math skills are being tested. For more on visit this
Today we are going to talk about an important topic of 12th grade algebra which is “ Types of Linear System”. Let's start with the topic of Algebra problems that are included in 12th grade math, they are basically divided in three parts, Algebra 1, Algebra 2 and linear algebra. Our topic today is a part of linear algebra and it involves a math problem in which the student has to manipulate more than one equation to solve the system. When all the equations of the system are in linear form than it may be a little easier to solve but the real challenge is solving the ones which are of non-linear form! Let us take an example of linear system:


Now all the equations above are of linear form with three variables. While solving linear system one point should always be kept in mind is that all the equations must have all variables in it as in above equations each of them includes x, y and z.
Suppose if any of the variable is missing in any equation then we have to assume it on our own as
Ax + z= c here y's derivative was missing and the equation can be rewritten as: Ax + 0(y) + z = c
This is done to make easier the comparison of all the equations in the system.
The solution of linear system is done by implementing two standard applications which are Gaussian Elimination and Gauss-Jordan method. Both of these methods prefer the matrix conversion of the linear system and ones the equations are converted into matrix form than the solution is found by calculating the rank of the matrix, determinant of the matrix and by inverse of the matrix.
Let us see how these fundamentals are executed on a linear system:
Firstly convert the linear system into matrix form, the formed matrix includes the integer coefficients of all the variables used to form that system:

1 1 1 | 0
1 -2 2 | 4
1 2 -1| 2
In above matrix the first column represents the value of x coefficient of all three equations, similarly second column represents y's value and third column is for z's value. The fourth column which is separated from a line is the value of the equations in their right side. To solve this linear system the Gaussian Algorithm defines some steps which are as:
to convert matrix into triangular form:
first: eliminate x from every row accept first one.
Second: eliminate y from every row below second row.
So now apply this steps on the above matrix as:
To eliminate x from 2nd and 3rd row you have to perform 2 elimination steps which are:
r3 → r3 – r1
here 'r' represents the row,


now r2-> r2 – r1


The above matrix form shows the proper executed form of first step of algorithm because all x values are removed from 2nd and 3rd rows.
Now for executing second step of the algorithm we have to perform following :
r2 → (r2 x 3) + r3



The above matrix form represents a triangular matrix, which is the resulted matrix of Gaussian algorithm. Now this matrix is further converted into linear equation form whose solution is much easy in comparison of the given linear system.
So converted linear system from Gaussian elimination is as:

The determination of all unknown variables x, y and z is now being done by solving these linear equations as:
x + y + z = 0....... 1
-3y + z = 4...........2
-5z = 10.........3
from the third equation
z= -10/ 5 = -2
Now put this value of z in equation number 2:
-3y + (-2) = 4
-3y = 6
-y = 6/-3 = -2
Use the value of 'y and 'z' to evaluate the value of 'x'.
x + (-2) + (-2)= 0
x = 4
So the solution matrix of the linear system is as;
x = 4
y = -2
z = -2
In Gaussian Elimination the solution of the linear system depends on the matrix Row elimination system but the other application Gauss-Jordan, prefers the matrix inverse program to normalize the system of equations into more simpler form by which the value of unknown variables in the equation easily can be calculated.
Matrix is an essential tool for 12th grade math problems because various complex problems are solved by implementing their related fundamentals and principles on corresponding matrix form. That's why to solve linear system and similar problems involving long calculations, students have to learn various matrix operations. Here we are going to elaborate some of the important matrix operations which are RANK determination and inverse matrix. First let us see what RANK determination actually means for a matrix?, Every student is aware of a matrix formation which includes certain number of rows and columns in it. If there are 'm' rows and 'n' columns in any matrix then the rank of the matrix is decided based on following rules. If both m and n are equal in other words, m = n than RANK is the equal to m or n. But if m>n than rank is n and vice versa. It means that the lowest value among the rows and column represents the RANK of the matrix. By calculating rank, number of solutions gets clarified for any linear system.
Now second most commonly applied operation on matrix is Inverse matrix calculations, if A is a matrix than A-1 represents its inverse form. To convert any matrix into its inverse form the use of Identity matrix is preferred.
Identity matrix is represented as [1 0]
[0 1]

If A is of m x n than an identity matrix of same dimensions is merged with it as:

AX= 1 3 3 | 1 0 0
1 4 3 | 0 1 0
1 3 4 | 0 0 1
the target is to convert the left side of the dashed line into identity by applying various row transformation as:
1 3 3 | 1 0 0 - R1 + R2 1 3 3 | 1 0 0
1 4 3 | 0 1 0 --------------> 0 1 0 | -1 1 0
1 3 4 | 0 0 1 - R1 + R2 0 0 1 | -1 0 1
-3 R2 + R1 ---------------------> 1 0 3 | 4 -3 0
0 1 0 | -1 1 0
0 0 1 | -1 0 1
-3 R3+ R1 ----------------------> 1 0 0 | 7 -3 -3
0 1 0 | -1 1 0
0 0 1 | -1 0 1
The right side of the final matrix represents the Inverse of the A matrix. To learn more matrix operations which are required to solve various types of linear system and other linear algebra problems students can rely on various online math tutoring services. These services deal with every student of various standards. The required term to be good in math is practice. If you are doing regular math practice than that will definitely excel your math skills and for proceeding help in this task, online tutors avail various math queries work sheets for students and also give their assistance in solving them whenever required. These online tutors make students to learn the optimized way of solving any problem which helps them to solve any complex query within a short while. In present time to move into college or in other higher education system , students have to go through various competitive entrance exams whose syllabus mostly includes 11th and 12th grade math queries. So if students are not able to sort out the queries in a short time than these exams become the toughest task to go through with. Because in these kind of exams the number of questions to be solved are large and time given to solve them is not enough if you preferred the standard implementation. The 12th grade includes vast math syllabus in which all math branches are scheduled into various units. The major part of these problems is that you need to convert every problem in such form where from you can apply your math standard principles on them to sort out. We have discussed above how to evaluate the problems of linear systems but this clause does not end here. The question arises “ when the equations of the system are not in linear form then how will be they solved?”. In that situation students need to implement various suitable Normalization techniques to convert non-linear equations into their corresponding linear form. To learn more about non-linear types of linear system students can attend the regular math sessions organized by Online math tutoring services.

In upcoming posts we will discuss about Syllabus of XII Grade and Discrete and continuous random variables. Visit our website for information on West Bengal class 12 syllabus

Conditional Probability for Grade XII

The probability of an event occurring when another event has already occurred is known as conditional probability. Also play conditional probability worksheets to improve your math skills.

The probability that event B occurs, given that event A has already occurred is

   P(B|A) = P(A and B) / P(A).

This formula is derived from the general multiplication principle and algebra.

Since, we know that event A has occurred, we conclude that the sample space has reduced. The sample space S, is reduced to a sample space of A since we know A has taken place. The number in the event divided by the number in the sample space continues. The value of both A and B should be in A since A has already taken place and divided by the number in A. Dividing numerator and denominator of the right hand side by the number in the sample space S, we have the probability of A and B divided by the probability of A.

Conditional Probability

Suppose we put a distribution function to a sample space and then learn that an event E has occurred.

Problem arise that how can we change the probabilities of the remaining events We shall call the

new probability for an event F the conditional probability of F given E and denote

it by P (F jE).

Example 1. Lets take an example of a die rolled once. Let X be the outcome.

Let F be the event fX = 6g, and let E be the event fX > 4g. We assign the

distribution function m(!) = 1=6 for ! = 1; 2; : : : ; 6. Thus, P (F ) = 1=6. Now

suppose that the die is rolled and we are told that the event E has occurred. This

leaves only two possible outcomes: 5 and 6. In the absence of any other information,

we would still regard these outcomes to be equal likely, so the probability of F

becomes 1/2, making P (F jE) = 1=2. 2

Example 2. Let us assume that, Out of 100,000 females, 89.835% can expect to live to age 60, while 57.062% can expect to live to age 80. Given that a woman is 60, what is the probability that she lives to age 80? This is an example of a conditional probability. In this case, the original sample space can be thought of as a set of 100,000 females. The events E and F are the subsets of the sample space consisting of all women who live at least 60 years, and at least 80 years, respectively. We consider E to be the new sample space, and note that F is a subset of E. Thus, the size of E is 89,835, and the size of F is 57,062.

So, the probability in question equals 57;062=89;835 = :6352. Thus, a woman who

is 60 has a 63.52% chance of living to age 80. 2.

Example 1:

The question, “ Do you know maths?" was asked to 100 people.

.	Yes	No	Total
Male	25	14	39
Female	24	37	61
Total	49	51	100

• What is the probability of a randomly selected person who is female and knows math? This is a joint probability. The number of "Female and Knows Math" divided by the total = 24/100 = 0.24
• What is the probability of a randomly selected person who is a male excluding whether he knows math or does not know the same? It can be calculated by taking the total for male divided by the total =39/100 = 0.39. Since the trait of knowing maths is not asked, male of each class can be mentioned.
• What is the probability of a randomly selected person knowing math? As gender is not mentioned this question becomes a marginal probability, the total who know maths divided by the total = 49/100 = 0.49.
• What is the probability of a randomly selected male knowing math? What is the probability of the males knowing math? So, now we have males knowing math of 39 males, so 25/39 = 0.641.
• What is the probability that a randomly selected person knowing maths is male? Now, we have a person knowing maths and asked to find the probability that the person knowing math is also male. There are 25 male who know math out of total 49 people knowing math, so 25/49 = 0.5104.

Example 2:

There are three major manufacturing companies that make a product: Aberations, Brochmailians, and Chompielians. Aberations has a 50% market share, and Brochmailians has a 30% market share. 5% of Aberations' product is defective, 7% of Brochmailians' product is defective, and 10% of Chompieliens' product is defective.

This information can be placed into a joint probability distribution

Company	Good	Defective	Total
Aberations	0.50-0.025 = 0.475	0.05(0.50) = 0.025	0.50
Brochmailians	0.30-0.021 = 0.279	0.07(0.30) = 0.021	0.30
Chompieliens	0.20-0.020 = 0.180	0.10(0.20) = 0.020	0.20
Total	0.934	0.066	1.00

The percent of the market share for Chompieliens wasn't given, but since the marginals must add to be 1.00, they have a 20% market share.

Notice that the 5%, 7%, and 10% defective rates don't go into the table directly. This is because they are conditional probabilities and the table is a joint probability table. These defective probabilities are conditional upon which company was given. That is, the 7% is not P(Defective), but P(Defective|Brochmailians). The joint probability P(Defective and Brochmailians) = P(Defective|Brochmailians) * P(Brochmailians).

The "good" probabilities can be found by subtraction as shown above, or by multiplication using conditional probabilities. If 7% of Brochmailians' product is defective, then 93% is good. 0.93(0.30)=0.279.

• What is the probability a randomly selected product is defective? P(Defective) = 0.066
• What is the probability that a defective product came from Brochmailians? P(Brochmailian|Defective) = P(Brochmailian and Defective) / P(Defective) = 0.021/0.066 = 7/22 = 0.318 (approx).
• Are these events independent? No. If they were, then P(Brochmailians|Defective)=0.318 would have to equal the P(Brochmailians)=0.30, but it doesn't. Also, the P(Aberations and Defective)=0.025 would have to be P(Aberations)*P(Defective) = 0.50*0.066=0.033, and it doesn't.

The second question asked above is a Bayes' problem. Again, my point is, you don't have to know Bayes formula just to work a Bayes' problem.

Bayes' Theorem

Bayes' formula can be well defined by the assumption given further.

Let's say, each event is taken as to A, B, C, and D.

Bayes' formula is used to find the reverse conditional probability P(B|D). And P(D|B) is not a Bayes problem as discussed in the given problem.

It is assumed that D is made of three parts, the part of D in A, the part of D in B, and the part of D in C.

  

                                        P(B and D)                   
   P(B|D) =       P(A and D)  + P(B and D)  + P(C and D)

Inserting the multiplication rule for each of these joint probabilities gives

                                      P (D|B)*P(B)                        
   P(B|D) =         P(D|A)*P(A) + P(D|B)*P(B) + P(D|C)*P(C)

It can be well accepted that, it is much easier to take the joint probability divided by the marginal probability.

Bayes Probabilities

Our original tree measure gave us the probabilities for drawing a ball of a given color, given the urn chosen. We have just calculated the inverse probability that a

particular urn was chosen, given the color of the ball. Such an inverse probability is called a Bayes probability and may be obtained by a formula that we shall develop later. Bayes probabilities can also be obtained by simply constructing the tree measure for the two-stage experiment carried out in reverse order.

The paths through the reverse tree are in one-to-one correspondence with those in the forward tree, since they correspond to individual outcomes of the experiment, and so they are assigned the same probabilities. From the forward tree, we find that the probability of a black ball is

1 . 2 + 1. 1 = 9
2 5 2 2 20

The probabilities for the branches at the second level are found by simple division. For example, if x is the probability to be assigned to the top branch at the second level, we must have ,

9 . x = 1

20 5

or x = 4=9. Thus, P (IjB) = 4=9, in agreement with our previous calculations. The reverse tree then displays all of the inverse, or Bayes, probabilities.

Independent Events:

It often happens that the knowledge that a certain event E has occurred has no effect on the probability that some other event F has occurred, that is, that P (F jE) = P (F ). One would expect that in this case, the equation P (EjF ) = P (E) would also be true. In fact (see Exercise 1), each equation implies the other. If these

equations are true, we might say the F is independent of E. For example, you would not expect the knowledge of the outcome of the first toss of a coin to change

the probability that you would assign to the possible outcomes of the second toss, that is, you would not expect that the second toss depends on the first. This idea

is formalized in the following definition of independent events. Two events E and F are independent if both E and F have positive probability and if

P (EjF ) = P (E) ;

and

P (F jE) = P (F ) :

As noted above, if both P (E) and P (F ) are positive, then each of the above equations imply the other, so that to see whether two events are independent, only

one of these equations must be checked.

In upcoming posts we will discuss about Types of Linear Systems in 12th Grade and elimination method calculator. Visit our website for information on West Bengal council of higher secondary education syllabus