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In algebra we come across certain products very frequently. For e.g., (a + b)2, (a + b)3 (a + b + c)2 etc. These are nothing but products of binomials or trinomials. We derive the formulae for these products and apply them whenever necessary.
A formula is formed by using:
(a) mathematical symbols and variables (b) given conditions, and (c) simplification.
If m is a positive integer, a x a x a .... m times is written as am. a is called the base and m is the power. We read it as "a raised to the power m". The power is also called "the index" or "the exponent".
An equation is an equality connecting some unknowns. The unknowns are represented by "letters" and are called "the variables". If the equation has only one unknown, it is called "an equal in one variable". The word "Linear" means "of degree one".
Solving two equations simultaneously means to find the common solution of both the equations, i.e., a solution which satisfies both the equations.
The following two methods are used to find a solution: (a) Method of elimination (b) Method of substitution.
A linear equation of the form ax + by + c = 0 where a = 0, b¹0 is called a linear equation in two variables.
An equation is called a Linear equation in one variable or an equation of degree one in one variable, if only a single variable with degree one occurs in the equation.
Writing a polynomials as the product of two or more polynomials is called factorisation. If A = B x C, B and C are called factors of A.
Methods of Factorisation: (i) Common factors (ii) By expressing as difference of squares (iii) By grouping (iv) Trinomials (v) Sum or difference of cubes.
An equation of the form ax2+bx+c=0 where a, b, c are real numbers and where "a" does not equal to zero(0).
The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term divided by the leading coefficient.
The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.
An inequation is said to be linear if each term of the algebraic expression (or expressions) of the inequation contains first degree variables (not the product of variables).
ax + by ³ 0, ax + by £ 0, ax + by ³ c, ax + by £ c where a, b not equals to 0 are linear inequations of two variables, (x, y) of degree 1. ax ³ c, ay ³ k are also linear inequations. They are single variable inequations.
A relation R is a non-empty sub-set of a cartesian product.
A relation is a set of ordered pairs, i.e., R Ì A x B where A and B are two non-empty sets.
Domain of a relation is the set of all first components.
Range of a relation is the set of all second components.
Left Hand Limit: Let f(x) tend to a limit l1 as x tends to a through values less than 'a', then l1 is called the left hand limit.
Right Hand Limit: Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit.
An ordered pair has a pair of elements which occur in a definite order.
Cartesian Product - Given two sets A and B, all possible ordered pairs (x, y) obtained such that xÎA and yÎB is called the cartesian product of the sets A and B and is denoted by A x B.
Relation - If A and B are two non-empty sets, then a relation R in A x B, is a subset of A x B.
A function is a relation on A x B is which (i) no two second elements have a common first element. (ii) every first element has a corresponding second element. Every function is either one-one onto or one-one into or many-one onto or many-one into.
Circles, ellipses, parabolas, and hyperbolas are called conic sections because they can be obtained as a intersection of a plane with a double- napped circular cone.
The knowledge of linear inequations is very helpful in solving problems in science, mathematics, engineering, linear programming etc. An equation in defined as a statement involving variable(s) and the sign of equality (=).
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