STD X – ALGEBRA – EINSTIEN

LINEAR EQUATION

Linear Inequations in one unknown for x ∈ N, W, Z, R.

Algebraically and writing the solution in set notation form.
Representation of solution on the number line.

QUADRATIC EQUATION

An equation of the form ax² + bx + c = 0 where a not equal to 0 is known as quadratic equation. a, b and c are constants. The name Quadratic comes from “quad” meaning square, because the variable gets squared (like x²).
It is also called an “Equation of Degree 2” (because of the “2” on the x)

RATIO AND PROPORTION

In certain situations, comparison by division makes better sense than comparison by taking the difference. The comparison by division is called the ratio of the two numbers. For example, for most mammals the ratio of legs to noses is 4:1, but for humans, the ratio of legs to noses is 2:1.

• A ratio is a comparison of two numbers by division.
• The ratio of two quantities (a and b) of the same kind and in the same units is a comparison obtained by dividing one quantity by the other

REMAINDER AND FACTOR THEOREM

Did you ever know that we can find out the remainder in polynomial division without actually performing the division? Well, it can be done with Remainder Theorem. Remainder theorem is an application of Euclidean division of polynomials. The polynomial remainder theorem states that the remainder of the division of a polynomial f(x) by a linear polynomial (x – a) is equal to f(a).x-a is a divisor of f(x) if and only if f (a) = 0.

The Factor theorem is used to link factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. Let p(x) be a polynomial of degree n>1 and ‘a’ be a real number. If p(a) = 0, then (x – a) is a factor of p(x). Conversely if (x – a) is a factor of p(x),p(a) = 0.

MATRICES

A matrix is a rectangular arrangement of numbers, arranged in rows and columns

• Each number or entity in a matrix is called its element
• In a matrix, the horizontal lines are called rows; whereas the vertical lines are called columns
• Matrices, in general, are denoted by capital letters, say A

ARITHMETIC PROGRESSION (A.P.)

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term, except the first term.

This fixed number is called the common difference of the AP. It can be positive, negative or zero Eg: 1, 2, 3, 4, . . . is an arithmetic progression. Each of the numbers in the list is called a term.The common difference in this case is equal to 1 = 2 – 1= 3 – 2 = 4 – 3 = ….

An arithmetic progression having finite number of terms is called a finite arithmetic progression. An arithmetic progression having infinite number of terms is called an infinite arithmetic progression

GEOMETRIC PROGRESSION (G.P.)

A Geometric progression is a list of numbers in which each term is obtained by multiplying or dividing a fixed number to the preceding term, except the first term.

This fixed number is called the common ratio of the GP. 

 Eg: 2, 4, 8, 16, . . . is a geometric progression as each term of it multiplied by 2 gives its next term. (Clearly, common ratio r = 2).