Direct Proportion and Indirect Proportion

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Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. That is if \( \frac{x}{y} = k \) [ k is a positive number], then x and y are said to vary directly. In such a case if \( y_1, y_2 \) are the values of y corresponding to the values \( x_1, x_2 \) of a respectively then \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \).

 

Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if \( xy = k \), then x and y are said to vary inversely. In this case if \( y_1, y_2 \) are the values of y corresponding to the values of x respectively then \( x_1y_1 = x_2y_2 \) or \( \frac{x_1}{x_2} = \frac{y_2}{y_1} \).

Polynomials

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Polynomial:- An algebraic expression consisting of one or more terms comprising of constants and / or variables and non-negative exponents of the variables.

e.g.            x + 1,         x2 + 2,    y2 - 2y + 1

Zero polynomial:- The constant polynomial 0 is called the zero polynomial.

Constant polynomials:- Polynomials which do not have terms with variables are termed as constant polynomials e.g. 2, -3, 14.

Polynomials in one variable:- e.g. x2 - 2,  y3 + 3y - 4

Each term of a polynomial has a coefficient. So in x2 - 2 the coefficient of x2 is 1, and that of x0 is -2.

Monomials:- Polynomials having only one term.

Binomials:- Polynomials having only two terms are called binomials.

Trinomials:- Polynomials having only three terms are called trinomials.

Degree of a polynomial:- The degree of a non-zero constant polynomial is zero.

Linear polynomial:- A polynomial of degree one is called a linear polynomial.

Quadratic polynomial:- A polynomial of degree two is called a quadratic polynomial.

Cubic polynomial:- A polynomial of degree three is called a cubic polynomial.

Degree of zero polynomial:- The degree of zero polynomial is not defined.

Quadrilateral

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Quadrilateral is a type of polygon with four sides.

Sum of all interior angles: Sum of all interior angles of a quadrilateral is 360°.

Types of quadrilaterals and their properties:

Quadrilateral Properties

Parallelogram

A quadrilateral with each pair of opposite sides equal.

(1) Opposite sides are equal.

(2) Opposite angels are equal.

(3) Diagonals bisect one another.

Rhombus

A parallelogram with sides of equal length.

(1) All the properties of a parallelogram.

(2) Diagonals are perpendicular to each other.

Rectangle

A parallelogram with a right angle.

(1) All the properties of a parallelogram.

(2) Each of the angles is a right angle.

(3) Diagonals are equal.

Square

A rectangle with sides of equal length.

All the properties of a parallelogram, rhombus and a rectangle.

Kite

A quadrilateral with exactly two pairs of equal consecutive sides.

(1) The diagonals are perpendicular to one another.

(2) One of the diagonals bisects the other.

(3) In the figure m∠B = m∠D but m∠A≠m∠C.

Polygons

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Polygon: A simple closed curve made up of only line segments is called a polygon.

Classification of polygons:

Number of Sides Classification
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
... ...
n n-gon

Diagonal: A diagonal is a line segment connecting two non-consecutive vertices of a polygon.

Convex polygons: Where all the interior angles are less than 180° and no part of its diagonals lie outside the polygon.

Concave polygons: Where at least one interior angle is more than 180° and some of the diagonals will lie outside the polygon.

Regular polygons: A regular polygon is both 'equiangular' and 'equilateral'. e.g. equilateral triangle.

Irregular polygons: Polygons which are not equiangular or equilateral are irregular polygons. e.g. right-angled triangle.

Angle sum property: Sum of all interior angles = \( (n - 2) \times 180° \) where n --> number of interior angles.

Algebriac Expressions

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Algebraic Expressions:

For e.g. 4x2+3, 7y2-2y+3, etc.

The combination of variables and constants when take part in the formation of mathematical expression, then such an expression is known as Algebraic Expression.

Terms of an expressions:

Terms are added to form expression.

Consider the first expression above, the terms are (4x2) and (3).

Now consider the send expression above, which can be written as

7y2 +(-2y) + 3

therefore, the terms of the expression are (7y2), (-2y) and (3). Note, the minus sign (-) is included in the term.