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.

Posted in Terms Defined

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.