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Quantitative Ability or Mathematics is the base of human civilization. From cutting vegetables to arranging books on the shelves, from tailoring clothes to motion of Planets - Mathematics applies everywhere. Mathematics as a discipline has its own importance as it prepares you to develop problems solving skills.

Sphere

Rotate a semicircle about its diameter. The solid so generated with this rotation is called a sphere.

Perimeter and Area of Sector

A part of a circular region enclosed between two radii of the corresponding circle is called a sector of the circle.

Circumference and Area of Circle

The circumference (perimeter) of a circle is 2πr and its area is πr², where r is the radius of the circle and π is a constant equal to the ratio of circumference of a circle to its diameter.

Perimeter and Area of Rectangle

The perimeter of rectangle ABCD as 2(AB + BC).

Perimeter and Area of Triangle

The distance covered around a closed figure on a plane is called its perimeter.

Area of Parallelogram

The area of parallelogram is equal to base multiplied by its corresponding height.

Perimeter and Area of Square

The square is a special rectangle in which length and breadth are equal.

Area of Trapezium

Area of a trapezium = (half the sum of parallel sides) × height

Centroid of a Triangle

The centroid of a triangle is the point of concurrency of its medians and divides each median in the ratio of 2 : 1.

Section Formula

To find the co-ordinates of a point, which divides the line segment joining two points, in a given ratio internally.

Mid Point Formula

The co-ordinates of the mid-point of the line segment joining two points (x₁, y₁) and (x₂, y₂) can be obtained by taking m = n in the section formula.

Distance Between Two Points

The distance between any two points P(x₁, y₁) and Q(x₂, y₂) in the plane is the length of the line segment PQ.

Coordinates of a Point

The position of a point is given by two numbers, called co-ordinates which refer to the distances of the point from these two axes. By convention the first number, the x-co-ordinate (or abscissa), always indicates the distance from the y-axis and the second number, the y-coordinate (or ordinate) indicates the distance from the x-axis.

Coordinate System

The position of a point in a plane is fixed w.r.t. to its distances from two axes of reference, which are usually drawn by the two graduated number lines XOX′ and YOY′, at right angles to each other at O.

Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples. For example, the common multiples of 4 and 6 are 12, 24, 36, ... . The lowest of these is 12. So, the lowest common multiple of 4 and 6 is 12.

Highest Common Factor (HCF)

You can find the common factors of any two numbers. For example, the common factors of 12 and 16 are 1, 2 and 4. The highest of these common factors is 4.

Tests for Divisibility of Numbers

You can find a pattern that can tell whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11.

Even and Odd Numbers

In the numbers 2, 4, 6, 8, 10, 12, 14, ... each of them is a multiple of 2. These are called even numbers.

Prime and Composite Numbers

There are numbers, having exactly two factors 1 and the number itself. Such number are 2, 3, 5, 7, 11, and so on. These numbers are prime numbers.

Irrational Numbers

The decimal expansion of a rational number is either terminating or is a non-terminating and repeating decimal. A decimal expansion which is neither terminating nor is repeating represents an irrational number.

Decimal Form of Rational Number

The process of expressing a rational number into decimal form is to carry out the process of long division using decimal notation.

Sum of First n Terms of AP

Carl Friedrich Gauss, the great German mathematician, was in elementary school, when his teacher asked the class to find the sum of first 100 natural numbers. While the rest of the class was struggling with the problem, Gauss found the answer within no time.

General (nth) Term of AP

A pattern in which each term except the first is obtained by adding a fixed number (positive or negative) to the previous term is called an Arithmetic Progression (A.P.).

Word Problems on Quadratic Equations

1. The sum of squares of two consecutive odd natural numbers is 74. Find the numbers.

Solution of Quadratic Equation

A zero of a polynomial is that real number, which when substituted for the variable makes the value of the polynomial zero. In case of a quadratic equation, the value of the variable for which LHS and RHS of the equation become equal is called a root or solution of the quadratic equation.

Quadratic Equations

A polynomial of degree two is called a quadratic polynomial. When a quadratic polynomial is equated to zero, it is called a quadratic equation.

Word Problems: Linear Equations in Two Variables

1. The perimeter of a rectangular garden is 20 m. If the length is 4 m more than the breadth, find the length and breadth of the garden.

Linear Equations in Two Variables

An equation which contains two variables and the exponents of each variable is one and has no term involving product of variables is called a linear equation in two variables.

Properties of Probability

Probability has many interesting properties.

Probability of Event

Suppose a coin is tossed at random. You have two possible outcomes, Head (H) and Tail (T). We may assume that each outcome H or T is as likely to occur as the other. In other words, we say that the two outcomes H and T are equally likely.

Random Experiment and Outcomes

The words may, likely, unlikely, chances, doubt show that the event, we are talking about , is not certain to occur. It may or may not occur. Theory of probability is a branch of Mathematics which has been developed to deal with situations involving uncertainty.

Mode

Mode is one of the measures of central tendency. The observation that occurs most frequently in the data is called mode of the data.

Median

The mean is affected by the extreme values of the observations in the data. This weakness of mean drives us to look for another average which is unaffected by a few extreme values. Median is one such a measure of central tendency.

Arithmetic Average or Mean

To calculate the mean of raw data, all the observations of the data are added and their sum is divided by the number of observations.

Histogram and Frequency Polygon

A continuous grouped frequency distribution can be represented graphically by a histogram. A histogram is a vertical bar graph with no space between the bars.

Graphical Representation of Data: Bar Graphs

Graphical representation is more convenient for the purpose of comparison among the individual items. Bar chart (graph) is one of the graphical representation of numerical data.

Presentation of Data

When the work of collection of data is over, the next step to the investigator is to find ways to condense and organise them in order to study their salient features. Such an arrangement of data is called presentation of data.

Collection of Data

Statistics is the science which deals with the collection, organisation, analysis and interpretation of the numerical data.

Angle of Elevation and Depression

Trigonometry can be used to determine the distance between the objects or the distance between the objects or the heights of objects.

T-Ratios for Angle of 60°

Let a ray OA start from OX and rotate in anti-clockwise direction and make an angle of 60° with x-axis. Take any point P on OA.

T-Ratios for Angle of 30°

Let a ray OA start from OX and rotate in the anti clockwise direction and make an angle of 30° with x-axis.

T-Ratios for Angle of 45°

Let a ray OA start from OX and rotate in the anti-clockwise direction and make an angle of 45° with the x-axis.

T-Ratios of Complementary Angles

Two angles are complementary if their sum is 90°. If the sum of two angles A and B is 90°, then ∠A and ∠B are complementary angles and each of them is complement of the other.

Trigonometric Identities

When equation involving a variable is true for all values of the variable, it is called an identity.

Trigonometric Ratios

Let there be a right triangle ABC, right angled at B. Here ∠A (∠CAB) is an acute angle, AC is hypotenuse, side BC is opposite to ∠A and side AB is adjacent to ∠A.

Heron's Formula

If the base and corresponding altitude of a triangle are known, you can find the area of a triangle. However, sometimes you are not given the altitude (height) corresponding to the given base of a triangle. Instead, you are given the three sides of the triangle.

Right Circular Cone

Rotate a right triangle ABC right angled at B about one of its side AB containing the right angle. The solid generated as a result of this rotation is called a right circular cone. In daily life, we come across many objects of this shape, such as Joker's cap, tent, ice cream cones.

Right Circular Cylinder

Rotate a rectangle ABCD about one of its edges say AB. The solid generated as a result of this rotation is called a right circular cylinder. In daily life, we come across many solids of this shape such as water pipes, tin cans, drums, powder boxes.

Cuboid and Cube

A brick, chalk box, geometry box, match box, a book, are all examples of a cuboid. A cuboid has six rectangular regions as its faces.

Natural Numbers

The counting numbers 1, 2, 3,... constitute the system of natural numbers. These are the numbers which are used in day-to-day life.

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