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INTERVALS

(1) Closed Interval: Let a and b are two given real numbers such that \(\text{a}\,\text{}\,\text{b}\) , then \(\left[ \text{a,}\,\text{b} \right]=\left\{ x:x\in \text{R},\,a\le x\le b \right\}\) (2) Open Interval: Let a and b are two given real numbers such that \(\text{a}\,\text{}\,\text{b,}\) then \(\left] a,b \right[\) or \(\left( a,\,\,b \right)=\left\{ x:x\in \text{R},a<x<b \right\}\) (3) Closed-open Interval: Let […]

SUBSETS

SUBSET: A set P is said to be a subset of a set Q if each element of P is also an element of Q. If P is subset of Q, we write \(\text{P}\subseteq \text{Q}\) PROPER SUBSET: If \(\text{P}\subseteq \,\text{Q}\) and \(\text{P}\ne \text{Q}\) then P is called a proper subset of Q and we write […]

MULTIPLICATION OF A TWO DIGIT NUMBER WITH ANOTHER TWO DIGIT NUMBER

Keep one digit from right hand side and carry forward remaining digits Step I : Multiply unit’s place with unit’s place Step II : Cross multiply and add Step III : Multiply ten’s place with ten’s place Step IV : Keep unit’s digit and carry forward the remaining.

RADIUS AND CENTRE OF CIRCLE

If the equation of circle is Then center = (–g, –f) and radius                   Or If the equation of circle is  By completing square,                                                         Example1: Find the centre and radius of the circle Solution: Comparing the given equation with the standard equation,   and c = 9 Centre Radius Therefore the circle has […]

DISTANCE FROM A POINT TO THE GIVEN LINE

Consider a point P(x1, y1) and a line in the line co – ordinate axes. The coordinates of points A and B are and . Area of   Example: Find the distance from the point (1, 2) to the line .   The required distance = units.

EQUATION OF A STRAIGHT LINE IN PARAMETRIC FORM

Consider a line AB makes an angle with the positive direction of x – axis and a point on the line is . Let a general point P on the line AB and the distance between P and Q is r. Draw a line QL, PM perpendicular to x-axis and draw. In a right angled […]

EQUATION OF A STRAIGHT LINE IN NORMAL FORM

Definition A normal to a line is a line segment drawn from a point perpendicular to the given line. Let p be the length of perpendicular form from the origin to a line, which subtends an anglewith the positive direction of x-axis as shown below: Let AB be a straight line on coordinate axes. p […]

EQUATION OF A LINE IN DEFFERENT FORMS

1. Slope-intercept form: Consider AB a straight line making an angle\(\displaystyle \theta \)with x-axis and cutting off an intercept OD=c from OY. As the line makes intercept OD=c on the y-axis, it is called as y-intercept. Consider the line AB intersect OX’ at T. Take a point P (x, y) on the line AB. Draw\(\displaystyle […]

EQUATION OF A LINE IN INTERCEPT FORM

The equation of a line which cuts off the given intercept on both the coordinate axes is determined as follows: Let PQ be a line meeting x-axis in A and y-axis in B. Consider OA = a, and OB = b. Then, the coordinates of A and B is. The equation of line joining A […]

ANGLE BETWEEN TWO LINES

Consider two lines: and , where , Angle between both lines are . Taking tan both sides,     Example Find the angle between the lines.   Solution The equation is written as follows: The angle between two lines is: we know that, substitute the values as follows: Therefore, the angle between two lines is. […]

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