2 DIGIT EASY MULTIPLY

EASY MULTIPICATION

This about multiplication of  (2 digit)x(2 digit). This will make you to multiply number fastly.

1^st step

This is multiplying the right end digit.

If the product is single digit directly take the number to right most one(one’s place) of the result.

If it is double digit, take the one’s place to the one’s place of the result and ten’s place digit keep it in reserve.

2^nd step

Multiply  the cross digit both and add them.

If there is any reserve number in 1^st step add that number also.

After going through this process the one’s digit obtained in the 2^nd step is the ten’s digit of the result.

And keep the ten’s digit of 2^nd step multiplication in reserve.

3^rd step

Now this is the last step and multiply the left most digits (ten’s digits of given numbers) and add if any reserve number from 2^nd step.

MOD (| |) ------------ curves

HOW  DO  MODULUS EFFECT THE CURVES?? (y=f(|x|),y=|f(x)|….)

WHAT DOES MODULUS DO FIRST?

Simply it changes some quantity  positive or gives absolute value.

Here lets ‘x’ be some real number

|x|=    1. x , where x = positive real number.

             2. –x, where x = negative real number.  

  Now,

Lets us take a curve y=f(x),

Firstly you have to know how to draw the y=f(x) curve.

To draw the curve y=f(|x|),

draw

Y=f(|x|)        1. Y=f(x), for x=+ve values.

                      2.y=f(-x),for x=-ve values.

To draw the curve y=|f(x)|

Draw

Y=|f(x)|          1.y=f(x) , for f(x)=+ve values.

                         2. y=-f(x), for f(x)=-ve values.

Examples

Now  we will take the curve y=sin(x).

Sketch the graph of y=sin(x)

Now lets draw the curve y=-sin(x) and the graph of y=sin(-x) will be same because sin(-x)=-sin(x)

By now we are ready to draw the curves y=f(|x|),y=|f(x)|.

1.Y=f(|x|)  i.e y=sin(|x|)

    Draw 1.sin(x) for x = positive

               2.  sin(-x)=-sin(x) for x=negative     

2.y=|f(x)|  i.e y=|sin(x)|

Draw 1.sin(x) for sin(x)=positive (y=positive)

           2.-sin(x) for sin(x)=negative(y=negative)

CHECKING SYMMETIC NATURE OF CURVES

Symmetric nature of curves

Here what does actually symmetric nature mean?

Symmetric nature of the curve mean that it is reflection or

Image  or replica of the curve about some line or point.

Here “about some line or point” usually we consider line x-axis(y=0) or y-axis (x=0)  and the point is origin(0,0)

Now  how to find a curve is symmetric about a some point

1.symmetric about origin

 we replace the y with –y  and the x with –x ,if the equation the curve doesn’t change then the curve is said to be symmetric about origin.

Example: line y=x,y=x³,y=x²ⁿ⁺¹where n is natural number N, etc..

Lets now check y=x³  holds this property good

Given equation y=x³

Replace by y by –y and x by –x, then the equation will become

(-y)=(-x)³

-y=-x³

Y=x³    hence the equation is symmetric about origin.

2.symmtric about some other point(x,y)

How to check the symmetric nature ?

First take the point(x,y) which is told curve is symmetric about

And choose any point which is on the curve and find the image of the point on the curve about (x,y)

If the image point lies on the curve , the curve is said to be symmetric about point(x,y) only when all points on the curve satisfies this property.

Symmetric nature about a line

1.     about  x-axis(y=0)

 we replace y by-y, if there is no change in the equation of cuvre

2.about y-axis(x=0)

We replace the xby –x, if there is no change in the equation .

3.about some other line

For all points on the curve if we find out the image about given line all the images of those points lies on curve .

For checking symmetric take some point on curve and check it out. 

SHIFTING CURVES

SHIFTING OF CURVE

The graph of the curve is  normally shifting of curve is adding a constant value to a standard curves like  y=x, y=X²,y=sin(x) etc..

Lets take a function f(x)=y and the constant be some ‘k’.

1.   If we add k to ‘x’ i.e y=f(x+k),then the curve will be shifted k units towards left.

2.   If we subtract k to ‘x’ i.e y=f(x-k),then the curve will be shifted k units towards right.

3.   If we add k to ‘y’ i.e (y+k)=f(x),then the curve will be shifted k units towards down.

4.   If we subtract k to ‘y’ i.e (y-k)=f(x),then the curve will be shifted k units towards up.

For example lets take a curve y=|x|

The graph of the curve is

For the 1^st one i.e f(x+k) 

For the 2^nd one i.e f(x-k)

 For the 3^rd one i.e (y+k)=f(x) 

 For the 4^th one i.e (y-k)=f(x) 

Same thing can be done for shifting of any curve.

congurent triangles

what does congurent mean in triangles?

two triangles are said to be congurent means that the  the dimensions(sides) and the angles are equal respectively

# actually  there are 6 dimensions i.e 3 sides and 3 angles for a triangle.

# if all the dimensions are same for both the triangle they are said to be congurent.

# it is difficult for us to check all the dimensions to conform the congurence

# so there are few property from which we can easily find out the congurence of the tirngles

#there are 4 types congurence properties through which we can find congurency

1) side-side-side(sss)

2)angle-side-angle(asa)

3)side-angle-side(sas)

4)right angle-hypotense-side

1.SSS property

if sides of the one triangle are equal to sides of the other triangle . both the the triangles are said to be congurent under sss property

2.ASA property

if any two angles of the triangle and the included side of both the triangles are equal . the two triangles are similar under asa property

.

included side: the side in between the two angles is included side

3. SAS property

if the two side and the included angle of both the triangles are equal they are under sas property

include angle: the angle between two sides

4.RHS property

this property is for the right angled(90 degrees) triangle only

if both the triangles have same length of hypotense and any side they are said to be congurent under RHS property