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Three Dimensional Geometry (part - 2) , The Planes | study material for IIT JEE | concept booster, chapter highlights/

THREE DIMENSIONAL GEOMETRY (PART - 2), THE PLANES










THE PLANESPlane is the surface such that if any two points are taken on it, then the line joining the two points lies on it.
General equation of plane is given as -  ax + by + cz + d = 0 , where a ,b and c is not equal to zero.
           POINTS TO BE REMEMBERED 1). a, b and c are the directions ratios of the normal to the plane ax + by + cz + d = 0.

2). Equation of the yz - plane is x = 0

3). Equation of the zx - plane is y = 0

 4). Equation of the xy - plane is z = 0

5). Equation of the any plane parallel to the xy - plane is z = c. Similarly for Planes parallel to yz and zx is x = c and y = c
EQUATION OF THE PLANE IN NORMAL FORMVECTOR FORMIf n̂ be a unit vector normal to a given plane and d be the length of the perpendicular from the origin to the plane, then the equation of the plane is given by -                           r.n̂ = d CARTESIAN FORMIf l, m, n are the directions cosines of the normal to the plane and d is the perpendicular distan…

MENSURATION fast track formulae on SQUARE , RECTANGLE , QUADRILATERAL , PARALLELOGRAM , TRAPEZIUM and RHOMBUS.

Today we are going to talk about some important arithmetic formulae on square , rectangle , quadrilateral , parallelogram , trapezium and rhombus. we are giving whole complete list of all the formulae here ,which are very helpful to solve the problems quickly and accurately. These are very important topics of mensuration , so it is advised to students that they learn these formulae and keep practicing with it.

SOME IMPORTANT SUTRAS
* SQUARE
1). Area of square = side^2 2). Area of square = diagonal^2/2 3). Diagonal of square = side√2 4). Perimeter of square = 4×side 5). Side of square = √Area 6). Side of square = diagonal/√2 7). Side of square = perimeter/4 8). Diagonal of square = area√2
* RECTANGLE
1). Area of rectangle = length × breath 2). Perimeter of rectangle = 2(l+b) 3). Diagonal of rectangle = √l^2+b^2 4). Length of rectangle = A/b 5). Breath of rectangle = A/l 6). Perimeter of rectangle = 2√(2A+d^2) 7). Longer side of rectangle =  √(2A+d^2)+(√d^2-2A)/2 8). Shorter side of rect…

Triangles quick arithmetic formulae for problem solving.

Today we are going to talk about some important arithmetic formulae on triangle. we are giving whole complete list of all the formulae here ,which are very helpful to solve the problems quickly and accurately. Triangle is a very important topic of mensuration , so it is advised to students that they learn these formulae and keep practicing with it.

SOME IMPORTANT SUTRAS

1). Area of triangle = (b×h)/2
2). Perimeter of the triangle = a+b+c
3). Semi-perimetre of the triangle = (a+b+c)/2
4). Area of right angled triangle = bp/2
5). Area of triangle = pbsinθ/2
6). Area of triangle = (area of rectangle or parallelogram)/2
7). Height of triangle = 2A/b
8). Area of isoceles right angled triangle =      (hypotenuse)^2/4
9). Area of isoceles right angled triangle = (equal side)^2/2
10). Hypotenuse of isoceles right angled triangle = 2√Area
11). Hypotenuse of isoceles right angled triangle= side√2
12). Equal side of isoceles right angled triangle = area√2
13). Equal side of isoceles right angled…

" PROPERTIES AND SOLUTIONS OF TRIANGLE for IIT-JEE (mains & advanced), AIEEE , NTSE , KVPY , and others engineering entrance examination.

PROPERTIES AND SOLUTIONS OF TRIANGLE
** POINTS TO BE REMEMBERED
*  LAWS OF SINE OR SINE RULE-     The sides of a triangle are proportional to the sines of the opposite angles. That is , In a ∆ABC, we have;                     a/sinA = b/sinB = c/sinC = k Where k is some constant.
* LAWS OF COSINES OR COSINES RULE-    In any ∆ABC , We have;  a). CosA = (b^2 +c^2 - a^2)/2bc  b). CosB = (c^2 + a^2 - b^2)/2ac  c). CosC = ( a^2 + b^2 - c^2)/2ab
* PROJECTION FORMULA - In any ∆ABC a). a = bCosC + cCosB b). b = aCosC + cCosA c). c = aCosB + bCosA
* LAWS OF TANGENT OR TANGENT RULES(Napier's Analogy)
a). tan(B-C/2) = (b-c/b+c).cot(A/2) b). tan(A-B/2) = (a-b/a+b).cot(C/2) c). tan(C-A/2) = (c-a/c+a).cot(B/2)
* HALF ANGLE FORMULAE OR SEMI-SUM FORMULAE
a). sin(A/2) = √(s-b)(s-c)/bc ,       sin(B/2) = √(s-c)(s-a)/ac       sin(C/2) = √(s-a)(s-b)/ab
b). cos(A/2) = √s(s-a)/bc       cos(B/2) = √s(s-b)/ac       cos(C/2) = √s(s-c)/ab
c). tan(A/2) = √(s-b)(s-c)/s(s-a)      tan(B/2) = √(s-c)(s-a)/s(s-b   …

"PROBLEM BASED ON AGES". | Important SHORTCUT and TRICKS for all competitive examination.

Today we are going to talk about some important arithmetic formulae on PROBLEM BASED ON AGES , which is very important for types of competitive examination. Students are advised to learn these formulae and keep practicing with it.

SOME IMPORTANT SUTRAS
1). If t yrs earlier , the age of the father is x times the age of his son. At present time , the age of father is y times the age of his son. Then their present age ; son age = t(x-1)/x-y , father age = y[t(x-1)/x-y] years respectively.
2). If the present age of the father is x time the age of his son, t years later , the age of father become y times the age of his son , then their present age ;    Son age = x[t(y-1)/x-y] and father age = t(y-1)/x-y   years respectively.
3). If t1 yrs earlier , the age of the father is x times the age of his son. and hence t2 years later, the age of father is y times the age of his son. Then their present age ;  Son age = t1(x-1)+t2(y-1)/x-y , and father age =    t1y(x-1)+t2x(y-1)/x-y
4). If the ratio …

NUMBER SYSTEM Shortcut and tricks for all competitive examination.

Today we are going to talk about some important arithmetic formulae on the chapter "number system". Which are very useful to all types of competitive examination. Readers are advised to learn all the formulae and keep practicing with it.

SOME IMPORTANT SUTRAS

1). Dividend = (divisor × quotient) + remainder 2). Quotient = (dividend - remainder)/divisor 3). Divisor = (dividend - remainder)/quotient 4). Remainder = dividend -(divisor × quotient) 5). Sum of n consecutive natural number=          n(n+1)/2
6). Sum of squares of n even numbers =    [n(n+1)/2]^2
7). Sum of multiples of x till to n = x.n(n+1)/2
8). Sum of squares of n odd numbers=                                                            n(2n-1)(2n+1)/3 9). Sum of squares of consecutive natural numbers till n =  n(n+1)(2n+1)/6
10). Sum of cubes of consecutive natural numbers till n = [n(n+1)/2]^2
11). Sum of consecutive even numbers till n =        n(n+1)/4
12). Sum of consecutive n odd numbers = n^2
13). Sum of con…

when and why do stars get collapse | what happened to the stars , after the collapsion.

Today we are going to talk about the struggle of the stars, not only the life of human beings are full of struggle ,whereas the star's life is also full of struggle. As same as living things , stars also born,  become old and finally get death.
When any living things get death, then what we call it, probably we called that his/her soul get escaped from their body. But what will we call in the context of star death? In the reference to stars ,it is little different. COLLAPSING  is the term which is used to describe the death of stars. So today we are going to understand that why do the stars get collapse? First of all let's understand the meaning of collapse and sometime it is also called Gravitational collapse. Collapsing means pulled up of all the mass into a small point due to high Gravitational force.
Gravitational collapse is process due to which an astronomical object like stars, get contracted towards its centre of gravity in the influence of own gravity. Gravitational …

The Simplified concepts of pendulums | Types of pendulums , and derivation of their Time Periods.

Today we are going to talk about the various types of pendulums and their time periods. Pendulums plays a very important role in simple harmonic motion physics that is also called oscillation. So for everyone who read and understand physics, it is necessary to understand the underlying concept of pendulums. So here we are going to cover a detailed talk on various types of pendulums. So stay tuned with us till end.
So let's starts...
First of all we have to understand that, what is pendulum? Pendulum is derived from  a Latin word 'pendulus'  which means hanging. So, A pendulum is nothing but it is only a arrangements in which, when a weight is suspended from a pivot with a inextensible cord such that it can swing freely back and forth after applying a small force on the weight.  When no any external force is applied in the bob, then  pendulum remains in rest position. And the position where it remains in rest is called mean position. When a pendulum is swinging back and fo…

CORONA VIRUS, history of origin , discovery , infection mechanism, symptoms and treatment.

Today we are going to talk about a virus , which is spreading very fastly all over the world. The virus which we are going to talk about is the CORONA VIRUS. So today we will talk about everything of this virus. So let's starts ...

OVERVIEW OF CORONAVIRUS
According to the biological study , Coronavirus is a cluster of viruses that causes diseases in birds and mammals. Therefore humans are also mammals then in human being this viruses cause respiratory infections , and one of the respiratory infections is mild common cold. Coronavirus can lead to diarrhea in cows and pigs but in chicken they can cause upper respiratory infections. Currently there is no vaccine or antiviral drugs for the treatment of diseases caused by Coronavirus.
BIOLOGICAL INTRODUCTION OF coV
The family of Coronavirus is coronaviridae, and it's subfamily is Orthocoronavirinae and order is Nidovirales, Coronavirus is a member of Orthocoronavirinae subfamily. All Coronavirus is coated with positive sense single …