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Lesson Posted on 23/09/2018 CBSE/Class 12/Science/Mathematics CBSE/Class 12/Science/Mathematics/Unit III: Calculus IIT JEE/IIT - JEE Advanced/Mathematics/Differential calculus +2 Exam Coaching/Diploma CET Coaching Exam Coaching/COMEDK Coaching less

Om Prakash Mishra

I am teaching Mathematics and Physics from past more than 12 years in coaching as well as home tuition...

To find maximum and minimum value of a given function. Differentiate the function first and then find the values of x at which the function becomes zero these are the possible points of Maximum and minimum values of the function then apply any of the following test (1) 1st derivative test: If at any... read more

To find maximum and minimum value of a given function.

Differentiate the function first and then find the values of x at which the function becomes zero these are the possible points of Maximum and minimum values of the function then apply any of the following test

(1) 1st derivative test: If at any of the above values of x dy/dX changes its sign from +ve to -ve then that is a point of local Maxima.

And if dy/dX changes sign from -ve to +ve that is a point of local minima.

(2) 2nd derivative test: Find the 2nd derivative and put the above values of x obtained if it comes -ve that is a point of local Maxima and if it comes +ve that's a point of local minima.

Note: there can be more than 1 point of local Maxima and Minima

Global Maximum and Minimum: put all the values of x obtained in the function and check which one is largest, that value correspond to Global Maxima and that vale is global maximum. Similarly check for Global minima.

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For writing NATA, from when I should start preparing?

Lesson Posted on 04/08/2018 IIT JEE/IIT - JEE Advanced/Physics/Mechanics

Abhilash K S

Faculty for Engineering Mathematics in a private tuition center. Faculty for CAT/GMAT/GRE/SAT. Faculty...

In kinematics, graph plays an important part in understanding various motions. Commonly taught graphs for beginners are distance Vs time and displacement Vs time graph of rectilinear motion, in particular when an object is thrown vertically upwards. The displacement Vs time graph in such a case... read more

In kinematics, graph plays an important part in understanding various motions.

Commonly taught graphs for beginners are distance Vs time and displacement Vs time graph of rectilinear motion, in particular when an object is thrown vertically upwards.

The displacement Vs time graph in such a case is, a parabola mouth opening downwards. The derivative of this graph gives the value of velocity which gradually decreases to zero at its highest point then increases with a negative sign. But the double derivative which gives the value of acceleration remains constant throughout which is the value of g, i.e. acceleration due to gravity.

The shape of distance Vs time graph is a bit different. It remains same as that of displacement Vs time till first half then the curvature of the graph changes. Mathematically, such a point is called the point of inflexion where the curvature of graph changes. The important point to note here is that at the point of inflexion the double derivative is zero.

If we compare this graph with the previous one, the value of double derivative remains the same at all points except at the point of infection. It is so because the double derivative of distance Vs time graph gives "tangential acceleration". At the highest point, the velocity of the body is zero so is the value of tangential acceleration. A point which otherwise missed, if graphs are not studied properly.

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Lesson Posted on 03/08/2018 IIT JEE/IIT - JEE Advanced/Physics/Modern Physics CBSE/Class 11/Science/Physics

Do you know that whenever energy is added to the system, the system gains mass? Is there any example of conversion of energy to mass and vice - versa? Yes. A spring's mass increases whenever it is put into compression or tension. Its added mass arises from the added potential energy stored within... read more

Do you know that whenever energy is added to the system, the system gains mass?

Is there any example of conversion of energy to mass and vice - versa? Yes.

A spring's mass  increases whenever it is put into compression or tension.  Its added  mass arises from the added potential energy stored within it, which is  bound in the stretched chemical (electron) bonds linking the atoms  within the spring.

Actually neither mass gets converted to energy nor energy gets converted to mass. Both are equivalent to each other. If you have some energy E, then you also have mass E/c^2. Similarly, if you have mass m, you have energy mc^2. So, when a high energy gamma photon converts to an electron and a positron in the presence of a nucleus, some books wrongly say that energy has got converted to matter. The right thing is: Total energy of the gamma photon, E= h nu has mass h nu/c^2 and is equal to 2 m c^2/sqrt(1-v^2/c^2) where m is the rest mass of the electron, and v is the speed of electron (positron) in the rest frame of the nucleus.

Let's look at the reverse example.

An atom is made up of number of protons, neutrons and electrons. Each element has a mass associated with it. So logically the mass of an atom should be equal to the sum total masses of all three elements multiplied by the number of such elements present in the atom. Surprisingly, it is observed that the mass of an atom is less than the sum total masses of all elements. Where does the mass go? This mass is converted in to Energy called Binding Energy of the atom and this energy keeps the protons bound together in a nucleus which otherwise should have dismantled in to the environment due to strong repulsive forces between protons. Thus mass is converted in to the energy.

There are many more such examples in the wonderful nature and physics does the job of discovering and explaining such phenomena. In short Physics is Life and Life is Physics.

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Lesson Posted on 12/07/2018 CBSE/Class 11/Science/Physics IIT JEE/IIT - JEE Advanced/Physics/Modern Physics IIT JEE/IIT - JEE Mains/Physics/Physics and Measurement

This is the foremost important physics topic. Students very often than not forget Conversion of Units of physical quantities to one system while solving numerical questions. This leads to wrong answers and failure in competitive exams. Such errors or failures also cause disaster in real life making the... read more

This is the foremost important physics topic. Students very often than not forget Conversion of Units of physical quantities to one system while solving numerical questions. This leads to wrong answers and failure in competitive exams. Such errors or failures also cause disaster in real life making the society pay a very heavy price.

Physics is fundamentally an empirical science. Measurements, units & uncertainties are at the heart of it. All laws of nature are derived empirically and expressed by mathematical equations. How otherwise scientists can express relationships between measurable physical quantities? It is equally vital to express the measurement in right unit; else invite disaster.

The Mars Climate Orbiter costing about US\$ 125 million came to a catastrophic conclusion because of confusion in UNITS. One engineering team used Metric Units while another team used English Units. As a result in September 1999 the spacecraft entered Martian atmosphere instead of reaching a stable orbit.

Equally important is Uncertainties in measurements e.g. pair of pants of size 34 that varies just 3 percent changes a full inch in waist size. It could end up a 35 and hang on the hips or a 33 making you wonder how you gained all that weight!

Measurements, Units and Uncertainties are crucially important and when properly implemented can create wonders in Life (and also exams); else a disaster. Students should pay utmost importance to Units, Conversion of Units and Uncertainties.

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Lesson Posted on 05/06/2018 CBSE/Class 11/Science/Physics Exam Coaching/Engineering Entrance Coaching/IIT JEE Coaching IIT JEE/IIT - JEE Advanced/Physics/Modern Physics

Vishesh Nigam

I am a Chemical Engineer and Khan Academy Talent Hunt Finalist (2017)(please watch my winning video below...

Linear Motion also called rectilinear motion is a one-dimensional motion along a straight line, and can, therefore, be described mathematically using only one dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; nonuniform linear motion... read more

Linear Motion also called rectilinear motion is a one-dimensional motion along a straight line, and can, therefore, be described mathematically using only one dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; nonuniform linear motion with non-zero acceleration or variable velocity. The motion of a particle along a line can be described by its position x, which varies with t (time). An example of linear motion is an athlete running 100m along a straight track.

Check out the webinar video, to understand in-depth.

Have a question? Post your question or comment below.

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Lesson Posted on 04/06/2018 CBSE/Class 10/Mathematics IIT JEE/IIT - JEE Advanced/Mathematics/Integral calculus IIT JEE/IIT - JEE Mains/Mathematics/Integral Calculus +2 CBSE/Class 9/Mathematics Integral Calculus less

Indrajit

I am successfully providing tuition of Mathematics in Class IX-XII of all Boards near Kavi Najrul Metro...

Before starting the discussion I would like to mention this problem The conventional way to solve this problem is to deal with trigonometric identities and to mange by parts, but this type of problem can be easily solved by applying results of... read more

Before starting the discussion I would like to mention this problem

$\int_0^{\pi/2}\cos^4x\sin^2x=?$

The conventional way to solve this problem is to deal with trigonometric identities and to mange by parts, but this type of problem can be easily solved by applying results of Beta and Gamma Functions.

———————————————————————————————————————————————————

Gamma Function:

The gamma function denoted by  $\Gamma&space;(n)$ is defined for positive values of $n$ by the integral

$\Gamma(n)=\int_0^{\infty}e^{-x}x^{n-1}dx,n>0$                                                          .......(1)

Now,

1. For any $a>0.$

$\int_0^\infty&space;e^{-ax}x^{n-1}dx=\frac{&space;\Gamma(n)}{a^n},n>0$

2.

${\color{Red}&space;\Gamma(n+1)=n\Gamma(n),n>0}$

Integration by parts gives

$\int&space;_{\epsilon}^{B}e^{-x}.x^{n-1}dx=&space;[e^{-x}&space;.\frac{x^n}{n}]_{\epsilon}^B+\frac{1}{n}\int_\epsilon^Be^{-x}x^ndx$

As $B\to&space;\infty$ and $\epsilon&space;\to&space;0+$, the integreted part vanishes at both limits and therefore,

$\int_0^\infty&space;e^{-x}x^{n-1}dx=\frac{1}{n}\int_0^{\infty}e^{-x}x^ndx$

i.e.

$\Gamma(n)=\frac{1}{n}\Gamma(n+1)$

3.     $\Gamma(1)=1.$

By direct computation   $\Gamma(1)$ converges

$\Gamma(1)=\int_0^{\infty}e^{-x}dx=\lim_{B\to\infty}\int_0^Be^{-x}dx=\lim_{B\to\infty}(1-e^{-B})=1$

4. $\Gamma(n+1)=n!$

Combining the above reletions,

$\Gamma(n+1)=n\Gamma(n)=n(n-1)\Gamma(n-1)=\cdots$

$=n(n-1)(n-2)\cdots3.2.1..\Gamma(1)=n!$

Beta Function:

The beta function denoted by  $B(m,n)$ is defined for positive values of m and n by the integral

$B(m,n)=\int_0^{1}x^{m-1}(1-x)^{n-1}dx,&space;m,n>0$

1.      $B(m,n)=B(n,m)$

This reletion can be established by giving the transformation  $x=1-y$

2.

$B(m,n)=\int_0^{\infty}&space;\frac{x^{m-1}}{(1+x)^{m+n}}dx=\int_0^{\infty}&space;\frac{x^{n-1}}{(1+x)^{m+n}}dx=B(n,m)$

Substituting

$x=\frac{1}{1+y}$

,then

$\int_{\epsilon}^{1-\delta}x^{m-1}(1-x)^{n-1}dx&space;=&space;\int_{\frac{1}{\epsilon}-1}^{\frac{\delta}{1-\delta}}\frac{1}{(1+y)^{m-1}}.\frac{y^{n-1}}{(1+y)^{n-1}}.(-1).\frac{1}{(1+y)^2}dy$

Now letting    $\epsilon\to\0+,\delta\to0+$, we have

$\int_0^1x^{m-1}(1-x)^{n-1}dx=B(m,n)=\int_0^{\infty}\frac{y^{n-1}}{(1+y)^{m+n}}dy$

also,

$B(m,n)=B(n,m)$.

3.

$B(m,n)=2\int_0^{\pi/2}&space;\sin^{2m-1}\theta&space;\cos^{2n-1}\theta&space;d\theta;&space;m,n>0$

This reletion can be established by by giving the transformation $x=\sin^2\theta$ in the definition of Beta Function.

4.

$B(\frac{1}{2},\frac{1}{2})=\pi$

This can be established by putting

$m=n=\frac{1}{2}$

in

$B(m,n)=2\int_0^{\pi/2}&space;\sin^{2m-1}\theta&space;\cos^{2n-1}\theta&space;d\theta;&space;m,n>0$

5.

$B(m,n)=\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)};&space;m,n>0$

so,

$\Gamma(\frac{1}{2})=\sqrt{\pi}$

—————————————————————————————————————————————————————

Now Lets solve the problem:

$\int_0^{\pi/2}\cos^4x\sin^2x=?$

Now to solve this problem plug

$m=\frac{3}{2},n=\frac{5}{2}$

in

$B(m,n)=2\int_0^{\pi/2}&space;\sin^{2m-1}\theta&space;\cos^{2n-1}\theta&space;d\theta;&space;m,n>0$

$\int_0^{\pi/2}\cos^4x\sin^2x=\frac{1}{2}B(\frac{3}{2},\frac{5}{2})=\frac{\Gamma(\frac{3}{2}).\Gamma(\frac{5}{2})}{\Gamma(\frac{3}{2}+\frac{5}{2})}$

=

$\frac{\frac{1}{2}.\Gamma(\frac{1}{2}).\frac{3}{2}.\Gamma(\frac{3}{2})}{\Gamma(4)}=\frac{\pi}{32}$

As

$\Gamma(\frac{1}{2})=\sqrt{\pi}$

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Lesson Posted on 23/05/2018 CBSE/Class 11/Science/Physics IIT JEE/IIT - JEE Advanced/Physics/Modern Physics Exam Coaching/Engineering Entrance Coaching/IIT JEE Coaching +1

Vishesh Nigam

I am a Chemical Engineer and Khan Academy Talent Hunt Finalist (2017)(please watch my winning video below...

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Lesson Posted on 05/05/2018 IIT JEE/IIT - JEE Advanced/Chemistry/Physical Chemistry

Prashant R.

I am a B.tech in Chemical Engineering from IIT Bombay & I have 5 years of experience in teaching & mentoring...

Mole is the foundation of Chemistry. It is the fundamental thing, yet many students get difficulty in understanding this concept & there are various reasons for different students.In this lesson, I am going to talk about mole concept in very basic detail. The mole is the unit of measurement for the... read more

Mole is the foundation of Chemistry. It is the fundamental thing, yet many students get difficulty in understanding this concept & there are various reasons for different students.

In this lesson, I am going to talk about mole concept in very basic detail.

The mole is the unit of measurement for the amount of substance in the International System of Units (SI). Just like kg for mass, litre for volume. we can relate mole to a dozen in an interesting way as we define dozen as an amount which contains 12 number of units of anything in the same way 1 mole is the amount of any substance which contains 6.022 × 10^23 entities

here entities can be anything like atoms, molecules, ions, electrons, protons or neutrons

e.g.                1 mole = 6.022×10^23 atoms

1 dozen = 12 atoms or bananas

Standard definition - it is the amount of substance which contains the same number of entities as there are in 12 gm of the C-12 atom.

further, it was found that number of entities are 6.022×10^23

Carbon-12 is the most abundant stable isotope of carbon, and it is used for measuring atomic masses of all nuclides.

Relation of moles with mass, Volume & Number of entities

as we know,

1 mole contains 6.022×10^23 entities

number of moles = Number of entities⁄6.022×10^23

in the same way as 12 gm of carbon contain 1 mole

then we can write

no of moles = mass in grams/molecular mass(atomic mass)

and by Avogadro's law which says the equal volume of gases contain an equal number of molecules irrespective of nature of gases

we can write

number of moles=Volume at STP(litre)/22.4

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Lesson Posted on 16/04/2018 Exam Coaching/Engineering Entrance Coaching/IIT JEE Coaching Exam Coaching/UPSC Exams Coaching/NDA (National Defence Acadamy and Naval Acadamy) Exam Coaching Exam Coaching/Engineering Entrance Coaching +2 IIT JEE/IIT JEE MAINS/Mathematics IIT JEE/IIT - JEE Advanced/Mathematics less

Deepak

I have 7 year teaching experience with IIT as well as board classes ( 10th , 11th , and 12th ) for maths...

These are my few suggestions for 2018 Jee advance aspirants on how to begin JEE-ADVANCED 2018 Preparation.My wishing you best of luck a day before the exam will not work, you will have to study in next 40 days as a commando because you are on a mission to launch your self into the real world.First solve... read more

These are my few suggestions for 2018 Jee advance aspirants on how to begin JEE-ADVANCED 2018 Preparation.
My wishing you best of luck a day before the exam will not work, you will have to study in next 40 days as a commando because you are on a mission to launch your self into the real world.
First solve last five years JEE(A) papers in first five days in examination like conditions, after papers continue your regular scheduled study.
It will give you exposure to the topics that you feel strongly, i.e. you will have a battle plan of about 75 % syllabus first.
Identify topics that you are weak and questions have been asked very frequently from it. Divide these into four units, read class notes ( whatever I taught and stuck yourself on the chapter which you have done yourself) on them, solve two basic exercise and JEE questions on them and then move on. It will ensure that if an easy problem comes from it, you will be able to crack it.
Make your daily routine a very disciplined one like a soldier. Get up by 7 am in the morning, get ready like you are going to school have to breakfast by 8 am. Now you are on a mission. Study continuously in three-hour sittings. Take proper lunch, avoid napping, the slot of 2 pm to 5 pm must be study hours. Try to take two full JEE LEVEL tests every week in examination conditions.

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