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Two distinct numbers x and y are chosen at random from the first 30 natural numbers. What is the probability that a2 - b2 is divisible by 3?
Well, instead of giving my clumsy solution, I'd provide the one from Karthik.
Hint: 3|(a2 - b2)
=> 3|(a - b)(a + b)
=> 3|(a - b) or 3|(a + b) or both
Find the doublets (a, b) such that either a - b or a + b or both are divisible by 3.
Class 1: {1, 4, 7, ... }, {2, 5, 8, ... }, {3, 6, 9, ... }. Each set has 10 numbers. Total number of such combinations is 3 * (10 * 9) / 2 = 135
Class 2: {1, (2, 5, 8, ..)}, {2, (4, 7, ...)}, ...
Now look at the three sets {1, 2, ... }, {2, 4, ... }, {3, 6, ... }. In these sets, we have covered all numbers in range [1, 30]. Forgetting about the third set, to find two numbers such that their sum is divisible by 3, all we need to do it pick one number from the first set and another from the second.
Total number of combinations = 10 * 10 = 100
Class 3: The combinations of {3, 6, 9, ... } will be repeated in both. So, we need to count them only once. We have already counted in Class 1, so we'll not count again.
Total number of picking two numbers is 30C2 = 435
Probability =
(135 + 100) / 435=235/435=470/870=47/87
Scenario: A body is moving in a circular motion with constant speed. A practical example can be -- a car moving in circular motion with constant speed. The surface of motion is horizontal.
Problem: In which direction does the friction force apply? Assume there is only pure rolling at the contact surface.
Solution: Have a look at the following diagram that demonstrates the change in velocity (Borrowed from Physics Classroom).
The magnitues of vi and vf are same and as the points A and B get closer, the direction of dv becomes radial.
Conclusion 1: The net acceleration of the body in this scenario is radial and directed towards the center. Or in other words, the net force on the body is towards the center.
Now, look at the following diagram.
I have not mentioned about gravitational forces and reaction from ground thereof since that is not of any importance here.
Fn denotes the net force (radial).Fm denotes the force by the motor (or engine) of the car.Ff denotes the frictional force.
Here's the simple relation between the three forces:
Fn = Fm + Ff
The magnitude of Ff depends on the magnitude of the speed and the force by the engine.
Conclusion 2: The frictional force is oblique to the direction of motion and directed inwards towards the center (not radially).
Thanks, Varun, for your mail to the group regarding the new format.
The pattern applicable for JEE 2006 is available here
As per the latest notification from the Joint Admission Board of the IITs, the students who have passed their qualifying examination in 2005 or earlier will be permitted to appear in JEE 2006.
A long story whose sweet and short summary is that as far as JEE 2006 is concerned, there is no change in the pattern. Not even as far as the format of the question paper is concerned. It remains same as of last year.
Full news can be read at NDTV news item.
More information about JEE 2006 can be found at the IIT Kanpur JEE website.
Thanks to the dedicated team of volunteers from whom the group has become a passion and may be on way to becoming an obsession, the Free JEE Coaching Group has made a news in newspapers.
A couple of citings are given below:
I should be seen back in action starting next week. I planned last week but had to rush out for a week and this weekend too... I'd be flying away. Would be back with some more books from home.
btw, I'm working on a (currently background) project -- putting all problems and solutions of JEE since 1982 online!!! Let me see how much success do I get in this.
Participate in discussion forum at YahooGroup. Chill out with your friends. Take a break before we start for an intensive JEE preparation. Ok!
I have been asked to write about Physics and Chemistry also. There has been a lot of Maths and also that I haven't written for about a month now. Aise to naheen chalega, boss!
Chemistry... I admit that I don't remember even a single word of it except for the C-H-O things. Physics was my strong point and I'll write on it.
I will start with mechanics and start with efficient use of vector in mechanics. In my opinion, as soon as you see something in motion - be it mechanics, electro-magnetics or a combination of them - make an origin, make 3 orthogonal axes, make vectors and then start putting the values as you read the question.
Of course, more details on it... but later.
I beg, DO NOT do this blunder. Well, it's under condition that that the coaching has more than 20-25 people in a batch / class. If not, you may join.
I know if the coaching people read this, they'd be behind me. But it's a fact. More often than not they would attempt to coach more than 100 students in a class. What if you have any doubts? I bet, you won't get an answer in the class. What more... if you get your basics wrong, they will always remain unanswered. Complex questions may be, sometimes, answered.
Join some correspondence test series, like of FIIT-JEE, Brilliant Tutorials etc. A lot more have come since I joined in 1996. The national scale test series give you two benefits:
Oh yes... if you need to join coaching, join a small center where the batch will not be big, the teacher will be enthusiatic to talk and work and your queries will be duly attended.
No... I am not talking about taking food but about her worrying about your board exams or JEE screening. Believe it or not, more often than not, you will be pressurized by your mother or grandmother or some other relative to qualify JEE. If you pay more attention towards JEE, you will get scolding on not studying for your board exams. It happened to me also.
Luckily, for me, my father was with me. He said to me just one thing, which I still remember and probably will never forget. He said, "If you qualify in JEE and pass boards with even 33% marks, nobody will complain. If you don't beat JEE, your board marksheet if everything." I always kept it in my mind.
I still remember it was January of 1998 and my board exams were to be in March / April, 1998. I had some idea of English course but a very little idea of Hindi. I took the board exams with no more than 15 days of exclusive study of these subjects and passed with handsome marks.
But who looks at them. All I have to say now is - "I am an IIT Kanpur Graduate / B.Tech.". That's it.
So, don't listen to anybody on this topic. But also don't overlook them completely. After all, you'll be giving your internal examinations.