There are two ways to solve the problem, one is of-course by going through the basic concepts and principles while there’s a one more method known as trial and error or approximation. Later method is more about applying common sense to solve a problem. Tough exams like IIT JEE often needs such capability to derive a solution in a very limited time period. It sounds scary but in reality its a fun and it makes solution easier.
Let me now explain above with an example. I am trying to find the formula for surface area of a frustum.
Conventional process is to make it as full cone and try to use Pythagoras theorem in various triangles to come up with a formula for surface area. Its good but quite a long way to solve this problem. Now let me tell you a shorter solution.
Surface area = perimeter x height – for any regular shape
How to use above simple formula for a irregular shape like frustum? Lets put it like below as an approximation:
Perimeter = average perimeter of frustum = (2 pi R + 2 pi r)/2 = 2 pi (R+r)/2
Height = slant height = 
So surface area of frustum = average perimeter X slant height = 2 pi (R+r)/2 X 
=> pi (R+r) X 
Isn’t it simple? Lets see how similar reasoning helps solving IIT questions. Below question is from IIT 2004 paper:
The sides a, b and c of a triangle are in the ratio 1 : √3 : 2. Then the angles A,B, and C of the triangle are in the ratio?
But how do we proceed further? One way is to draw a triangle and try solving it but its a long and cumbersome process.Another easier way is:
We all know that SinA: SinB: SinC :: a:b:c
or SinA: SinB: SinC :: 1 : √3 : 2
=> SinA: SinB: SinC :: 1/2 : √3/2 : 2/2 – Dividing by 2
=> Above gives A = 30, B = 60 and C = 90
Is above correct? one way to verify is A+B+C = 30+60+90 = 180, yes it looks right!
So answer is angles are in ratio of 1:2:3.
So next time you have limited time and trying to answer a rigorous question then it makes sense to give a thought on approximation method which yields a faster result.
MATH MADE EASY 