One of the things that weighs constantly on CAT test-takers is how they can increase their speed and accuracy. More often than not test-takers discover the right way to solve a problem mid-way through the problem – after they have spent about 2 minutes following one approach. So what is the best way to increase speed and accuracy?
Identify the right way to approach a problem before you start solving it.
Let us examine how to do this by taking an old SimCAT problem.
Given that f(x) = sec x – tan x. If f(x) = k, then what is the value of cosec x?
Rule # 1: Do not leave this problem thinking — “It is Functions + Trigonometry!“
As we discussed in a previous post, CAT problems are deliberately designed to be seemingly tough or unapproachable! This problem has nothing to do with functions, it is just a fancy way of saying if sec x – tan x = k then cosec x = ?
Rule # 2: Do not starting to solving the problem hoping something would happen.
Usually those who are not scared by Trigonometry and are know their way around the formulae involved, start by simplifying it into sin and cos. Now once they do this they get (1 – sin x)/cos x (if you are doing this you should do this part done without putting pen on paper!).
They then think about what they CAN do — say multiply and divide numerator and denominator by (1 + sin x) and go ahead and do it. Now as far as the approach goes, there has been no effort to concretely connect it to arriving at the solution or linking it to what is required, cosec x. The steps are carried out more with hope — may be things will get cancelled in the end — rather than with conviction. In short they keep trying what they CAN do and not what they SHOULD do!
You should start taking a particular line of solving only after you have examined where it will lead you.
A better way of thinking/talking your way through the problem is to ask yourself:
- Do I know if in any way the expression sec x – tan x is connected to cosec x?
- I know sec2 x – tan2 x = 1 but that does not really have to do anything with cosec x, does it?
- So if from the question stem, I don’t know which path to take, what should I turn to?
The answer to the last question is obviously not to turn to the next question — you have answer options, remember?
Once you take a look at them you see that 3 out of 4 options revolve around k2. So it makes sense to first see what k2 will turn out to be.
Even for this step do not write the value of k, draw brackets around it and write a 2 in superscript! Straight away expand it in your head and write down the final expansion.
Now see what happens when you do 1 + k2 and when you do 1 – k2. Simplify both of these expressions with your end goal and answer options in view:
- If k =(1 – sin x)/cos x, then when you know you can ignore the denominator since it will get cancelled in all options.
- Since you need cosec x, which is 1/sin x, simplify the numerator to contain only sin x forms by writing all cos x forms that you get in terms of sin x by using sin2 x + cos2 x = 1
Once you do this you will get 1 + k2 = 2 – 2 sin x and 1 – k2 = 2 sin x – 2 sin2 x = sin x (2 – 2 sin x).
From this you can see that to get cosec x, which is 1/sin x, you have to divide the 1 + k2 by 1 – k2. Hence option (2).
To improve speed & accuracy not only should you connect not only your approach but also your solving process to what you need calculate or what the question is asking for. This will ensure that you are optimizing you solving process at every stage to reach the answer in the shortest possible time.
Remember solving a CAT problem is not just about using logic to figure out how to solve it but using logic to optimize the solving process as well.