Wednesday, October 15, 2014

CAT Preparation Online - A simple question from Quadratic Equations

Questions

x^2 - 17x + |p| = 0 has integer solutions. How many values can p take?

Explanation

To begin with, if we assume roots to be a and b, sum of the roots is 17 and product of the roots is |p|. Product of the roots is positive and so is the sum  of the two roots. 

So, both roots need to be positive.

So, we are effectively solving for 

Number of positive integer solutions for a + b = 17.

We could have (1,16), (2, 15), (3, 14)......(8, 9) - There are 8 sets of pairs of roots. Each of these will yield a different product.

So, |p| can take 8 different values. Or, p can take 16 different values.

is that it? Or, are we missing something? Can p be zero? What are the roots of x^2 - 17x = 0. This equation also has integer solutions. 

So, p can also be zero.

So, number of possible values of p = 16 + 1 = 17.

Wonderful question - chiefly because there are two really good wrong answers one can get. 8 and 16. So, pay attention to detail. No point telling yourself "Just missed, I just didnt think of zero. I deserve this mark". Being just wrong, will earn us a -1 instead of the honourable 0.


Tuesday, October 14, 2014

CAT Preparation Online - Simple one from Quadratic Equations

Question

How many integer solutions exist for the equation x2 - 8|x| - 48 =0?

Explanation


One approach is to solve when x > 0 and then solve for x < 0. However, there is a slightly simpler method.

Note that x2 is the same as |x|2, so we can treat the equation as a quadratic in |x|.

Or, |x|2 – 8|x| - 48 = 0
(|x| - 12|) (|x| + 4) = 0
|x| could be 12. |x| cannot be -4.

If |x| could be 12, x can be -12 and 12.


Two possible solutions exist. 

CAT - Question from Pipes and Cisterns

Question

There are n pipes that fill a tank. Pipe 1 can fill the tank in 2 hours, Pipe 2 in 3 hours, Pipe 3 in 4 hours and so on. Pipe 1 is kept open for 1 hour, pipe 2 for 1 hour, then pipe 3 and so on. In how many hours will the tank get filled completely?

Explanation


Almost all questions can be approached well by asking the question "What happens in 1 hour" (or 1 day, or 1 minutes)


So, let us start with that


In 1 hour, pipe 1 fills 1/2 of the tank. So, in the first hour, the tank will not be filled

In 1 hour, pipe 2 fills 1/3 of the tank. So, in two hours we would have filled 1/2 + 1/3 of the tank, or 5/6 of the tank. So, by the end of the second hour, the tank would still not be filled.
Let us move to the third hour. In 1 hour, pipe 3 fills 1/4 of the tank. So, by the end of the third hour, we should have filled 5/6 + 1/4 = 13/12 of the tank.

Oops, one cannot fill 13/12 of a tank. What this tells us is that the tank gets filled in the 3rd hour. 


When exactly during the third hour?


At the beginning of the third hour, we still have 1/6 of the tank still to fill. Pipe 3 can fill at the rate of 1/4 of the tank per hour. Or, pipe 3 will take 2/3 hours to fill the tank.


Or, the tank will be filled in 2 hours and 40 minutes.


This is an example of a sequence that is a harmonic progression. The formulae for HP are needlessly confusing. So, simple step-by-step approach works best. 

Friday, October 10, 2014

CAT Permutation Combination and Functions

Question

How many functions can be defined from Set A -- {1, 2, 3, 4} to Set B = {a, b, c, d} that are neither one-one nor onto? 

Explanation

To start with, if you do not know the meaning of one-one or onto, look these up. For good measure know the meanings of the terms surjective, injective etc also. CAT tests these terms.

Let us start by answering a far simpler question. How many functions are possible from Set A to Set B. This is equal to 4^4 = 256. 

Note that any function from Set A to Set B that is one-one will also be onto and vice versa. How? Why? - Think about this. Remember, tutors can only take the horse to the pond. :-)

So, we need to subtract only those functions that are one-one AND onto. Or, effectively we have to eliminate those functions where {1, 2, 3, 4} are mapped to {a, b, c, d} such that each is mapped to a different element. This is effectively same as the number of ways of rearranging 4 elements. Or, number of ways of doing this is 4! .

So, total number of functions that are neither one-one nor onto = 256 - 24 = 232.