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 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. 

CAT Coordinate Geometry Question and Solution

Question

What is the area enclosed by the region defined by y = |x -1| + 2, the line x = 1; X-axis and Y-axis?

Explanation

Solution is available on the video given below. 

Functions Question and Solution

Question

Consider set A with 'a' elements, set B with 'b' elements, set C with 'c' elements. We can define a function that is one-one but not onto from set A to set B, a function that is onto but not one-one from set B to set C; and a function that is injective but not surjective from C to A. Arrange a, b, c in ascending order.

Explanation