Integration By Partial Fractions

If


On comparing the coefficient of different terms
and
, so

Answer (c)

Given,





on putting , we get

on putting , we get

on putting , we get

on putting , we get

We are given the function Step 1: Checking if the function is one-one
A function is one-one (injective) if distinct values of produce distinct values of .
The function involves absolute value terms, and for certain intervals of , the absolute values might produce the same value for different 's.
For example:
- If , .
- If , .
- However, values for different 's could also repeat based on symmetry in the absolute terms, which shows that the function is not one-one.

Step 2: Checking if the function is onto
A function is onto (surjective) if every value in the target set (the range) is the output of some input in the domain. Here, we are given that the range of is . By testing extreme values and considering the nature of the absolute values, we can see that can take any value greater than or equal to , so the function is onto.

Thus, the function is onto but not one-one.

Step 1: Understanding the given curves The given equations represent conic sections. To find the enclosed area, we subtract the two equations: Step 2: Simplifying the expression Step 3: Finding the enclosed area Using the standard formula for enclosed area between two curves: Solving the integral, we obtain: Thus, the enclosed area is .