Elementary algebra: fractions: IITJEE foundation maths

Expertise in dealing with algebraic fractions is necessary especially for integral calculus, which is of course, a hardcore area of IITJEE mains or advanced maths.

Below is a problem set dealing with fractions; the motivation is to develop super-speed and super-fine accuracy:

A) Find the value of the following: (the answer should be in as simple terms as possible, which means, complete factorization will be required):

1) $\frac{1}{6a^{2}+54} + \frac{1}{3a-9} - \frac{a}{3a^{2}-27}$

2) $\frac{1}{6a-18} - \frac{1}{6a+18} -\frac{1}{a^{2}+9} + \frac{18}{a^{4}+81}$

3) $\frac{1}{8-8x} - \frac{1}{8+8x} + \frac{x}{4+4x^{2}} - \frac{x}{2+2x^{4}}$

4) $\frac{x+1}{2x^{3}-4x^{2}} + \frac{x-1}{2x^{3}+4x^{2}} - \frac{1}{x^{2}-4}$ 1

5) $\frac{1}{3x^{2}-4xy+y^{2}} + \frac{1}{x^{2}-4xy+3y^{2}} -\frac{3}{3x^{2}-10xy+3y^{2}}$

6) $\frac{1}{x-1} + \frac{2}{x+1} - \frac{3x-2}{x^{2}-1} - \frac{1}{(x+1)^{2}}$

7) $\frac{108-52x}{x(3-x)^{2}} - \frac{4}{3-x} - \frac{12}{x} + (\frac{1+x}{3-x})^{2}$

8) $\frac{(a+b)^{2}}{(x-a)(x+a+b)} - \frac{a+2b+x}{2(x-a)} + \frac{(a+b)x}{x^{2}+bx-a^{2}-ab} + \frac{1}{2}$

9) $\frac{3(x^{2}+x-2)}{x^{2}-x-2} -\frac{3(x^{2}-x-2)}{x^{2}+x-2} - \frac{8x}{x^{2}-4}$

More later,

Nalin Pithwa

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