Category Archives: IITJEE Foundation mathematics

You and your research ( You and your studies) : By Richard Hamming, AT and T, Bell Labs mathematician;

Although the title is grand (and quite aptly so)…the reality is that it can be applied to serious studies for IITJEE entrance, CMI entrance, highly competitive math olympiads, and also competitive coding contests…in fact, to various aspects of student life and various professional lifes…

Please read the whole article…apply it wholly or partially…modified or unmodified to your studies/research/profession…these are broad principles of success…


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

References for IITJEE Foundation Mathematics and Pre-RMO (Homi Bhabha Foundation/TIFR)

  1. Algebra for Beginners (with Numerous Examples): Isaac Todhunter (classic text): Amazon India link: 
  2. Algebra for Beginners (including easy graphs): Metric Edition: Hall and Knight Amazon India link:
  3. Elementary Algebra for School: Metric Edition:
  4. Higher Algebra: Hall and Knight: Amazon India link:
  5. Plane Trigonometry: Part I: S L Loney:

The above references are a must. Best time to start is from standard VII or standard VIII.

-Nalin Pithwa.

Binomial Theorem: part 2: algebra for math olympiads or IITJEE maths

Binomial Theorem: Part I: A Primer for mathematical olympiads and IITJEE/RMO

Arithmetic Progressions: A Primer for mathematical olympiads and IITJEE mathematics

a gentle introduction…it reminds us that math is about detecting and discerning patterns…whether in numbers, symbols, shapes/figures, or even logical reasoning…regards, Nalin Pithwa.

Harmonic Progressions: A Primer for IITJEE maths and olympiad mathematics

Simultaneous Quadratic Equations: part III: IITJEE algebra

Simultaneous Quadratic Equations: part II: IITJEE maths

Simultaneous Quadratic Equations: part I: IITJEE algebra or mathematics