Author Archives: Nalin Pithwa

I am a DSP Engineer and a mathematician working in closely related areas of DSP, Digital Control, Digital Comm and Error Control Coding. I have a passion for both Pure and Applied Mathematics.

A motivation for Math and some Math competitive exams in India

what motivates you to keep going in math?

Mathematics Hothouse

Sometime back, there was a tremendous publicity in the Indian media to two Fields medallists of Indian origin. They also talked about what motivated them towards Math when they were young. One should  not do Math just lured by its glamorous applications in IT or other engineering disciplines. But, one can develop both aptitude and attitude  towards it if one works from a young age.

What you need is intrinsic motivation. In this context, I like to quote the following words of a famous mathematician:

“And, a final observation. We should not forget that the solution to any worthwhile problem very rarely comes to us easily and without hard work; it is rather the result of intellectual effort of days or weeks or months. Why should the young mind be willing to make this supreme effort? The explanation is probably the instinctive preference for certain values, that is, the attitude…

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Skill Check II: IITJEE foundation maths

Set Theory Primer/basics/fundamentals/preliminaries:

I. Represent the following sets in Venn Diagrams: (a) \Xi = \{ x : x = n, n <40, n \in \mathcal{N} \} (b) \mu = \{ x : x = 6n, n < 6. n \in \mathcal{N}\} (c) \alpha = \{ x: x = 3n, n<8, n \in \mathcal{N} \}

2. If x = \{ x: x<29 \hspace{0.1in}and \hspace{0.1in}prime\} and A = \{ x: x \hspace{0.1in} is \hspace{0.1in} a \hspace{0.1in} prime \hspace{0.1in} factor \hspace{0.1in} of \hspace{0.1in} 210\}, represent A in Venn diagram and find \overline{A}.

3. 95 boys of a school appeared for a physical for selection in NCC and Boy Scouts. 21 boys got selected in both NCC and Boy Scouts, 44 boys were not selected in Boy Scouts and 20 boys were not selected only in boy scouts. Draw Venn diagram and find : (i) how many boys did not get selected in NCC and boy scouts. (ii) how many boys did not get selected only in NCC (iii) how many boys got selected in NCC (iv) how many boys got selected in boy Scouts (v) How many boys got selected in NCC not in boy Scouts?


Nalin Pithwa.

Skill Check I: IITJEE Foundation Maths

  1. Simplify: (+1) \times (-1) + (+1) \div (-1) -()-1 +(-1) \div (-1) \times (-1) +(-1)
  2. Simplify: \{ 7 \hspace{0.1in} of \hspace{0.1in} 6  \div 2 - 4(8 \times 12 \div 3 + \overline{-3 \hspace{0.1in} of \hspace{0.1in}+6 -2 })\} \div (-3)
  3. Simplify: 24 \div (-8) + 3 \times (-3)
  4. Simplify: (+7) - (-3) + (+4) \times (-3) \div (+3) of (-2)
  5. Simplify: (-3) of (-5) \div (-3) \times (-2) + (-5) - (-2) \div (+2)
  6. Simplify: (-7) + (-8) - (-3) \hspace{0.1in} of \hspace{0.1in} (-6) \div (+2) - (-4) \times (-4) \div (+2)
  7. Simplify: (+24) \div (-3) \hspace{0.1in} of \hspace{0.1in} (+4)  - (-25) \times (-6) \div (-3) + (-15) \div (-3) \times (-10)
  8. Simplify: (-3) \hspace{0.1in} of \hspace{0.1in} (-8) \div (-6)  - (-8 +4-3)
  9. Simplify: (-5) [ (-6) - \{ -5 + (-2 + 1 - \overline{3-2})  \} ]
  10. Simplify: (-3) [ (-8) - \{ +7 - (4-5 - \overline{2-5-1})\}] \div (-11)
  11. Simplify: (+8) \times (-3) \times (+2) \div [ -1 - \{ -3 + 8 - (6 -2 - \overline{3+5-4}) \} ]
  12. Simplify: (+32) \div (+2) of (-4) \div [(-7) of 3 \div \{4 - 5(3 - \overline{4 of 2 - 2 of 5}) \}]
  13. Simplify: (-30) + (-8) \div (-4) \time 2
  14. Simplify: (-3) \times (-6) \div (-2) + (-1)
  15. Simplify: 56 \div (16 + \overline{4-6}) + (6-8)
  16. Simplify: (7+6) \times [19 + \{ (-15) + \overline{6-1}\}]


Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai, Maharastra 400101

B.S. in Mathematics: IIT Bombay program:

Note that the admission is through IITJEE Advanced only.

Nalin Pithwa.

Math Basics Division by Zero

the eleventh commandment of Moses: Though shall not divide by zero !!

Mathematics Hothouse

Let’s pause Geometry for a little time and start thinking of some basic rules of the game of Math. Have you ever asked “why is division by zero not allowed in Math?” Try to do 1/2 in a calculator and see what you get!!

This was also a question an immortal Indian math genius, Srinivasa Ramanujan had asked his school teacher when he was a tiny tot. Note the following two arguments against the dangers of division by zero:

(a) Suppose there are 4 apples and two persons want to divide them equally. So, it is 4/2 apples per person, that is, 2 apples per person. But, now consider a scenario in which there are 4 apples and 0 persons. So, how can you divide 4 apples amongst (or by) 0 persons? You can think of any crazy answer and keep on arguing endlessly about it!!!!

(b) The cancellation law…

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IITJEE Foundation Maths: Tutorial Problems IV

  1. Resolve into factors: (a) 2x^{2}-3ab+(a-6b)x (b) 4x^{2}-4xy-15y^{2}
  2. In the expression, x^{3}-2x^{2}+3x-4, substitute a-2 for x, and arrange the result according to the descending powers of a.
  3. Simplify: (i) \frac{x}{1-\frac{1}{1+x}} (ii) \frac{x^{2}}{a+\frac{x^{2}}{a+\frac{x^{2}}{a}}}
  4. Find the HCF of 3x^{3}-11x^{2}+x+15 and 5x^{4}-7x^{3}-20x^{2}-11x-3
  5. Express in the simplest form: (i) \frac{\frac{x}{y}-\frac{y}{x}}{\frac{x+y}{y}-\frac{y+x}{x}} (ii) (\frac{x^{3}-1}{x-1} + \frac{x^{3}+1}{x+1})\div (\frac{1}{x-1} + \frac{1}{x+1})
  6. A person possesses Rs. 5000 stock, some at 3 per cent; four times as much at 3.5 %, and the rest at 4 %; find the amount of each kind of stock when his income is Rs. 178.
  7. Simplify the expression: -3[(a+b)-{(2a-3b) -( 5a+7b-16c)  - (-13a +2b -3c -5d)}], and find its value when a=1, b=2, c=3, d=4.
  8. Solve the following equations : (i) 11y-x=10 and 11x-101y=110 (ii) x+y-z=3, and x+z-y=5, and y+z-x=7.
  9. Express the following fractions in their simplest form: (i) \frac{32x^{3}-2x+12}{12x^{5}-x^{4}+4x^{2}} (ii) \frac{1}{x + \frac{1}{1+ \frac{x+3}{2-x}}}
  10. What value of a will make the product of 3-8a and 3a+4 equal to the product of 6a+11 and 3-4a?
  11. Find the LCM of x^{3}-x^{2}-3x-9 and x^{3}-2x^{2}-5x-12
  12. A certain number of two digits is equal to seven times the sum of its digits; if the digit in the units’ place be decreased by two and that in the tens place by one, and if the number thus formed be divided by the sum of its digits, the quotient is 10. Find the number.
  13. Find the value of \frac{6x^{2}-5xy-6y^{2}}{2x^{2}+xy-y^{2}} \times \frac{3x^{2}-xy-4y^{2}}{2x^{2}-5xy+3y^{2}} \div \frac{9x^{2}-6xy-8y^{2}}{2x^{2}-3xy+y^{2}}
  14. Resolve each of the following expressions into four factors: (i) 4a^{4}-17a^{2}b^{2}+4b^{4}; (ii) x^{8}-256y^{8}
  15. Find the expression of highest dimensions which will divide 24a^{4}b -2a^{2}b^{2}-9ab^{4} and 18a^{6}+a^{4}b^{2}-6a^{3}b^{3} without remainder.
  16. Find the square root of : (i) x(x+1)(x+2)(x+3)+1 (ii) (2a^{2}+13a+15)(a^{2}+4a-5)(2a^{2}+a-3)
  17. Simplify: x - \frac{2x-6}{x^{2}-6x+9} - 3 + \frac{x^{2}+3x-4}{x^{2}=x-12}
  18. A quantity of land, partly pasture and partly arable, is sold at the rate of Rs. 60 per acre for the pasture and Rs. 40 per acre for the arable, and the whole sum obtained at Rs. 10000. If the average price per acre were Rs. 50, the sum obtained would be 10 per cent higher; find how much of the land is pasture and how much is arable.


Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai, Maharastra 400101

Math from scratch

Most important out of all blogs…explains what is math and the nature of its rigour

Mathematics Hothouse

Math from scratch!

Math differs from all other pure sciences in the sense that it can be developed from *scratch*. In math jargon, it is called “to develop from first principles”. You might see such questions in Calculus and also Physics.

This is called Axiomatic-Deductive Logic and was first seen in the works of Euclid’s Elements (Plane Geometry) about 2500 years ago. The ability to think from “first principles” can be developed in high-school with proper understanding and practise of Euclid’s Geometry.

For example, as a child Albert Einstein was captivated to see a proof from *scratch* in Euclid’s Geometry that “the three medians of a triangle are concurrent”. (This, of course, does not need anyone to *verify* by drawing thousands of triangles and their mediansJ) That hooked the child Albert Einstein to Math, and later on to Physics.

An axiom is a statement which means “self-evident truth”. We accept…

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IITJEE Foundation Maths : Tutorial Problems III

  1. When a=-3, b=5, c=-1, d=0, find the value of 26c\sqrt[3]{a^{3}-c^{2}d+5bc-4ac+d^{2}}
  2. Solve the equations: (a) \frac{1}{3}x - \frac{2}{7}y = 8 -2x and \frac{1}{2}y - 3x =3-y (b) 1 = y+z =2(z+x)=3(x+y)
  3. Simplify: (a) \frac{a-x}{a+x} - \frac{4x^{2}}{a^{2}-x^{2}} +  \frac{a-3x}{x-a} (b) \frac{b^{2}-3b}{b^{2}-2b+4} \times \frac{b^{2}+b-30}{b^{2}+3b-18} \div \frac{b^{3}-3b^{2}-10b}{b^{2}+8}
  4. Find the square root of : 9-36x+60x^{2}-\frac{160}{3}x^{3}+\frac{80}{3}x^{4}-\frac{64}{3}x^{5}+\frac{64}{81}x^{6}
  5. In a cricket match, the extras in the first innings are one-sixteenth of the score, and in the second innings the extras are one-twelfth of the score. The grand total is 296, of which 21 are extras; find the score in each innings.
  6. Find the value of \frac{a^{2}-x^{2}}{\frac{1}{a^{2}} - \frac{2}{ax} + \frac{1}{x^{2}}} \times \frac{\frac{1}{a^{2}x^{2}}}{a+x}
  7. Find the value of : \frac{1}{3}(a+2) -3(1-\frac{1}{6}b) - \frac{2}{3}(2a-3b+\frac{3}{2}) + \frac{3}{2}b - 4(\frac{1}{2}a-\frac{1}{3}).
  8. Resolve into factors: (i) 3a^{2} -20a-7 (ii) a^{4}b^{2}-b^{4}a^{2}
  9. Reduce to lowest terms: \frac{4x^{3}+7x^{2}-x+2}{4x^{3}+5x^{2}-7x-2}
  10. Solve the following equations: (a) x-6 -\frac{x-12}{3}= \frac{x-4}{2} + \frac{x-8}{4}; (b) x+y-z=0, x-y+z=4, 5x+y+z=20; (c) \frac{ax+b}{c} + \frac{dx+e}{f} =1
  11. Simplify: \frac{x+3}{x^{2}-5x+6} - \frac{x+2}{x^{2}-9x+14} + \frac{4}{x^{2}-10x+21}
  12. A purse of rupees is divided amongst three persons, the first receiving half of them and one more, the second half of the remainder and one more, and the third six. Find the number of rupees the purse contained.
  13. If h=-2, k=1, l=0, m=1, n=-3, find the value of \frac{h^{2}(m-l)-\sqrt{3hn}+hk}{m(l-h)-2hm^{2}+\sqrt[3]{4hk}}
  14. Find the LCM of 15(p^{3}+q^{3}), 5(p^{2}-pq+q^{2}), 4(p^{2}+pq+q^{2}), 6(p^{2}-q^{2})
  15. Find the square root of (i) \frac{4x^{2}}{9} + \frac{9}{4x^{2}} -2; (ii) 1-6a+5a^{2}+12a^{3}+4a^{4}
  16. Simplify \frac{20x^{2}+27x+9}{15x^{2}+19x+6} + \frac{20x^{2}+27x+9}{12x^{2}+17x+6}
  17. Solve the equations: (i) \frac{a(x-b)}{a-b} + \frac{b(x-a)}{b-a} =1 (ii) \frac{9}{x-4} + \frac{3}{x-8} = \frac{4}{x-9} + \frac{8}{x-3}
  18. A sum of money is to be divided among a number of persons; if Rs. 8 is given to each there will be Rs. 3 short, and if Rs. 7.50 is given to each there will be Rs. 2 over; find the number of persons.


Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai , Maharastra 400101

IITJEE Foundation Maths: Tutorial Problems II

  1. A, B, C start from the same place at the rates a, a+b, a+2b kilometres per hour respectively. B starts n hours after A, how long after B must C start in order that they may overtake A at the same instant, and how far will they then have walked?
  2. Find the distance between two towns when by increasing the speed 7 kilometres per hour a train can perform the journey in 1 hour less, and by reducing the speed 5 kilometres per hour can perform the journey in 1 hour more.
  3. A person buys a certain quantity of land. If he had bought 7 hectares more each hectare would have cost Rs 80 less; and if each hectare had cost Rs. 360 more, he would have obtained 15 hectares less, how much did he pay for the land?
  4. A can walk half a kilometre per hour faster than B; and three quarters of a kilometre per hour faster than C. To walk a certain distance C takes three-quarters of an hour more than B, and two hours more than A; find their rates of walking per hour.
  5. A person spends Rs. 15 in buying goods; if each kg had cost 25 paise more he would have got 5 kg less, but if each kg had cost 15 paise less, he would have received 5 kg more; what weight did he buy?
  6. Five silver coins weight 125 gm and are worth Rs. 6. Ten bronze coins weigh 500 gm and are worth 80 paise. A number of silver and bronze coins which are worth Rs. 134 weigh 11 kg and 250 gm. How many coins of each kind are there?
  7. A and B are playing for money; in the first game, A loses one half of his money, but in the second he wins one-quarter of what B then has. When they cease playing, A has won Rs. 6 and B has still Rs. 14.50 more than A; with what amounts did they begin?
  8. A, B, C each spend the same amount in buying different qualities of the same commodity. B pays 36 paise per kg less than A and obtains 750 gm more; C pays 60 paise per kg more than A and obtains one kg less; how much does each spend?

IITJEE Foundation practice or training problem sheet: I

  1. If the numerator of a fraction is increased by 5, it reduces to \frac{2}{3}, and if the denominator is increased by 9, it reduces to \frac{1}{3}. Find the fraction.
  2. Find a fraction such that it reduces to \frac{3}{5} if 7 is subtracted from its denominator, and reduces to \frac{3}{8} on subtracting 3 from its numerator.
  3. If unity is taken from the denominator of a fraction, it reduces to \frac{1}{2}; if 3 is added to the numerator it reduces to \frac{4}{7}, find the required fraction.
  4. Find a fraction which becomes \frac{3}{4} on adding 5 to the numerator and subtracting 1 from its denominator; and, reduces to \frac{1}{3} on subtracting 4 from the numerator and adding 7 to the denominator.
  5. If 9 is added to the numerator a certain fraction will be increased by \frac{1}{3}; if 6 is taken from the denominator the fraction reduces to \frac{2}{3}; find the required fraction.
  6. At what time between 9 and 10 o’clock are the hands of a watch together?
  7. When are the hands of a clock 8 minutes apart between the hours at 5 and 6 ?
  8. At what time between 10 and 11 o’clock is the hour hand six minutes ahead of the minute hand?
  9. At what time between 1 and 2 o’clock are the hands of a watch in the same straight line?
  10. At what times between 12 and 1 o’clock are the hands of a watch at right angles?
  11. A person buys 20 m of cloth and 25 m of canvas for Rs. 22.50. By selling the cloth at a gain of 15 per cent, and the canvas at a gain of 20 per cent, he clears Rs. 3.75. Find the price of each per metre.
  12. A dealer spends Rs. 6950 in buying horses at Rs, 250/- each and cows at Rs. 200/- each; through disease, he loses 20 percent of the horses and 25 % of the cows. By selling the animals at the price he gave for them, he receives Rs. 5400/-. Find how many of each kind he bought.
  13. The population of a certain district is 53000, of whom 835 can neither read nor write. These consists of 2 %, of all the males and 3 % of all the females; find the number of males and females.
  14. Two persons C and D start simultaneously from two places a kilometre apart, and walk to meet each other; if C walks p kilometres per hour, and D one kilometre per hour faster than C, how far will D have walked when they meet?
  15. A can walk a kilometres per hour faster than B; supposing that he gives B a start of c kilometres, and that B walks a kilometres per hour, how far will A have walked when he overtakes B?


Nalin Pithwa