Category Archives: IITJEE Foundation Math IITJEE Main and Advanced Math and RMO/INMO of (TIFR and Homibhabha)

Maths in a minute! Thanks to PlusMath!

June 23, 2017 – 2:02 pm

https://plus.maths.org/content/maths-in-a-minute?nl=0

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Petronet Kashmir Super 30: Nine crack IITJEE 2017 exams

June 17, 2017 – 1:45 pm

(This news is a bit delayed here on this blog, nevertheless, inspirational.)

Reference: DNA, Mumbai print edition, June 15 2017:

Srinagar: Petronet LNG Ltd. (PLL), which started its CSR voyage from western India and traversing South and East, have now reached Northern state of Jammu and Kashmir.

Considering the lack of facilities/faculty in J and K to impart coaching for Engineering Entrance Examination to facilitate admissions to IITs/NITs/other institutions of repute, Petronet LNG Ltd. sponsored 40 underprivileged students in 2016-17 to fulfill their dream of higher engineering education (35 boys and 5 girls) under Petronet Kashmir Super 30 programme in association with Indian Army, CSRL. This 11 months’ residential programme was attended by 40 students (kargil 7, Pulwama 5, Bandipura 6, Baramulla 4, Anantnag 7, Ganderbal 4, Kulgam 1, Tangmarg 2, Ramban 1, Shopian 1, Sopora 1, Nagrota 1).

Despite hardships in Kashmir and educational institutions being closed last year, the project continued with its mission. The declaration of IITJEE results on April 27 2017 saw selection of 28 students (including 2 girls) who were coached under Petronet Kashmir Super 30 cracking the IITJEE mains exams. Six other students will join Regional Engineering Colleges.

Shri Dharmendra Pradhan, Hon’ble Minister of State (I/C) Petroleum and Natural Gas and Mr. Chowdhary Zulfkar Ali ji, Hon’ble Minister for Dept. of Food, Civil Supplies and Consumer Affairs and Information Dept., J and K, interacted with students of Petronet Kashmir Super 30 at Delhi and felicitated them for their hard work and achievement.

Shri Pradhan expressed happiness that students from underprivileged backgrounds from places like Kupwara, Pulwama, Anantnag, Shopian etc. have shown their determination and got selected in prestigious institutes. He also announced on behalf of Petronet LNG Ltd. that 100 students will be sponsored from Kashmir for engineering entrance next year.

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🙂 🙂 🙂 Congratulations to Petronet (Kashmir) LNG Team including their faculties and the students from Nalin Pithwa !

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Something Cute I Never Noticed Before About Infinite Sums

May 31, 2017 – 7:21 pm

Source: Something Cute I Never Noticed Before About Infinite Sums

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Arithmetic Puzzle

May 28, 2017 – 6:34 pm

Gaurish4Math

Following is a very common arithmetic puzzle that you may have encountered as a child:

Express any whole number $latex n&bg=ffffff$ using the number 2 precisely four times and using only well-known mathematical symbols.

This puzzle has been discussed on pp. 172 of Graham Farmelo’s “The Strangest Man“, and how Paul Dirac solved it by using his knowledge of “well-known mathematical symbols”:

$latex displaystyle{n = -log_{2}left(log_{2}left(2^{2^{-n}}right)right) = -log_{2}left(log_{2}left(underbrace{sqrt{sqrt{ldotssqrt{2}}}}_text{n times}right)right)}&bg=ffffff$

This is an example of thinking out of the box, enabling you to write any number using only three/four 2s. Though, using a transcendental function to solve an elementary problem may appear like an overkill.  But, building upon such ideas we can try to tackle the general problem, like the “four fours puzzle“.

This post on Puzzling.SE describes usage of following formula consisting of  trigonometric operation $latex cos(arctan(x)) = frac{1}{sqrt{1+x^2}}&bg=ffffff$ and $latex tan(arcsin(x))=frac{x}{sqrt{1-x^2}}&bg=ffffff$ to obtain the square…

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Somewhere — find out where !!

April 29, 2017 – 10:06 pm

You have been left in the middle of an island on which there are two villages. In village A, all of the residents, no matter where they are, always tell the truth. In village B, all of the residents, no matter where they are, always tell lies. After walking a few miles from the middle of the island, you find yourself in a village square where there is a resident of one of the villages sitting on some stone steps. What one question would you ask the resident so that you would know which of  the two villages you were in?

PS: Of course, there is no GPS device with you! Or if you have, there is no connectivity!

Have fun!

Nalin Pithwa.

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Growing Up Gifted

April 25, 2017 – 9:52 pm

Source: Growing Up Gifted

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Matchstick Problem

February 25, 2017 – 7:35 pm

Determination is like a match stick !!! 🙂

Singapore Maths Tuition

matchstick quiz

Translation: How do we move only 1 matchstick to make the equation valid?

(“Not Equals” sign $latex neq$ is not allowed)

This is a really tricky question… Hint: Need to think in Chinese, this is a Chinese joke 😛

Featured Post:

Motivational Books for Students (Educational)

The Motivational Books recommended in the above post seems to be very popular with readers on this site. Many readers (presumably parents) have bought the books! Buy a Christmas present for yourself this Xmas. 🙂

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Non-linear equations for IITJEE Mathematics training

November 28, 2016 – 6:13 pm

(Non-linear) equations involving three or more variables can only be solved in special cases. We shall here consider some of the most useful methods of solution.

Example 1.

Solve the following system of equations:

x+y+z=13 \ldots \ldots \ldots Equation I

x^{2}+y^{2}+z^{2}=65 \ldots \ldots \ldots Equation II

xy=10 \ldots \ldots\ldots Equation III

Solution I:

From II and III, we get (x+y)^{2} + z^{2}=85

Plug in u=x+y; then this equation becomes u^{2}+z^{2}=85.

Also, from I, we get u+z=13

hence, we obtain u=7 \hspace{0.1in}or \hspace{0.1in}u=6, z=6 \hspace{0.1in} or \hspace{0.1in}7.Thus,we have

x+y=7, xy=10; and x+y=6, xy=10. Hence, the solutions are

x=5, \hspace{0.1in} or \hspace{0.1in} 2; y=2, \hspace{0.1in} or \hspace{0.1in} 5; z=6;

or,

x=3 \pm \sqrt{-1}; y=3 \mp \sqrt{-1}; z=7.

Example 2:

Solve the following system of equations:

(x+y)(x+z)=30 Equation I

(y+z)(y+x)=15 Equation II

(z+x)(z+y)=18 Equation III

Solution: 2:

Write u, v, w for y+z, z+x, x+y respectively; thus

vw=30, uw=15, vu=18…..call this equations A

Multiplying these equations together, we have

u^{2}v^{2}w^{2}=30 \times 15 \times 18 = 8100. Hence, uvw=\pm 90.

Combining this result with each of the equations in A, we have

u=3, v=6. w=5; or, u=-3, v=-6, w=-5; therefore, we get two sets of linear equations, as follows:

y+z=3

z+x=6

x+y=5

for which the solution set is: x=4, y=1, z=2; and the other set is:

y+z=-3

z+x=-6

x+y=-5

for which the solution set if x=-4, y=-1, z=-2.

Example 3:

Solve the following system of equations:

y^{2}+yz+z^{2}=49…call this equation (1)

z^{2}+xz+x^{2}=19…call this equation (2)

x^{2}+xy+y^{2}=39…call this equation (3)

Subtracting (2) from (1), we get the following:

y^{2}-x^{2}+z(y-x)=30, that is, (y-x)(x+y+z)=30…call this equation (4).

Similarly, from (1) and (3), we get the following:

(z-x)(x+y+z)=10…call this equation (5).

Hence, from equations (4) and (5), by division, we get the following:

\frac{y-x}{z-x}=3, hence, y=3x-2z

Substituting in Equation (3), we obtain x^{2}-3xz+3z^{2}=13.

From (2), we get the following: x^{2}+xz+z^{2}=19

Solving these homogeneous equations (hint: put z=mx, where m is parameter to be found), we get the following:

x=\pm 2, z=\pm 3; and therefore, y=\pm 5

or, x=\pm \frac{11}{\sqrt{7}}; z=\pm \frac{1}{\sqrt{7}}; and therefore, y=\mp \frac{19}{\sqrt{7}}.

Example 4:

Solve the following system of equations:

x^{2}-yz=a^{2}

y^{2}-zx=b^{2}

z^{2}-xy=c^{2}

Solution 4:

Multiply the equations by y, z, and x respectively, and add; then,

c^{2}x+a^{2}y+b^{2}z=0…call this equation I

Multiply the equations by z, x and y respectively; and add; then

b^{2}x+c^{2}y+a^{2}z=0…call this equation II

From (I) and (II), by cross multiplication,

\frac{x}{a^{4}-b^{2}c^{2}}=\frac{y}{b^{4}-c^{2}a^{2}}=\frac{x}{c^{4}-a^{2}b^{2}}=k, suppose.

Substitute in any one of the given equations; then,

k^{2}(a^{6}+b^{6}+c^{6}-3a^{2}b^{2}c^{2})=1; hence, we get the following solution:

\frac{x}{a^{4}-b^{2}c^{2}}=\frac{y}{b^{4}-c^{2}a^{2}}=\frac{z}{c^{4}-a^{2}b^{2}}=\pm \frac{1}{\sqrt{a^{6}+b^{6}+c^{6}-3a^{2}b^{2}c^{2}}}.

More stuff in the pipeline,

Nalin Pithwa

 

By nkpithwa | Also posted in algebra | Comments (0)

Kids of pushy parents ‘face higher risk of depression’: NUS study

July 2, 2016 – 10:25 pm

The greatest gift we can give our children is the gift of enthusiam. (Charles Schwab). I came across this on Professor Terence Tao’s blog.

Singapore Maths Tuition

Recently there are two articles on “Tiger Moms” and “Kiasu (translated as “overly afraid of losing”) Parents” in Singapore. Interesting to read.

Parents in Singapore are indeed at a dilemma, overly pushing their child will lead to negative consequences (as mentioned in the articles), but not pushing their child may lead to falling behind academically.

This quote sums it up:

A housewife, who wanted to be known only as Mrs Lim, 43, said: “In Singapore, the pressure to do well starts early. Parents have no choice but to set high expectations of their kids’ performance.

“But I will be more mindful of the way I speak to my kids, so that they won’t feel bad about making mistakes in their work.”

The solution, ideally, is for children to be self-motivated rather than being pushed by parents. Check out some motivational educational books here.

The worst consequence of pushing children…

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Ivy University Myths

March 5, 2016 – 2:42 pm

Excellent lecture by a Princeton Professor: “Where you go is not where you’ll be ” This is the universal anxiety for all parents and students in Asian countries where there are li…

Source: Ivy University Myths

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