Crack the Problem

(59) All the roots of the equation x⁶ - 6x⁵ + 15x⁴ - ax³ + 15x² - bx + 1 = 0 are positive. Find a and b.

Submission Closed

Cracked by:

1. Prateek, 12th, St. Francis School, Anpara, Sonebhadra.
2. Varun Agrawal, 12th, DPS Vindhyanagar, Shaktinagar, Sonebhardra.
3. Ritesh Kumar, 2nd Year, B.Tech., KIIT Deemed University, Bhubaneswar.
4. Ashwini Kumar, 12th, DPS Vindhyanagar, Distt. Sidhi.

Prateek's Solution (.pdf)

(58)

Submission Closed

Cracked by:

1.Mukund, 12'th, Army Public School, Dhaula Kuan, New Delhi. 2.Ankit Dhingra, 12'th, Jai Hind College, Mumbai. 3.Danish Khan, 12'th, Hartmann College, Bareilly. 4.Ankul Garg, 11'th, Bal Bharati Public School, Rohini, Delhi. 5.Prateek, 12'th, St. Francis School, Anpara, Sonebhadra. 6.Ashwini Kumar, 12'th, DPS Vindhyanagar, Distt. Sidhi. 7.Suvarthi Datta, 1'st Year, Electronics & Communication Engg., NIT Durgapur, Durgapur. 8.Mohd Saad Bhamla, 1'st Year, Computer Science, VESIT (Vivekanand College), Mumbai. 9.Arya Shamang, 12'th Pass from DPS Dwarka, New Delhi.

Answer:

Let r be the radius of the pulley.

We have T₂r - T₁r = Ia,

Since pulley is light, I is negligible.

Hence, T₂ = T₁.

Note that generally frictionless pulley and frictionless axle mean the same things. The surface of the pulley is generally not frictionless. If the surface were frictionless, the string should slip over it which generally does not happen in typical applications of Newton's Laws. However, if string slips over the pulley, then also the tension should be the same for the light string. This can be proved in the following manner:

Consider a small length of the string. Let tensions be T and T' on its ends.

Net force on this length = T - T'= ma.

However, for light string mass is negligible.

Hence, T - T'=0.

Or, T = T'.

(57) Ethanol is made unfit for drinking is some places by adding Methanol to it. What is the reason that Methanol is more toxic than Ethanol?

Submission Closed

Cracked by:

1. Abhishek Dey Das, 12th, Bholananda National Vidyalaya, Barrackpore.
2. Ashwini Kumar, 12th, DPS Vindhyanagar, Distt. Sidhi.
3. Gaurav Kalra, 11th, S. M. Arya Public School, New Delhi.
4. Prateek, 12th, St. Francis School, Anpara, Sonebhadra.
5. Siddartha G.L., 12th, DVS Independent College, Shimoga.
6. Anubhav Sharma, 12th, Kendriya Vidyalaya, A.G.C.R. Colony, Delhi.

Answer:

Same group of enzymes digest both methanol and ethanol yielding CO₂ and water as the final products in stepwise degradation. It is the metabolites produced during this stepwise breakdown in the body that differ for these alcohols. These intermediates are more dangerous than methanol in case of methanol and the process prevents ethanol from building up toxic levels in the body. There are two key processes as far as methanol is concerned.

In the first enzymatic reaction, methanol is broken down into formaldehyde. Formaldehyde reacts with the amino acids in proteins. Proteins are chains of amino acids that fold to form unique structures. The way these chains fold gives proteins the proper shape and the flexibility to interact with other molecules. Formaldehyde diffuses into tissues and cells where it forms cross links between different amino acids. The protein is stuck rigidly in whatever conformation it was in and is no longer able to carry out any reactions.

Long-term exposure to formaldehyde may also cause cancer. However, it does not stay too long in the body because it is rapidly metabolized to formic acid by the second enzyme in this metabolic pathway. Formic acid is also extremely toxic to humans. It disrupts the function of a cell's mitochondria. Mitochondria normally serve as the "powerhouse of the cell" and disrupting their function is like abruptly shutting down a power plant. Not only do all the cellular processes stop for lack of energy, but the cells themselves are blown apart by a massive accumulation of different molecules involved in energy production. The cells that make up the optic nerve are exquisitely sensitive to formic acid, which is why blindness is so closely associated with methanol poisoning.

(56) If p, q, r, s are natural numbers, prove that:

Submission: Closed

Cracked by:

1. Ashwini Kumar, 12th, DPS Vindhyanagar, Distt. Sidhi.
2. Raunak Agarwal, 12th, Holy Cross Senior Secondary School Kapa, Raipur.
3. Prateek, 12th, St. Francis School, Anpara, Sonebhadra.

(55)

A scientist created a two dimensional circuit made of uniform wire, which he called manav during his research to find the lowest resistance path in human bodies. He experimented with it by connecting it across various points across a battery. Can you help him to find the resistance between the ear (A) of this, manav and the toe (B)? Take the resistance per unit length of the wire=l , Radius of head = a, Length of neck=n, Length of one arm=h, Length of the mid section=m, Length of one leg=l.

Submission: Closed

Ankul's Solution (.pdf)

Cracked by:

1. Varun Agrawal, 12th, DPS Vindhyanagar, Sonebhardra.
2. Amarpreet Singh, 12th Pass, Govt. Mahindra College, Patiala. Amarpreet has qualified in IIT-JEE 2005 Screening.
3. Ankul Garg, 11th, Bal Bharati Public School, Rohini, Delhi. A quality solution!
4. Pallav Vyas, 12th , Golden Jubilee School, Jalna.
5. R. V. Pradeep, 12th. Hyderabad.

(54) A hydrogen electrode and a calomel half-cell are used to determine the pH of a solution where the pressure is 500 mm of Hg. The H₂ was allowed to bubble out of the electrode at the atmospheric pressure prevailing there. If the pH is found to be 4.00 by a visiting scientist who wrongly assumed P(H₂)=1 atm, what is the correct pH? Temperature is 25^oC.

Submission: Closed

Cracked by:

Nikhil kumar, 12th, Thiruvalla, St. Marys residential central school.

Nikhil's Solution Edited by INDIANET

Let E be the potential of the hydrogen electrode.

Then E = E^o - RT/nF log ([H⁺]²/ P(H₂))

pH = -log [H⁺]

From the equation we have pH = -log [H⁺] = (E/0.0295 - log P(H₂))/2 (E^o = 0)

i.e. 4 = (E/0.0295 - log 1)/2

i.e. 8 = (E/0.0295 +0)

Therefore E = 0.236 Volt

Now P(H₂) = 500/760 atm =0.65789 atm

Therefore, pH = (0.236/0.0295 + 0.182 ) / 2

Therefore, pH = 4. 091

(53) Find the fallacy in the following argument:

Let us solve the equation, 2cos²x = sinx + sin3x - 5cosx.

This is same as, 2cos²x = 2sin2xcosx - 5cosx.

Hence, 2cosx = 2sin2x - 5.

Which does not yield any real solution on solving (check for yourself!).

But x= p/2 satisfies the original equation and hence must be a solution.

Submission: Closed

Cracked by:

Ankul's Solution (.pdf)

(52)

Four large metal plates are given charges as shown in the figure. The middle two are then connected through a wire. Find the charge that will flow through the wire.

Submission: Closed

Varun's solution Edited by INDIANET (.pdf)

Cracked by:

Varun Agrawal, 12th, DPS Vindhyanagar, Sonebhardra.

(51) P is a radioactive element with decay constant l_P having some nuclei initially. It decays to its daughter nucleus D which is also radioactive having decay constant l_D. Show that the time at which the number of radioactive daughter nuclei is maximum is

.

Submission: Closed

Vipul's solution Edited by INDIANET (.pdf)

Cracked by:

1. Vivek Saxena, 12th pass, Dr Virendra Swarup Education Centre, Kanpur.
2. Vipul Jain, 12th pass, The Lawrence Public School, Venkateshwara College, New Delhi. Vipul has qualified in JEE 2005.
3. Ankit Dhingra, 11th, Jai Hind College, Mumbai.