Solution - Career Point

5. Let z 1 , z 2 and z 3 are unimodular complex numbersthen find the greatest value of |z 1 – z 2 | 2 + |z 2 – z 3 | 2 +|z 3 – z 1 | 2 .Sol. |z 1 – z 2 | 2 + |z 2 – z 3 | 2 + |z 3 – z 1 | 2= 2[|z 1 | 2 + |z 2 | 2 + |z 3 | 2 ] – [z 1 z 2 + z1z 2 + z1z 3+ z3z 1 + z 2 z 3 + z2z 3 ]= 6 – [z 1 z 2 + z 2 z 1 + z 1 z 3 + z 3 z 1 +z 2 z 3 + z 3 z 2 ]...(1)Now |z 1 + z 2 + z 3 | 2 ≥ 0⇒ z 1 z 2 + z 2 z 3 + z 2 z 1 + z 3 z 2 + z 1 z 3 + 1From (1) & (2)max m value of |z 1 – z 2 | 2 + |z 2 – z 3 | 2 + |z 3 – z 1 | 2= 6 – (–3) = 9z z 3 ≥ –3...(2)6. Consider following two infinite series in real θ and rC = 1 + r cos θ +22r cos 2θ r 3 cos3θ + + ....2! 3!r 2 sin 2θ r 3 sin 3θ S = r sin θ + + + .....2! 3!If a remains constant and r varies the prove thatdC dS(i) C + S = (C 2 + S 2 ) cos θdr dr⎛ dC ⎞(ii) ⎜ ⎟⎠⎝ drSol. We have2C + iS =C – iS =Now,C 2 + S 2 =⎛ dS ⎞+ ⎜ ⎟⎠⎝ dr2= C 2 + S 2iθree ...(1)−iθree ...(2)re i2e θ =ercosθ.e2i rsin θ= e 2r cos θ ...(3)Differentiating (1) w.r.t.r, we getdC dS + i = e iθ iθr.e. e ....(4)dr dr2⎛ dC ⎞ ⎛ dS ⎞iθiθr.e∴ ⎜ ⎟⎠ + ⎜ ⎟⎠ = e .e = e⎝ dr ⎝ dr= C 2 + S 2 (Form 3)multiply (2) and (4)⎛ dC dS ⎞⎜ + i ⎟ (C – iS) = e iθ iθr.e. e . e⎝ dr dr ⎠= e iθ i ir(e e )[eθ − θ+]2−iθr.e= e iθ 2r cosθ. eNow equating real parts in both sidesdC dSC + S = (C 2 + S 2 ) cosθdr drHence proved.2r cos θScience FactsSkin Deep StorageChip implants that keeptrack of personalinformation seem like anovelty but do they have amore useful future?These days, some people are following theirpets and getting tagged. Radio frequency identification(RFID) chips are the size of a grain of rice and can beloaded up with personal information like passwordsand implanted under the skin. Instead of having toremember a login code, an RFID reader can be set upto automatically detect it and grant you access to arange of things from your computer to your frontdoor. It seems like it could be useful to people withexceptionally poor memories, but right now thesechips are being snapped up by technology geeks likeAmal Graafstra. The 29-year-old businessman fromVancouver, Canada, is one of the first people to havean RFID implant and so far is happy with the results. "Ijust don't want to be without access to the things thatI need to get access to. In his chip, he has stored aunique identification number which can be used to loghim into various electronic devices. It didn't cost himan arm and a leg either: he got the whole set-up onthe internet for about $50 (£30), including the $2 costof the chip itself.The procedure to implant the chip is quitesimple and painless. Amal's chip was implanted underthe skin of his left hand while he was under a localanesthetic. It is possible to inject the chips using alarge enough needle, but in Amal's case the chip wasinserted by simply cutting through his skin with ascalpel. Other than complaining of sensitivity in thearea of the implant, Amal said that it doesn't hurt andhe expects that eventually the chip will be completelyunobtrusive.A hand implanted with anRFID chip and the chip reader.The chip is made of silicon and isdigitally encoded with information..A RFID reader, which is installed in a computer or anelectronic device like a reader by a front door, emits aradio signal of a particular frequency, just like radiostations each broadcast on their own frequency. Thechip acts passively when it is within 3 inches of thereader: the right incoming radio signal induces justenough energy in the antenna of the chip for a circuitin the chip to power up and produce a response. Thereader can then access the information on the chip andpass it on to the computer or device that requires it.XtraEdge for IIT-JEE 45 APRIL 2010