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Welcome to Ctanujit Institute of Statistics & Mathematics...An Initiative by 'ISI'ian

Urgent Notice:

Due to unavoidable reason, Ctanujit Classes will be closed from 1st November to 30th November 2018.
Postal Package will be available from 1st December 2018.

SHORTCUT TRICKS IN MATHS FOR ISI,CMI,TIFR ENTRANCE EXAM

This is a kind of initiative from me to state a trick & solve problems using those tricks from tifr,isi,cmi,nbhm papers. This will be like a series. This is the beginning one.

Trick 1:-
If the sum of two positive quantities is a constant(given),then their product is maximum when two quantities are equal.
Ex. Let a+b=12,then max(ab)=36 , i.e. When a equals b.

Problem1. (CMI-2010 M.Sc entrance) True/False- For x<0 , e^x (1 - e^x) ≤ 1/4 .
Sol:- True. e^x,1-e^x both are positive. Now e^x +(1 - e^x)=1
So their product is max when e^x=1-e^x=1/2,so max[e^x (1 - e^x)] is 1/4.

Problem2. (ISI MMA PAPER 2010) If a,b are positive real variables whose sum is a constant k, then the minimum value of root of{(1+1/a)(1+1/b)} is
A. k-1/k         B. k+2/k        C. 1+2/k         D. none
Sol: (C). The given root is minimum when ab is maximum. Now ab is maximum when a=b=k/2. So value of
root of{(1+1/a)(1+1/b)}=(1+2/k).