IIT JEE 2005 MAINS

If \(f(x)\) is a differentiable function and \(g(x)\) is a double differentiable function such that \(|f(x)| \leq 1\) and \(f'(x)=g(x)\), where,\(f^{2}(0)+g^{2}(0)=9\) then prove that there exists some \(c \in(-3,3)\) such that \(g(c) \circ g^{n}(c) < 0\).

An element crystallises in fcc lattice having edge length 400 pm. Calculate the maximum diameter of the atom which can be placed in interstitial site such that the structure is not distorted.

An unknown resistance X is to be determined using resistances R\(_1\), R\(_2\) or R\(_3\). Their corresponding null points are A, B and C. Find which of the above will give the most accurate reading and why ?

If total number of runs scored in \(n\) matches is \(\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)\) where \(n > 1\), and the runs scored in the \(k^{\text {th }}\) match are given by \(k .2^{n+1-k}\), where \(1 \leq k \leq n\). Find, \(n\).

Nitrogen gas is absorbed on 20% surface sites. On heating N\(_2\) gas evolved from sites and was collected at 0.01 atm and 298 K in a container of volume 2.46 cm\(^3\). Find out the number of surface sites occupied per molecule of N, if the density of surface sites is \(6.023\times10^{14}/\mathrm{cm^3}\) and surface area is 1000 cm\(^2\).

A whistling train approaches a junction. An observer standing at the junction observes the frequency to be 2.2 kHz and 1.8 kHz of the approaching and the receding train. Find the speed of the train (speed of sound = 300 m/s).