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### Due to cold weather a 1 m water pipe of cross-sectional area ....

Due to cold weather a 1 m water pipe of cross-sectional area $1 cm^2$ is filled with ice at $-10^\circ C$. Resistive heating is used to melt the ice. Current of 0.5 A is passed through $4 k\Omega$ resistance. Assuming that all the heat produced is used for melting, what is the minimum time required?
(Given latent heat of fusion for water/ice $=3.33 \times 10^5 J Kg^{-1}$, specific heat of ice $=2\times 10^3 J Kg^{-1}$ and density of ice $=10^3 Kg m^{-3}$ )

(A) 3.53 s
(B) 0.353 s
(C) 35.3 s
(D) 70.65 s

Solution

Electrical Energy = Thermal Energy

$\therefore I^2 Rt = Cm\Delta T + mL = m (C\Delta T + L)$

$\Rightarrow 0.5^2 \times 4000 \times t = \rho V (2 \times 10^3 \times 10 + 3.33 \times 10^5 )$

$\Rightarrow 1000 t = 10^3 \times Al (2 \times 10^4 + 33.3 \times 10^4 )$

$\Rightarrow t = 1 \times 10^{-4} \times 1 (35.3 \times 10^4 ) = 35.3 s$

### Sum of the coefficients in the expansion of $(x+y)^n$ ....
If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$