 JEE Mains Previous Years Questions with Solutions

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1

AIEEE 2007

One end of a thermally insulated rod is kept at a temperature ${T_1}$ and the other at ${T_2}$. The rod is composed of two sections of length ${L_1}$ and ${L_2}$ and thermal conductivities ${K_1}$ and ${K_2}$ respectively. The temperature at the interface of the two section is A
${{\left( {{K_1}{L_1}{T_1} + {K_2}{L_2}{T_2}} \right)} \over {\left( {{K_1}{L_1} + {K_2}{L_2}} \right)}}$
B
${{\left( {{K_2}{L_2}{T_1} + {K_1}{L_1}{T_2}} \right)} \over {\left( {{K_1}{L_1} + {K_2}{L_2}} \right)}}$
C
${{\left( {{K_2}{L_1}{T_1} + {K_1}{L_2}{T_2}} \right)} \over {\left( {{K_2}{L_1} + {K_1}{L_2}} \right)}}$
D
${{\left( {{K_1}{L_2}{T_1} + {K_2}{L_1}{T_2}} \right)} \over {\left( {{K_1}{L_2} + {K_2}{L_1}} \right)}}$

Explanation ${{{K_1}A\left( {{T_1} - T} \right)} \over {{\ell _1}}} = {{{K_2}A\left( {T - {T_2}} \right)} \over {{\ell _2}}}$
$\therefore$ $T = {{{K_1}{T_1}{\ell _2} + {K_2}{T_2}{\ell _1}} \over {{K_2}{\ell _1} + {K_1}{\ell _2}}}$
$= {{{K_1}{\ell _2}{T_1} + {K_2}{\ell _1}{T_2}} \over {{K_1}{\ell _2} + {K_2}{\ell _1}}}$
2

AIEEE 2007

When a system is taken from state $i$ to state $f$ along the path iaf, it is found that $Q=50$ cal and $W=20$ $cal$. Along the path $ibf$ $Q=36$ $cal.$ $W$ along the path $ibf$ is A
$14$ $cal$
B
$6$ $cal$
C
$16$ $cal$
D
$66$ $cal$

Explanation

For path iaf, $\Delta U = Q - W = 50 - 20 = 30\,cal.$
For path ibf, $W = Q - \Delta U = 36 - 30 = 6\,cal.$
3

AIEEE 2007

If ${C_p}$ and ${C_v}$ denote the specific heats of nitrogen per unit mass at constant pressure and constant volume respectively, then
A
${C_p} - {C_v} = 28R$
B
${C_p} - {C_v} = R/28$
C
${C_p} - {C_v} = R/14$
D
${C_p} - {C_v} = R$

Explanation

According to Mayer's relationship ${C_p} - {C_v} = R$
$\therefore$ ${{{C_p}} \over M} - {{{C_v}} \over M} = {R \over M}$ $\,\,\,\,\,\,$ Here $M=28.$
4

AIEEE 2007

A Carnot engine, having an efficiency of $\eta = 1/10$ as heat engine, is used as a refrigerator . If the work done on the system is $10$ $J$, the amount of energy absorbed from the reservoir at lower temperature is
A
$100$ $J$
B
$99$ $J$
C
$90$ $J$
D
$1$ $J$

Explanation

The efficiency $\left( \eta \right)$ of a Carnot engine and the coefficient of performance $\left( \beta \right)$ of a refrigerator are related as
$\beta = {{1 - \eta } \over \eta }$
$\,\,\,\,\,\,$ Here, $\eta = {1 \over {10}}$
$\,\,\,\,\,\,$ $\therefore$ $\beta = {{1 - {1 \over {10}}} \over {\left( {{1 \over {10}}} \right)}} = 9.$

Also, Coefficient of performance $\left( \beta \right)$ is given by $\beta = {{{Q_2}} \over W},$
where ${Q_2}$ is the energy absorbed from the reservoir.
or, $9 = {{{Q_2}} \over {10}}$
$\therefore$ ${Q_2} = 90\,J.$