### JEE Mains Previous Years Questions with Solutions

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1

### AIEEE 2011

If a wire is stretched to make it $0.1\%$ longer, its resistance will:
A
increase by $0.2\%$
B
decrease by $0.2\%$
C
decrease by $0.05\%$
D
increase by $0.05\%$

## Explanation

Resistance of wire

$R = {{\rho l} \over A} = {{\rho {l^2}} \over C}$ (where $Al=C$ )

$\therefore$ Fractional charge in resistance

${{\Delta R} \over R} = 2{{\Delta l} \over l}$

$\therefore$ Resistance will increase by $0.2\%$
2

### AIEEE 2010

Two conductors have the same resistance at ${0^ \circ }C$ but their temperature coefficients of resistance are ${\alpha _1}$ and ${\alpha _2}.$ The respective temperature coefficients of their series and parallel combinations are nearly
A
${{{\alpha _1} + {\alpha _2}} \over 2},\,{\alpha _1} + {\alpha _2}$
B
${\alpha _1} + {\alpha _2},\,{{{\alpha _1} + {\alpha _2}} \over 2}$
C
${\alpha _1} + {\alpha _2},\,{{{\alpha _1}{\alpha _2}} \over {{\alpha _1} + {\alpha _2}}}$
D
${{{\alpha _1} + {\alpha _2}} \over 2},\,{{{\alpha _1} + {\alpha _2}} \over 2}$

## Explanation

${R_1} = {R_0}\left[ {1 + {\alpha _1}\Delta t} \right];$

${R_2} = {R_0}\left[ {1 + {\alpha _2}\Delta t} \right]$

$R = {R_1} + {R_2}$

$= {R_0}\left[ {2 + \left( {{\alpha _1} + {\alpha _2}} \right)\Delta t} \right]$

$= 2{R_0}\left[ {1 + \left( {{{{\alpha _1} + {\alpha _2}} \over 2}} \right)\Delta t} \right]$

${\alpha _{eq}} = {{{\alpha _1} + {\alpha _2}} \over 2}$

In Parallel, ${1 \over R} = {1 \over {{R_1}}} + {1 \over {{R_2}}}$

$= {1 \over {{R_0}\left[ {1 + {\alpha _1}\Delta t} \right]}} + {1 \over {{R_0}\left[ {1 + {\alpha _2}\Delta t} \right]}}$

$\Rightarrow {1 \over {{{{R_0}} \over 2}\left( {1 + {\alpha _{eq}}\Delta t} \right)}} = {1 \over {{R_0}\left( {1 + {\alpha _1}\Delta t} \right)}} + {1 \over {{R_0}\left( {1 + {\alpha _2}\Delta t} \right)}}$

$2\left( {1 - {a_{eq}}\Delta t} \right) = \left( {1 - {\alpha _1}\Delta t} \right)\left( {1 - {\alpha _2}\Delta t} \right)$

$\therefore$ ${\alpha _{eq}} = {{{\alpha _1} + {\alpha _2}} \over 2}$
3

### AIEEE 2010

Let $C$ be the capacitance of a capacitor discharging through a resistor $R.$ Suppose ${t_1}$ is the time taken for the energy stored in the capacitor to reduce to half its initial value and ${t_2}$ is the time taken for the charge to reduce to one-fourth its initial value. Then the ratio ${t_1}/{t_2}$ will be
A
$1$
B
${1 \over 2}$
C
${1 \over 4}$
D
$2$

## Explanation

Initial energy of capacitor, ${E_1} = {{q_1^2} \over {2C}}$

Final energy of capacitor, ${E_2} = {1 \over 2}{E_1} = {{q_1^2} \over {4C}} = {\left( {{{{{{q_1}} \over {\sqrt 2 }}} \over {2C}}} \right)^2}$

$\therefore$ ${t_1}=$ time for the charge to reduce to ${1 \over {\sqrt 2 }}$ of its initial value

and ${t_2} =$ time for the charge to reduce to ${1 \over 4}$ of its initial value

We have, ${q_2} = {q_1}{e^{ - t/CR}}$

$\Rightarrow \ln \left( {{{{q_2}} \over {{q_1}}}} \right) = - {t \over {CR}}$

$\therefore$$\ln \left( {{1 \over {\sqrt 2 }}} \right) = {{ - {t_1}} \over {CR}}...\left( 1 \right)$

and $\ln \left( {{1 \over 4}} \right) = {{ - {t_2}} \over {CR}}\,\,...\left( 2 \right)$

By $(1)$ and $(2),$ ${{{t_1}} \over {{t_2}}} = {{\ln \left( {{1 \over {\sqrt 2 }}} \right)} \over {\ln \left( {{1 \over 4}} \right)}}$

$= {1 \over 2}{{\ln \left( {{1 \over 2}} \right)} \over {2\ln \left( {{1 \over 2}} \right)}} = {1 \over 4}$
4

### AIEEE 2008

A $5V$ battery with internal resistance $2\Omega$ and a $2V$ battery with internal resistance $1\Omega$ are connected to a $10\Omega$ resistor as shown in the figure.

The current in the $10\Omega$ resistor is

A
$0.27A{P_2}\,\,to\,\,{P_1}$
B
$0.03A{P_1}\,\,to\,\,{P_2}$
C
$0.03A{P_2}\,\,to\,\,{P_1}$
D
$0.27A{P_1}\,\,to\,\,{P_2}$

## Explanation

Applying kirchoff's loop law in $AB\,{P_2}{P_1}A$ we get

$- 2i + 5 - 10\,{i_1} = 0\,\,\,\,\,\,\,\,...\left( i \right)$

Again applying kirchoffs loop law in ${P_2}$ $CD{P_1}{P_2}$ we get,

$10{i_1} + 2 - i + {i_1} = 0\,\,\,\,\,\,...\left( {ii} \right)$

From $\left( i \right)$ and $\left( {ii} \right)$ $11{i_1} + 2 - \left[ {{{5 - 10{i_1}} \over 2}} \right] = 0$

$\Rightarrow {i_1} = {1 \over {32}}$ A from ${P_2}$ to ${P_1}$