 JEE Mains Previous Years Questions with Solutions

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1

AIEEE 2008

Suppose an electron is attracted towards the origin by a force ${k \over r}$ where $'k'$ is a constant and $'r'$ is the distance of the electron from the origin. By applying Bohr model to this system, the radius of the ${n^{th}}$ orbital of the electron is found to be $'{r_n}'$ and the kinetic energy of the electron to be $'{T_n}'.$

Then which of the following is true?

A
${T_n} \propto {1 \over {{n^2}}},{r_n} \propto {n^2}$
B
${T_n}$ independent of $n,{r_n} \propto n$
C
${T_n} \propto {1 \over n},{r_n} \propto n$
D
${T_n} \propto {1 \over n},{r_n} \propto {n^2}$

Explanation

When $F = {k \over r} =$ centripetal force, then ${k \over r} = {{m{v^2}} \over r}$

$\Rightarrow m{v^2} =$ constant $\Rightarrow$ kinetic energy is constant

$\Rightarrow T$ is independent of $n.$
2

AIEEE 2008

This question contains Statement- 1 and Statement- 2. Of the four choices given after the statements, choose the one that best describes the two statements:
Statement- 1:
Energy is released when heavy nuclei undergo fission or light nuclei undergo fusion and

Statement- 2:
For heavy nuclei, binding energy per nucleon increases with increasing $Z$ while for light nuclei it decreases with increasing $Z.$

A
Statement - $1$ is false, Statement - $2$ is true
B
Statement - $1$ is true, Statement - $2$ is true; Statement - $2$ is a correct explanation for Statement - $1$
C
Statement - $1$ is true, Statement - $2$ is true; Statement - $2$ is not a correct explanation for Statement - $1$
D
Statement - $1$ is true, Statement - $2$ is false

Explanation

We know that energy is released when heavy nuclei undergo fission or light nuclei undergo fusion. Therefore statement $(1)$ is correct.

The second statement is false because for heavy nuclei the binding energy per nucleon decreases with increasing $Z$ and for light nuclei, B.E/nucleon increases with increasing $Z$ and for light nuclei, $B.E/$nucleon increases with increasing $Z.$
3

AIEEE 2007

Which of the following transitions in hydrogen atoms emit photons of highest frequency ?
A
$n = 1$ to $n=2$
B
$n = 2$ to $n=6$
C
$n = 6$ to $n=2$
D
$n = 2$ to $n=1$

Explanation

We have no find the frequency of emitted photons. For emission of photons the transition must take place from a higher energy level to a lower energy level which are given only in options $(c)$ and $(d)$.

Frequency is given by

$hv = - 13.6\left( {{1 \over {n_2^2}} - {1 \over {n_1^2}}} \right)$

For transition from $n=6$ to $n=2,$

${v_1} = {{ - 13.6} \over h}\left( {{1 \over {{6^2}}} - {1 \over {{2^2}}}} \right)$

$= {2 \over 9} \times \left( {{{13.6} \over h}} \right)$

For transition from $n=2$ to $n=1,$

${v_2} = {{ - 13.6} \over h}\left( {{1 \over {{2^2}}} - {1 \over {{1^2}}}} \right)$

$= {3 \over 4} \times \left( {{{13.6} \over h}} \right).$

$\therefore$ ${v_1} > {v_2}$
4

AIEEE 2007

The half-life period of a ratio-active element $X$ is same as the mean life time of another ratio-active element $Y.$ Initially they have the same number of atoms. Then
A
$X$ and $Y$ decay at same rate always
B
$X$ will decay faster than $Y$
C
$Y$ will decay faster than $X$
D
$X$ and $Y$ have same decay rate initially

Explanation

According to question,

Half life of $X,\,{T_{1/2}} = {\tau _{av}},\,\,\,$ average life of $Y$

$\Rightarrow {{0.693} \over {{\lambda _X}}} = {1 \over {{\lambda _Y}}} \Rightarrow {\lambda _X} = \left( {0.693} \right).{\lambda _Y}$

$\therefore$ ${\lambda _X} < {\lambda _Y}.$

Now, the rate of decay is given by

$- {{dN} \over {dt}} = \lambda N$

$\therefore$ $Y$ will decay faster than $X.$ [ as $N$ is same ]