Jee Main 2024 31 01 2024 Shift 2
Question 1
For the block shown, $F _1$ is the minimum force required to move block upwards and $F _2$ is the minimum force required to prevent it from slipping, find $\left|\vec{F} _1-\vec{F} _2\right|$
(1) $50 \sqrt{3} N$
(2) $5 \sqrt{3} N$
(3) $25 \sqrt{3} N$
(4) $\frac{5 \sqrt{3}}{2} N$
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Answer: (2)
Solution:
$f _K=\mu m g \cos \theta$
$ \begin{aligned} & =0.1 \times \frac{50 \times \sqrt{3}}{2} \\ & =2.5 \sqrt{3} N \end{aligned} $
$F _1=m g \sin \theta+f _K$
$ =25+2.5 \sqrt{3} $
$F _2=m g \sin \theta-f _K$
$ =25-2.5 \sqrt{3} $
$\therefore F _1-F _2=5 \sqrt{3} N$
Question 2
Force on a particle moving in straight line is given by $\vec{F}=6 t^{2} \hat{i}-3 t \hat{j}$ and velocity is $\vec{v}=3 t^{2} \hat{i}+6 t \hat{j}$. Find power at $t=2$.
(1) $216 W$
(2) $108 W$
(3) $0 W$
(4) $54 W$
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Answer: (1)
Solution:
$P=\vec{F} \cdot \vec{V}$
$ \begin{aligned} & =18 t^{4}-18 t^{2} \\ \Rightarrow & P(t=2)=18[16-4]=216 W \end{aligned} $
Question 3
If $E=\frac{A-x^{2}}{B t}$ where $E$ is energy, $x$ is displacement and $t$ is time. Find dimensions of $A B$
(1) $\left[M^{-1} L^{2} T\right]$
(2) $\left[ML^{2} T^{-1}\right]$
(3) $\left[M^{-1} L^{2} T^{-2}\right]$
(4) $\left[ML^{2} T^{-2}\right]$
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Answer: (1)
Solution:
$[A]=L^{2}$
$ B=\frac{x^{2}}{t E} \equiv \frac{L^{2}}{TML^{2} T^{-2}}=\frac{1}{MT^{-1}} $
$[B]=M^{-1} T$
$[A B]=\left[M^{-1} L^{2} T\right]$
Question 4
Unpolarised light incident on transparent glass at incident angle $60^{\circ}$. If reflected ray is completely polarised, then angle of refraction is
(1) $45^{\circ}$
(2) $60^{\circ}$
(3) $30^{\circ}$
(4) $37^{\circ}$
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Answer: (3)
Solution:
By Brewsters law
$ \begin{aligned} & \mu=\tan i \\ & \mu=\sqrt{3} \\ & \therefore \quad 1 \times \frac{\sqrt{3}}{2}=\sqrt{3} \times \sin r \\ & \sin r=\frac{1}{2} \\ & r=30^{\circ} \end{aligned} $
Question 5
Two solid spheres each of mass $2 kg$ and radius 75 $cm$ are arranged as shown. Find MOI of the system about the given axis.
(1) $3.15 kg m^{2}$
(2) $31.5 kg m^{2}$
(3) $0.9 kg m^{2}$
(4) $9 kg m^{2}$
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Answer: (1)
Solution:
$I=\left(\frac{2}{5} M R^{2}+M R^{2}\right) \times 2$
$=\frac{14}{5} \times 2 \times \frac{9}{16}$
$=\frac{63}{20}$
$=3.15 kg m^{2}$
Question 6
If the current through an incandescent lamp decreases by $20 \%$, how much change will be there in its illumination?
(1) $36 \%$
(2) $64 \%$
(3) $20 \%$
(4) $40 \%$
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Answer: (1)
Solution:
$p=i^{2} R$
$ p^{\prime}=0.64 i^{2} R $
Question 7
Find the speed of sound in oxygen gas at STP.
(1) $300 m / s$
(2) $350 m / s$
(3) $330 m / s$
(4) $400 m / s$
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Answer: (3)
Solution:
$v=\sqrt{\frac{\gamma R T}{M}}=330 m / s$
Question 8
Find average power in electric circuit if source voltage $(V)=20 \sin (100 \omega t)$ and current in the circuit
$(I)=2 \sin \left(100 \omega t+\frac{\pi}{3}\right)$
(1) $10 W$
(2) $20 W$
(3) $5 W$
(4) $15.5 W$
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Answer: (1)
Solution:
$\langle P\rangle=I V \cos \phi$
$ \begin{aligned} & =\frac{20}{\sqrt{2}} \times \frac{2}{\sqrt{2}} \times \cos 60^{\circ} \\ & =10 W \end{aligned} $
Question 9
In a photoelectric experiment, frequency $f=1.5 f _0$ ( $f _0$ : threshold frequency). If the frequency of light is changed to $f / 2$, then photocurrent becomes (intensity of light has doubled)
(1) Zero
(2) Doubled
(3) Same
(4) Thrice
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Answer: (1)
Solution:
Since $\frac{f}{2}<f _0$
$ \Rightarrow \text { current }=0 $
Question 10
Radius of curvature of equiconvex lens is $20 cm$. Material of lens is having refractive index of 1.5. Find image distance from lens if an object is placed $10 cm$ away from the lens.
(1) $20 cm$
(2) $10 cm$
(3) $40 cm$
(4) $5 cm$
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Answer: (1)
Solution:
$\frac{1}{f}=(\mu-1)\left(\frac{2}{R}\right) \quad f=20 cm$
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
$\frac{1}{v}+\frac{1}{10}=\frac{1}{20}$
Question 11
Draw truth table of given gate circuit.
(1)
| $A$ | $B$ | $X$ |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
(2)
| $A$ | $B$ | $X$ |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
(3)
| $A$ | $B$ | $X$ |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
(4)
| $A$ | $B$ | $X$ |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
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Answer: (2)
Solution:
$X=\overline{(A+\bar{B})+(\bar{A}+B)}$
$ \begin{aligned} & (\overline{A+\bar{B}}) \cdot(\overline{\bar{A}+B}) \\ & (\bar{A} \cdot \overline{\bar{B}}) \cdot(\overline{\bar{A}} \cdot \bar{B}) \end{aligned} $
$(\bar{A} \cdot B) \cdot(A \cdot \bar{B})=\bar{A} \cdot B \cdot A \cdot \bar{B}=0$
Question 12
The magnetic flux through a loop varies with time as $\phi=5 t^{2}-3 t+5$. If the resistance of loop is $8 \Omega$, find the current through it at $t=2 s$
(1) $\frac{15}{8} A$
(2) $\frac{5}{8} A$
(3) $\frac{17}{8} A$
(4) $\frac{13}{8} A$
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Answer: (3)
Solution:
$\frac{d \phi}{d t}=10 t-3$
at $t=2, V=17$
$i=\frac{V}{R}=\frac{17}{8} A$
Question 13
8 moles of oxygen and 4 moles of nitrogen are at same temperature $T$ and are mixed. The total internal energy is
(1) $60 R T$
(2) $15 R T$
(3) $30 R T$
(4) $90 R T$
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Answer: (3)
Solution:
$U=n C _v T$
$ \begin{aligned} \Rightarrow & U=n _1 C _{v _1} T+n _2 C _{V _2} T \\ \Rightarrow & 8 \times \frac{5 R}{2} \times T+4 \times \frac{5 R}{2} \times T \\ & =30 R T \end{aligned} $
Question 14
In the system shown below, the pulley 4 string are ideal. If the acceleration of blocks is $\frac{g}{8}$, find $\frac{m _1}{m _2}$
(1) $\frac{9}{7}$
(2) $\frac{8}{7}$
(3) $\frac{5}{7}$
(4) $\frac{9}{8}$
Show Answer
Answer: (1)
Solution:
$a=\frac{\left(m _1-m _2\right) g}{\left(m _1+m _2\right)}=\frac{g}{8}$
$ \begin{aligned} & 8 m _1-8 m _2=m _1+m _2 \\ & 7 m _1=9 m _2 \\ & \frac{m _1}{m _2}=\frac{9}{7} \end{aligned} $
Question 15
The force between two charged particle placed in air at separation $x$ is $F _0$. Both the charged particle immerged in a medium of dielectric constant $K$ without changing separation between two charge, then net force on one of the particle is now
(1) $\frac{F _0}{K}$
(2) $\frac{F _0}{2 K}$
(3) $\frac{2 F _0}{K}$
(4) $F _0$
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Answer: (1)
Solution:
In air $F=\frac{1}{4 \pi \in _0} \frac{q _1 q _2}{r _2}$
In medium $F^{\prime}=\frac{1}{4 \pi\left(k \in \in _0\right)} \frac{q _1 q _2}{r^{2}}$
$F^{\prime}=\frac{F _0}{K}$
Question 16
Two vector each of magnitude $A$ are inclined at angle $\theta$ with each other, then magnitude of resultant vector is
(1) $A \cos ^{2} \frac{\theta}{2}$
(2) $2 A \cos \frac{\theta}{2}$
(3) $2 A \cos \theta$
(4) $A \cos \frac{\theta}{2}$
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Answer: (2)
Solution:
The magnitude of resultant vector $=\sqrt{a^{2}+b^{2}+2 a b \cos \theta}$
$(R)$
here $a=b=A$
$ \text { then } \begin{aligned} R & =\sqrt{A^{2}+A^{2}+2 A^{2} \cos \theta} \\ & =A \sqrt{2} \sqrt{1+\cos \theta} \\ & =\sqrt{2} A \sqrt{2 \cos ^{2} \frac{\theta}{2}} \\ = & 2 A \cos \frac{\theta}{2} \end{aligned} $
Question 17
Statement 1 : Electric and magnetic energy density in electromagnetic waves are equal.
Statement 2 : Electromagnetic waves exert pressure on a surface.
(1) Statement 1 is true \& Statement 2 is true and is correct explanation of Statement 1
(2) Statement 1 is true \& Statement 2 is true but is not correct explanation of Statement 1
(3) Statement 1 is true but Statement 2 is false
(4) Statement 1 is false but Statement 2 is true
Show Answer
Answer: (2)
Solution:
$\frac{1}{2} \varepsilon _0 E^{2}=\frac{B^{2}}{2 \mu _0}$
$\because E=C B$ and $C=\frac{1}{\mu _0 \varepsilon _0}$
Question 18
A pendulum completes 50 oscillations in 40 seconds. If the length of pendulum is $(20 \pm 0.2) cm$ and resolution of watch is 1 second, find the percentage error in calculation of $g$.
(1) $7 \%$
(2) $3 \%$
(3) $6 \%$
(4) $4 \%$
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Answer: (3)
Solution:
$T=2 \pi \sqrt{\frac{l}{g}}$
$g=\frac{4 \pi^{2} I}{T^{2}}$
$ \begin{aligned} \frac{\Delta g}{g} & =\frac{\Delta l}{l}+\frac{2 \Delta T}{T} \\ & =\frac{0.2}{20}+2\left(\frac{1}{40}\right) \\ & =6 \% \end{aligned} $
Question 19
The period of oscillation of system shown below is $\pi \sqrt{\frac{\alpha m}{5 k}}$ then $\alpha$ is
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Answer: (12)
Solution:
$k _{e q}=\frac{2 k \cdot k}{3 k}+k=\frac{5 k}{3}$
Angular frequency of oscillation $(\omega)=\sqrt{\frac{k _{e q}}{m}}$
$\omega=\sqrt{\frac{5 k}{3 m}}$
Period of oscillation $(\tau)=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{3 m}{5 k}}$
$ =\pi \sqrt{\frac{12 m}{5 k}} $
Question 20
In the given circuit, $r=2 \Omega$. The power dissipated in the circuit is W.
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Answer: (2)
Solution:
$R _{eq}=r$
$\therefore \quad P=\frac{V^{2}}{r}=\frac{4}{2}=2 W$
Question 21
A body of mass $m$ is projected with speed $u$ at angle $45^{\circ}$ with horizontal. The angular momentum of body, about point of projection when body is at highest point, is $\frac{\sqrt{2} m u^{3}}{x g}$ find $x$,
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Answer: (8)
Solution:
$L=m u \cos \theta \frac{u^{2} \sin ^{2} \theta}{2 g}$
$ =m u^{3} \frac{1}{4 \sqrt{2} g} \Rightarrow x=8 $
Question 22
Mass of moon is $\frac{1}{81}$ times the mass of a planet and radius is $\frac{1}{9}$ times the radius of the planet. The ratio of escape speed from planet to escape speed from moon is
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Answer: (3)
Solution:
$v _{\text {esc }}=\sqrt{\frac{2 G M}{R}}$
$ \Rightarrow \text { Ratio }=\sqrt{\frac{81}{9}}=3 $
Question 23
Find the mass number of an atom whose radius is half of that of a given atom of mass number 192.
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Answer: (24)
Solution:
$r=R _0(192)^{\frac{1}{3}}$
$ \begin{aligned} & \frac{r}{2}=R _0(m)^{\frac{1}{3}} \\ & m=\frac{192}{8}=24 \end{aligned} $
