JEE Main 2022 Question Paper (28-Jun-2022) (Shift 2)

Question 1

Velocity $ \left(v\right) $ and acceleration $ \left(a\right) $ in two systems of units 1 and 2 are related as $ v_{2} = \frac{n}{m^{2}} v_{1} $ and $ a_{2} = \frac{a_{1}}{mn} $ respectively. Here $ m $ and $ n $ are constants. The relations for distance and time in two systems respectively are

Accepted Answer

$ \frac{n^{3}}{m^{3}} L_{1} = L_{2} $ and $ \frac{n^{2}}{m} T_{1} = T_{2} $

Answer

${L}_{1} = \frac{n^{4}}{m^{2}} {L}_{2}$ and ${T}_{1} = \frac{n^{2}}{m} {T}_{2}$

Answer

${L}_{1} = \frac{n^{2}}{m} {L}_{2}$ and ${T}_{1} = \frac{n^{4}}{m^{2}} {T}_{2}$

Answer

$ \frac{n^{2}}{m} L_{1} = L_{2} $ and $ \frac{n^{4}}{m^{2}} T_{1} = T_{2} $

Question 2

A ball is spun with angular acceleration $ \alpha = 6t^{2} - 2t $ where $ t $ is in seconds and $ \alpha $ is in $ \mathrm{rad} \, \mathrm{s}^{-2} $. At $ t = 0 $, the ball has an angular velocity of $ 10 \, \mathrm{rad} \, \mathrm{s}^{-1} $ and an angular position of $ 4 \, \mathrm{rad} $. The most appropriate expression for the angular position of the ball is

Accepted Answer

$ \frac{t^{4}}{2} - \frac{t^{3}}{3} + 10t + 4 $

Answer

$\frac{3}{2} t^{4} - t^{2} + 10t$

Answer

$ \frac{2t^{4}}{3} - \frac{t^{3}}{6} + 10t + 12 $

Answer

$2t^{4} - \frac{t^{3}}{2} + 5t + 4$

Question 3

A block of mass $2  \mathrm{kg}$ moving on a horizontal surface with speed of $4  \mathrm{m}  \mathrm{s}^{-1}$ enters a rough surface ranging from $x = 0.5   \mathrm{m}$ to $x = 1.5   \mathrm{m}$. The retarding force in this range of rough surface is related to distance by $F = -kx$ where $k = 12  \mathrm{N}  \mathrm{m}^{-1}$. The speed of the block as it just crosses the rough surface will be

Accepted Answer

$2  \mathrm{m}  \mathrm{s}^{-1}$

Answer

$2.5  \mathrm{m}  \mathrm{s}^{-1}$

Answer

$1.5  \mathrm{m}  \mathrm{s}^{-1}$

Answer

zero

Question 4

A $\sqrt{34} \mathrm{m}$ long ladder weighing $10 \mathrm{kg}$ leans on a frictionless wall. Its feet rest on the floor $3 \mathrm{m}$ away from the wall as shown in the figure. If $F_{f}$ and $F_{w}$ are the reaction forces of the floor and the wall, then the ratio of $\frac{F_{w}}{F_{f}}$ will be:
(Use $g = 10 \mathrm{m} \mathrm{s}^{-2}$.)

Accepted Answer

$\frac{3}{\sqrt{109}}$

Answer

$\frac{6}{\sqrt{110}}$

Answer

$\frac{3}{\sqrt{113}}$

Answer

$\frac{2}{\sqrt{109}}$

Question 5

Water falls from a $40 \quad \mathrm{m}$ high dam at the rate of $9 \times 10^{4} \quad \mathrm{kg}$ per hour. Fifty percent of gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy, the number of $100 \quad \mathrm{W}$ lamps that can be lit is:
(Take $g = 10 \quad \mathrm{m} \quad \mathrm{s}^{-2}$)

Accepted Answer

$50$

Answer

$25$

Answer

$100$

Answer

$18$