Back to Units & Measurements QuestionsQuestion Details

JEE PYQ: Units & Measurements - Question ID 850114c549cc (JEE Main 2022)

ID: 850114c549ccJEE Main 2022Single Correct MCQ

Consider the efficiency of carnot's engine is given by $\eta=\frac{\alpha \beta}{\sin \theta} \log_e \frac{\beta x}{k T}$ , where $\alpha$ and $\beta$ are constants. If T is temperature, k is Boltzmann constant, $\theta$ is angular displacement and x has the dimensions of length. Then, choose the incorrect option :

Select Option

Step-by-step Explanation

Since, dimensions trigonometric function and logarithmic function are dimensionless quantities.

\therefore[\eta]=\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}^0\right] <br/><br/>

Also, dimensions of temperature,

[T]=\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}{ }^0 \mathrm{~K}\right]

Dimensions of Boltzmann constant,

[k]=\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]

Dimension of

x=\left[\mathrm{M}^0 \mathrm{LT}^0\right]

(A) $[\beta ] = \left[ {{{kT} \over x}} \right] = \left[ {{E \over x}} \right] = [ML{T^{ - 2}}] = [F]$

(B) $[\alpha \beta ] = [{M^0}{L^0}{T^0}]$

${[\alpha ]^{ - 1}} = [\beta ] = \left[ {{{kT} \over x}} \right]$

So ${[\alpha ]^{ - 1}}[x] = [kT] = [M{L^2}{T^{ - 2}}]$

So $[\eta \sin \theta ] = [\alpha \beta ]$

$[\eta ] = [{M^0}{L^0}{T^0}]$ it is dimensionless quantity

(D) $[\alpha ] \ne [\beta ]$

Related Questions from Units & Measurements

ID: 32f827e3012fJEE Main 2026

In a Vernier calipers, when both jaws touch each other, zero of the Vernier scale is shifted to the right of zero of the main scale and $7^{\text {th }}$ Vernier division coincides with a main scale reading. If the value of 1 main scale division is 1 mm and there are 10 Vernier scale divisions, then the Vernier caliper has

View Solution →

ID: 4c5472dca7e2JEE Main 2026

Dimensions of universal gravitational constant ( $G$ ) in terms of Planck's constant ( $h$ ), distance ( $L$ ), mass ( $M$ ) and time ( $T$ ) are _______.

View Solution →

ID: 605eaee8ed80JEE Main 2026

The time period of a simple harmonic oscillator is $T = 2\pi \sqrt{\frac{k}{m}}$ . Measured value of mass $(m)$ of the object is 10 g with an accuracy of 10 mg and time for 50 oscillations of the spring is found to be 60 s using a watch of 2 s resolution. Percentage error in determination of spring constant $(k)$ is ________%.

View Solution →

ID: 5561befb0f22JEE Main 2026

When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4{ }^{\text {th }}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes 15 divisions on main scale and $5^{\text {th }}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is $\_\_\_\_$ mm.

(Least count of Vernier calliper $=0.1 \mathrm{~mm}$ )

View Solution →