Units, Dimensions & Errors · JEE Main previous year questions

1. A.A, B and C only
2. B.A, C and D only
3. C.A, B and D only
4. D.A and B only

A particle moves along the $x$-axis and has its displacement $x$ varying with time $\mathrm{t}$ according to the equation: $x=c_0\left(t^2-2\right)+c(t-2{)}^2$ where $c_0$ and $c$ are constants of appropriate dimensions. Then, which of the following statements is correct?

1. A.the acceleration of the particle is $2(c+c_0)$
2. B.the acceleration of the particle is $2c_0$
3. C.the acceleration of the particle is $2c$
4. D.the initial velocity of the particle is $4c$

A motor operating on $100\mathrm{V}$ draws a current of $1\mathrm{A}$. If the efficiency of the motor is $91.6\%$, then the loss of power in units of $cal/s$ is

1. A.$6.2$
2. B.$2$
3. C.$8.4$
4. D.$4$

A physical quantity C is related to four other quantities $\mathrm{p},\mathrm{q},\mathrm{r}$ and $\mathrm{s}$ as follows $$C=\frac{p{q}^2}{r^3\sqrt{s}}$$ The percentage errors in the measurement of $\mathrm{p},\mathrm{q},\mathrm{r}$ and $\mathrm{s}$ are $1\%,2\%,3\%$ and $2\%$, respectively. The percentage error in the measurement of $C$ will be__________\%

The equation for real gas is given by $\left(\mathrm{P}+\frac{\mathrm{a}}{{\mathrm{V}}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}$, where $\mathrm{P}$, $\mathrm{V}$, $\mathrm{T}$ and $\mathrm{R}$ are the pressure, volume, temperature and gas constant, respectively. The dimension of ${\mathrm{ab}}^{-2}$ is equivalent to that of :

1. A.Compressibility
2. B.Planck's constant
3. C.Energy density
4. D.Strain

If ${\mu }_0$ and ${ϵ}_0$ are the permeability and permittivity of free space, respectively, then the dimension of $\left(\frac{1}{{\mu }_0{ϵ}_0}\right)$ is :

1. A.${\mathrm{T}}^2/L$
2. B.${\mathrm{L}}^2/{\mathrm{T}}^2$
3. C.${\mathrm{T}}^2/{\mathrm{L}}^2$
4. D.$L/{\mathrm{T}}^2$

Given a charge q , current I and permeability of vacuum ${\mu }_o$. Which of the following quantity has the dimension of momentum ?

1. A.$qI/{\mu }_o$
2. B.$q^2{\mu }_{o}I$
3. C.$q{\mu }_0/I$
4. D.$q{\mu }_o\mathrm{I}$

1. A.Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
2. B.Both (A) and (R) are true and (R) is the correct explanation of (A)
3. C.(A) is false but (R) is true
4. D.(A) is true but (R) is false

A physical quantity $Q$ is related to four observables $a,b,c,d$ as follows : $$Q=\frac{a{b}^4}{cd}$$ where, $\mathrm{a}=(60\pm 3)\mathrm{Pa}$; $\mathrm{b}=(20\pm 0.1)\mathrm{m}$; $\mathrm{c}=(40\pm 0.2)\mathrm{Ns}{\mathrm{m}}^{-2}$ and $\mathrm{d}=(50\pm 0.1)\mathrm{m}$, then the percentage error in Q is $\frac{x}{1000}$ , where $x=$ ____ .

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of $[M^a{L}^b{T}^c]$. If $b=-3$, the value of $c$ is ____.

A tiny metallic rectangular sheet has length and breadth of $5\mathrm{mm}$ and $2.5\mathrm{mm}$, respectively. Using a specially designed screw gauge which has pitch of $0.75\mathrm{mm}$ and $15$ divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be $\frac{x}{100}$ where $x$ is ____.

For an experimental expression $y=\frac{32.3\times 1125}{27.4}$, where all the digits are significant. Then to report the value of $y$ we should write

1. A.$y=1326.186$
2. B.$y=1326.2$
3. C.$y=1326.19$
4. D.$y=1330$

The least count of a screw gauge is $0.01\mathrm{mm}$ . If the pitch is increased by $75\%$ and number of divisions on the circular scale is reduced by $50\%$, the new least count will be ____ $\times 10^{-3}\mathrm{mm}$

The energy of a system is given as $\mathrm{E}(t)={\alpha }^3{\mathrm{e}}^{-\beta t}$, where $t$ is the time and $\beta =0.3{\text{ s}}^{-1}$. The errors in the measurement of $\alpha$ and $t$ are $1.2\%$ and $1.6\%$, respectively. At $t=5\text{ s}$, maximum percentage error in the energy is :

1. A.$6\%$
2. B.$11.6\%$
3. C.$4\%$
4. D.$8.4\%$

If $B$ is magnetic field and ${\mu }_0$ is permeability of free space, then the dimensions of $(B/{\mu }_0)$ is

1. A.${\mathrm{MT}}^{-2}{\mathrm{A}}^{-1}$
2. B.${\mathrm{L}}^{-1}\mathrm{A}$
3. C.${\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{A}}^{-1}$
4. D.${\mathrm{LT}}^{-2}{\mathrm{A}}^{-1}$

Given below are two statements : Statement I: In a vernier callipers, one vernier scale division is always smaller than one main scale division. Statement II : The vernier constant is given by one main scale division multiplied by the number of vernier scale divisions. In the light of the above statements, choose the correct answer from the options given below.

1. A.Both Statement I and Statement II are false
2. B.Statement I is true but Statement II is false
3. C.Both Statement I and Statement II are true
4. D.Statement I is false but Statement II is true

The dimensional formula of latent heat is :

1. A.$[{\mathrm{M}}^0\mathrm{L}{\mathrm{T}}^{-2}]$
2. B.$[\mathrm{ML}{\mathrm{T}}^{-2}]$
3. C.$[{\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^{-2}]$
4. D.$[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-2}]$

One main scale division of a vernier caliper is equal to $\mathrm{m}$ units. If ${\mathrm{n}}^{\text{th}}$ division of main scale coincides with $(n+1{)}^{\text{th}}$ division of vernier scale, the least count of the vernier caliper is :

1. A.$\frac{1}{(n+1)}$
2. B.$\frac{m}{(n+1)}$
3. C.$\frac{n}{(n+1)}$
4. D.$\frac{m}{n(n+1)}$

The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $h/\sqrt{2mE}$. The dimensional formula for Planck's constant is :

1. A.$\left[{\mathrm{M}}^2{\mathrm{L}}^2{\mathrm{T}}^{-2}\right]$
2. B.$\left[\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\right]$
3. C.$\left[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-1}\right]$
4. D.$\left[{\mathrm{MLT}}^{-2}\right]$

Young's modulus is determined by the equation given by $Y=49000\frac{m}{l}\frac{\text{dyne}}{{\text{cm}}^2}$ where $M$ is the mass and $l$ is the extension of wire used in the experiment. Now error in Young modules ($Y$) is estimated by taking data from $M-l$ plot in graph paper. The smallest scale divisions are $5\text{ g}$ and $0.02\text{ cm}$ along load axis and extension axis respectively. If the value of $M$ and $l$ are $500\text{ g}$ and $2\text{ cm}$ respectively then percentage error of $Y$ is :

1. A.$2\%$
2. B.$0.02\%$
3. C.$0.5\%$
4. D.$0.2\%$

The diameter of a sphere is measured using a vernier caliper whose 9 divisions of main scale are equal to 10 divisions of vernier scale. The shortest division on the main scale is equal to $1\text{ mm}$. The main scale reading is $2\text{ cm}$ and second division of vernier scale coincides with a division on main scale. If mass of the sphere is $8.635\text{ g}$, the density of the sphere is:

1. A.$2.2{\text{ g/cm}}^3$
2. B.$2.0{\text{ g/cm}}^3$
3. C.$1.7{\text{ g/cm}}^3$
4. D.$2.5{\text{ g/cm}}^3$

There are $100$ divisions on the circular scale of a screw gauge of pitch $1\mathrm{mm}$. With no measuring quantity in between the jaws, the zero of the circular scale lies $5$ divisions below the reference line. The diameter of a wire is then measured using this screw gauge. It is found that $4$ linear scale divisions are clearly visible while $60$ divisions on circular scale coincide with the reference line. The diameter of the wire is :

1. A.$4.65\mathrm{mm}$
2. B.$4.60\mathrm{mm}$
3. C.$4.55\mathrm{mm}$
4. D.$3.35\mathrm{mm}$

Least count of a vernier caliper is $\frac{1}{20N}\mathrm{cm}$. The value of one division on the main scale is $1\mathrm{mm}$. Then the number of divisions of main scale that coincide with $N$ divisions of vernier scale is :

1. A.$\left(\frac{2N-1}{2N}\right)$
2. B.$\left(\frac{2N-1}{20N}\right)$
3. C.$(2N-1)$
4. D.$\left(\frac{2N-1}{2}\right)$

Given below are two statements: Statement (I) : Dimensions of specific heat is $[{\mathrm{L}}^2{\mathrm{T}}^{-2}{\mathrm{K}}^{-1}]$. Statement (II) : Dimensions of gas constant is $[{\mathrm{ML}}^2{\mathrm{T}}^{-1}{\mathrm{K}}^{-1}]$. In the light of the above statements, choose the most appropriate answer from the options given below.

1. A.Statement (I) is incorrect but statement (II) is correct
2. B.Both statement (I) and statement (II) are incorrect
3. C.Both statement (I) and statement (II) are correct
4. D.Statement (I) is correct but statement (II) is incorrect