Units, Dimensions & Errors · JEE Main previous year questions

In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its $4^{\mathrm{th}}$ division coincides exactly with a certain division on main scale. If $50$ vernier scale divisions equal to $49$ main scale divisions and zero error in the instrument is $0.04\mathrm{mm}$ then how many main scale divisions are there in $1\mathrm{cm}$ ?

1. A.5
2. B.40
3. C.10
4. D.20

In finding out refractive index of glass slab the following observations were made through travelling microscope $50\text{ vernier scale division}=49\mathrm{MSD}$; $20\text{ divisions on main scale in each }\mathrm{cm}$ For mark on paper $\mathrm{MSR}=8.45\mathrm{cm},\mathrm{VC}=26$ For mark on paper seen through slab $\mathrm{MSR}=7.12\mathrm{cm},\mathrm{VC}=41$ For powder particle on the top surface of the glass slab $\mathrm{MSR}=4.05\mathrm{cm},\mathrm{VC}=1$ ($\mathrm{MSR}=$ Main Scale Reading, $\mathrm{VC}=$ Vernier Coincidence) Refractive index of the glass slab is :

1. A.1.52
2. B.1.35
3. C.1.24
4. D.1.42

What is the dimensional formula of $a{b}^{-1}$ in the equation $(\mathrm{P}+\frac{\mathrm{a}}{{\mathrm{V}}^2})(\mathrm{V}-\mathrm{b})=\mathrm{RT}$, where letters have their usual meaning.

1. A.$[{\mathrm{M}}^0{\mathrm{L}}^7{\mathrm{T}}^4]$
2. B.$[{\mathrm{M}}^{-1}{\mathrm{L}}^5{\mathrm{T}}^3]$
3. C.$[{\mathrm{M}}^0{\mathrm{L}}^3{\mathrm{T}}^{-2}]$
4. D.$[{\mathrm{ML}}^2{\mathrm{T}}^{-2}]$

A vernier callipers has 20 divisions on the vernier scale, which coincides with $19^{\text{th}}$ division on the main scale. The least count of the instrument is $0.1\mathrm{mm}$. One main scale division is equal to ____ $\mathrm{mm}$.

1. A.5
2. B.2
3. C.1
4. D.0.5

In an experiment to measure focal length ($f$) of convex lens, the least counts of the measuring scales for the position of object (u) and for the position of image (v) are $\Delta u$ and $\Delta v$, respectively. The error in the measurement of the focal length of the convex lens will be:

1. A.$2f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]$
2. B.$f\left[\frac{\Delta u}{u}+\frac{\Delta v}{v}\right]$
3. C.$f^2\left[\frac{\Delta u}{u^2}+\frac{\Delta v}{v^2}\right]$
4. D.$\frac{\Delta u}{u}+\frac{\Delta v}{v}$

Applying the principle of homogeneity of dimensions, determine which one is correct, where $T$ is time period, $G$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.

1. A.$T^2=\frac{4{\pi }^{2}r}{G{M}^2}$
2. B.$T^2=4{\pi }^2{r}^3$
3. C.$T^2=\frac{4{\pi }^2{r}^3}{GM}$
4. D.$T^2=\frac{4{\pi }^2{r}^2}{GM}$

10 divisions on the main scale of a Vernier calliper coincide with 11 divisions on the Vernier scale. If each division on the main scale is of 5 units, the least count of the instrument is :

1. A.$\frac{5}{11}$
2. B.$\frac{10}{11}$
3. C.$\frac{50}{11}$
4. D.$\frac{1}{2}$

The dimensional formula of angular impulse is :

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

A particle is moving in one dimension (along $x$ axis) under the action of a variable force. Its initial position was $16\mathrm{m}$ right of origin. The variation of its position $(x)$ with time $(t)$ is given as $x=-3t^3+18t^2+16t$, where $x$ is in $\mathrm{m}$ and t is in $\mathrm{s}$. The velocity of the particle when its acceleration becomes zero is ____ $m/s$.

Match List - I with List - II. | List I (Number) | List II (Significant figure) | | :--- | :--- | | (A) 1001 | (I) 3 | | (B) 010.1 | (II) 4 | | (C) 100.100 | (III) 5 | | (D) 0.0010010 | (IV) 6 | Choose the correct answer from the options given below :

1. A.(A)-(II), (B)-(I), (C)-(IV), (D)-(III)
2. B.(A)-(IV), (B)-(III), (C)-(I), (D)-(II)
3. C.(A)-(I), (B)-(II), (C)-(III), (D)-(IV)
4. D.(A)-(III), (B)-(IV), (C)-(II), (D)-(I)

A force is represented by $F=a{x}^2+b{t}^{\frac{1}{2}}$ where $x=$ distance and $t=$ time. The dimensions of $b^2/a$ are:

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

Consider two physical quantities $A$ and $B$ related to each other as $E=\frac{B-x^2}{At}$ where $E,x$ and $t$ have dimensions of energy, length and time respectively. The dimension of $AB$ is

1. A.$L^{-2}{M}^1{T}^0$
2. B.$L^2{M}^{-1}{T}^1$
3. C.$L^0{M}^{-1}{T}^1$
4. D.$L^{-2}{M}^{-1}{T}^1$

A physical quantity $Q$ is found to depend on quantities $a,b,c$ by the relation $Q=\frac{a^4{b}^3}{c^2}$. The percentage error in $a,b$ and $c$ are $3\%,4\%$ and $5\%$ respectively. Then, the percentage error in $Q$ is :

1. A.$43\%$
2. B.$34\%$
3. C.$66\%$
4. D.$14\%$

Given below are two statements : Statement (I) : Planck's constant and angular momentum have same dimensions. Statement (II) : Linear momentum and moment of force have same dimensions. In the light of the above statements, choose the correct answer from the options given below :

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

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : In Vernier calliper if positive zero error exists, then while taking measurements, the reading taken will be more than the actual reading. Reason (R) : The zero error in Vernier Calliper might have happened due to manufacturing defect or due to rough handling. In the light of the above statements, choose the correct answer from the options given below :

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

The equation of state of a real gas is given by $(\text{P}+\frac{\text{a}}{{\text{V}}^2})(\text{V}-\text{b})=\text{RT}$, where P, V and T are pressure, volume and temperature respectively and R is the universal gas constant. The dimensions of $\frac{\text{a}}{{\text{b}}^2}$ is similar to that of :

1. A.$\text{P}$
2. B.$\text{RT}$
3. C.$\text{PV}$
4. D.$\text{R}$

Given below are two statements: one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$ Assertion A : A spherical body of radius $(5\pm 0.1)\text{mm}$ having a particular density is falling through a liquid of constant density. The percentage error in the calculation of its terminal velocity is $4\%$. Reason R : The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius. In the light of the above statements, choose the correct answer from the options given below

1. A.A is false but $\mathbf{R}$ is true
2. B.A is true but $\mathbf{R}$ is false
3. C.Both $\mathbf{A}$ and $\mathbf{R}$ are true but $\mathbf{R}$ is NOT the correct explanation of $\mathbf{A}$
4. D.Both $\mathbf{A}$ and $\mathbf{R}$ are true and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$

If force (F), velocity (V) and time (T) are considered as fundamental physical quantities, then dimensional formula of density will be :

1. A.$\mathrm{F}{\mathrm{V}}^{-2}{\mathrm{T}}^2$
2. B.$\mathrm{F}{\mathrm{V}}^4{\mathrm{T}}^{-6}$
3. C.${\mathrm{F}}^2{\mathrm{V}}^{-2}{\mathrm{T}}^6$
4. D.$\mathrm{F}{\mathrm{V}}^{-4}{\mathrm{T}}^{-2}$

A physical quantity P is given as $$P=\frac{a^2{b}^3}{c\sqrt{d}}$$ The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4% respectively. The percentage error in the measurement of quantity P will be

1. A.12%
2. B.13%
3. C.16%
4. D.14%

In an experiment with vernier callipers of least count $0.1\text{ mm}$, when two jaws are joined together the zero of vernier scale lies right to the zero of the main scale and 6th division of vernier scale coincides with the main scale division. While measuring the diameter of a spherical bob, the zero of vernier scale lies in between $3.2\text{ cm}$ and $3.3\text{ cm}$ marks, and 4th division of vernier scale coincides with the main scale division. The diameter of bob is measured as

1. A.$3.18\text{ cm}$
2. B.$3.22\text{ cm}$
3. C.$3.26\text{ cm}$
4. D.$3.25\text{ cm}$

Dimension of $\frac{1}{{\mu }_0{ϵ}_0}$ should be equal to

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

A cylindrical wire of mass $(0.4\pm 0.01)\text{g}$ has length $(8\pm 0.04)\text{cm}$ and radius $(6\pm 0.03)\text{mm}$. The maximum error in its density will be:

1. A.$1\%$
2. B.$5\%$
3. C.$4\%$
4. D.$3.5\%$

A $2\text{meter}$ long scale with least count of $0.2\mathrm{cm}$ is used to measure the locations of objects on an optical bench. While measuring the focal length of a convex lens, the object pin and the convex lens are placed at $80\mathrm{cm}$ mark and $1\mathrm{m}$ mark, respectively. The image of the object pin on the other side of lens coincides with image pin that is kept at $180\mathrm{cm}$ mark. The $\%$ error in the estimation of focal length is:

1. A.$1.70$
2. B.$0.51$
3. C.$1.02$
4. D.$0.85$

$(P+\frac{a}{V^2})(V-b)=RT$ represents the equation of state of some gases. Where $P$ is the pressure, $V$ is the volume, $T$ is the temperature and $a,b,R$ are the constants. The physical quantity, which has dimensional formula as that of $\frac{b^2}{a}$, will be:

1. A.Energy density
2. B.Bulk modulus
3. C.Modulus of rigidity
4. D.Compressibility

If $\mathrm{R},{\mathrm{X}}_L$, and ${\mathrm{X}}_C$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless :

1. A.$\frac{\mathrm{R}}{{\mathrm{X}}_L{\mathrm{X}}_C}$
2. B.$\mathrm{R}{\mathrm{X}}_L{\mathrm{X}}_C$
3. C.$\frac{\mathrm{R}}{\sqrt{{\mathrm{X}}_L{\mathrm{X}}_C}}$
4. D.$\mathrm{R}\frac{{\mathrm{X}}_L}{{\mathrm{X}}_C}$