Units, Dimensions & Errors · JEE Main previous year questions

Given below are two statements : Statement I : For transmitting a signal, size of antenna ( $l$ ) should be comparable to wavelength of signal (at least $l=\frac{\lambda }{4}$ in dimension) Statement II: In amplitude modulation, amplitude of carrier wave remains constant (unchanged). 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 correct
3. C.Both Statement I and Statement II are incorrect
4. D.Statement I is correct but Statement II is incorrect

In a screw gauge, there are 100 divisions on the circular scale and the main scale moves by $0.5\text{ mm}$ on a complete rotation of the circular scale. The zero of circular scale lies 6 divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, 4 linear scale divisions are clearly visible while $46^{\text{th}}$ division the circular scale coincide with the reference line. The diameter of the wire is _________ $\times 10^{-2}\text{ mm}$.

In a cuboid of dimension $2\mathrm{L}\times 2\mathrm{L}\times \mathrm{L}$, a charge $q$ is placed at the center of the surface 'S' having area of $4{\mathrm{L}}^2$. The flux through the opposite surface to 'S' is given by

1. A.$\frac{q}{2{ϵ}_0}$
2. B.$\frac{q}{3{ϵ}_0}$
3. C.$\frac{q}{12{ϵ}_0}$
4. D.$\frac{q}{6{ϵ}_0}$

Match List I with List II : $$\begin{array}{llll}& \text{List I (Physical Quantity)}& & \text{List II (Dimensional Formula)}\\ \text{A.}& \text{Pressure gradient}& \text{I.}& [{\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^{-2}]\\ \text{B.}& \text{Energy density}& \text{II.}& [{\mathrm{M}}^1{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}]\\ \text{C.}& \text{Electric Field}& \text{III.}& [{\mathrm{M}}^1{\mathrm{L}}^{-2}{\mathrm{T}}^{-2}]\\ \text{D.}& \text{Latent heat}& \text{IV.}& [{\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-3}{\mathrm{A}}^{-1}]\end{array}$$ Choose the correct answer from the options given below:

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

The equation of a circle is given by $x^2+y^2=a^2$, where a is the radius. If the equation is modified to change the origin other than (0, 0), then find out the correct dimensions of A and B in a new equation : $(x-At{)}^2+{\left(y-\frac{t}{B}\right)}^2=a^2$. The dimensions of t is given as $[{\mathrm{T}}^{-1}]$.

1. A.$\mathrm{A}=[{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}],\mathrm{B}=[\mathrm{L}{\mathrm{T}}^{-1}]$
2. B.$\mathrm{A}=[{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}],\mathrm{B}=[\mathrm{LT}]$
3. C.$\mathrm{A}=[\mathrm{LT}],\mathrm{B}=[{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}]$
4. D.$\mathrm{A}=[{\mathrm{L}}^{-1}\mathrm{T}],\mathrm{B}=[\mathrm{L}{\mathrm{T}}^{-1}]$

In an experiment of measuring the refractive index of a glass slab using travelling microscope in physics lab, a student measures real thickness of the glass slab as 5.25 mm and apparent thickness of the glass slab as 5.00 mm. Travelling microscope has 20 divisions in one cm on main scale and 50 divisions on vernier scale is equal to 49 divisions on main scale. The estimated uncertainty in the measurement of refractive index of the slab is $\frac{x}{10}\times 10^{-3}$, where $x$ is ____

Given below are two statements : One is labelled as Assertion (A) and other is labelled as Reason (R). Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension (S), if density of the liquid is $\rho$ and radius of the drop is $\mathrm{r}$, then $\mathrm{T}=\mathrm{K}\sqrt{\rho {\mathrm{r}}^3/{\mathrm{S}}^{3/2}}$ is dimensionally correct, where $\mathrm{K}$ is dimensionless. Reason (R) : Using dimensional analysis we get R.H.S. having different dimension than that of time period. In the light of above statements, choose the correct answer from the options given below.

1. A.Both (A) and (R) are true and (R) is the correct explanation of (A)
2. B.Both (A) and (R) are true 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

Two metallic wires of identical dimensions are connected in series. If ${\sigma }_1$ and ${\sigma }_2$ are the conductivities of these wires respectively, the effective conductivity of the combination is :

1. A.$\frac{{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$
2. B.$\frac{2{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}$
3. C.$\frac{{\sigma }_1+{\sigma }_2}{2{\sigma }_1{\sigma }_2}$
4. D.$\frac{{\sigma }_1+{\sigma }_2}{{\sigma }_1{\sigma }_2}$

A travelling microscope has 20 divisions per $\mathrm{cm}$ on the main scale while its vernier scale has total 50 divisions and 25 vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling microscope?

1. A.$0.001\mathrm{cm}$
2. B.$0.002\mathrm{mm}$
3. C.$0.002\mathrm{cm}$
4. D.$0.005\mathrm{cm}$

1. A.$2.92\mathrm{mm}$
2. B.$2.54\mathrm{mm}$
3. C.$2.98\mathrm{mm}$
4. D.$3.45\mathrm{mm}$

The dimensions of $\left(\frac{{\mathrm{B}}^2}{{\mu }_0}\right)$ will be : (if ${\mu }_0$ : permeability of free space and $B$ : magnetic field)

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

The distance of centre of mass from end A of a one dimensional rod (AB) having mass density $\rho ={\rho }_0\left(1-\frac{x^2}{L^2}\right)\text{kg/m}$ and length L (in meter) is $\frac{3L}{\alpha }\text{m}$. The value of $\alpha$ is _________. (where x is the distance from end A)

The one division of main scale of Vernier callipers reads $1\text{ mm}$ and 10 divisions of Vernier scale is equal to the 9 divisions on main scale. When the two jaws of the instrument touch each other, the zero of the Vernier lies to the right of zero of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the zero of the Vernier scale lies in between $4.1\text{ cm}$ and $4.2\text{ cm}$ and $6^{\text{th}}$ Vernier division coincides scale division. The diameter of the bob will be _________ $\times 10^{-2}\text{ cm}$.

An expression of energy density is given by $u=\frac{\alpha }{\beta }\sin \left(\frac{\alpha x}{kt}\right)$, where $\alpha ,\beta$ are constants, $x$ is displacement, $k$ is Boltzmann constant and $t$ is the temperature. The dimensions of $\beta$ will be :

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

A screw gauge of pitch $0.5\mathrm{mm}$ is used to measure the diameter of uniform wire of length $6.8\mathrm{cm}$, the main scale reading is $1.5\mathrm{mm}$ and circular scale reading is $7$ . The calculated curved surface area of wire to appropriate significant figures is : [Screw gauge has 50 divisions on its circular scale]

1. A.$6.8\mathrm{c}{\mathrm{m}}^2$
2. B.$3.4\mathrm{c}{\mathrm{m}}^2$
3. C.$3.9\mathrm{c}{\mathrm{m}}^2$
4. D.$2.4\mathrm{c}{\mathrm{m}}^2$

In an experiment to determine the Young's modulus of wire of a length exactly $1\mathrm{m}$, the extension in the length of the wire is measured as $0.4\mathrm{mm}$ with an uncertainty of $\pm 0.02\mathrm{mm}$ when a load of $1\mathrm{kg}$ is applied. The diameter of the wire is measured as $0.4\mathrm{mm}$ with an uncertainty of $\pm 0.01\mathrm{mm}$. The error in the measurement of Young's modulus $(\Delta \mathrm{Y})$ is found to be $x\times 10^{10}\mathrm{N}{\mathrm{m}}^{-2}$. The value of $x$ is ____. $(\text{take }g=10\mathrm{m}{\mathrm{s}}^{-2})$

If n main scale divisions coincide with (n + 1) vernier scale divisions. The least count of vernier callipers, when each centimetre on the main scale is divided into five equal parts, will be :

1. A.$\frac{2}{n+1}\mathrm{mm}$
2. B.$\frac{5}{n+1}\mathrm{mm}$
3. C.$\frac{1}{2n}\mathrm{mm}$
4. D.$\frac{1}{5n}\mathrm{mm}$

A student in the laboratory measures thickness of a wire using screw gauge. The readings are $1.22\mathrm{mm}$, $1.23\mathrm{mm}$, $1.19\mathrm{mm}$ and $1.20\mathrm{mm}$. The percentage error is $\frac{x}{121}\%$. The value of x is ____.

A silver wire has a mass $(0.6\pm 0.006)\mathrm{g}$, radius $(0.5\pm 0.005)\mathrm{mm}$ and length $(4\pm 0.04)\mathrm{cm}$. The maximum percentage error in the measurement of its density will be :

1. A.$4\%$
2. B.$3\%$
3. C.$6\%$
4. D.$7\%$

The SI unit of a physical quantity is pascal-second. The dimensional formula of this quantity will be :

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

If L, C and R are the self inductance, capacitance and resistance respectively, which of the following does not have the dimension of time?

1. A.$RC$
2. B.$\frac{L}{R}$
3. C.$\sqrt{LC}$
4. D.$\frac{L}{C}$

An expression for a dimensionless quantity P is given by $P=\frac{\alpha }{\beta }{\log }_e\left(\frac{kt}{\beta x}\right)$; where $\alpha$ and $\beta$ are constants, x is distance; k is Boltzmann constant and t is the temperature. Then the dimensions of $\alpha$ will be :

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

The elongation of a wire on the surface of the earth is $10^{-4}\text{ m}$. The same wire of same dimensions is elongated by $6\times 10^{-5}\text{ m}$ on another planet. The acceleration due to gravity on the planet will be ________ ${\text{ms}}^{-2}$. (Take acceleration due to gravity on the surface of earth $=10{\text{ ms}}^{-2}$)

In a vernier callipers, each cm on the main scale is divided into 20 equal parts. If tenth vernier scale division coincides with nineth main scale division. Then the value of vernier constant will be ________ $\times 10^{-2}\text{ mm}$.

The dimension of mutual inductance is :

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