Units, Dimensions & Errors · JEE Main previous year questions

2022 23 questions

1. Q312022 · 29 Jul 2022 · Shift 1

   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
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   Answer: D

2. Q422022 · 29 Jul 2022 · Shift 1

   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.$\frac{{\sigma }_1+{\sigma }_2}{{\sigma }_1{\sigma }_2}$
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   Answer: B

3. Q482022 · 29 Jul 2022 · Shift 1

   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.$0.005\mathrm{cm}$
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   Answer: C

4. Q492022 · 29 Jul 2022 · Shift 1
   
   1. A.$2.92\mathrm{mm}$
   2. B.$2.54\mathrm{mm}$
   3. C.$2.98\mathrm{mm}$
   4. D.$3.45\mathrm{mm}$
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   Answer: C

5. Q312022 · 28 Jul 2022 · Shift 1

   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.$\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{A}}^{-1}\right]$
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   Answer: C

6. Q522022 · 28 Jul 2022 · Shift 2· Numerical answer

   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)

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   Answer: 8

7. Q522022 · 27 Jul 2022 · Shift 1· Numerical answer

   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 _________ $\times 10^{-2}\text{ cm}$.

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   Answer: 412

8. Q312022 · 27 Jul 2022 · Shift 2

   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.$\left[{\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^0\right]$
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   Answer: D

9. Q482022 · 26 Jul 2022 · Shift 1

   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.$2.4\mathrm{c}{\mathrm{m}}^2$
   Show answer

   Answer: B

10. Q532022 · 26 Jul 2022 · Shift 1· Numerical answer

    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 ____. $(\text{take }g=10\mathrm{m}{\mathrm{s}}^{-2})$

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    Answer: 2

11. Q322022 · 30 Jun 2022 · Shift 1

    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.$\frac{1}{5n}\mathrm{mm}$
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    Answer: A

12. Q512022 · 28 Jun 2022 · Shift 2· Numerical answer

    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 ____.

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    Answer: 150

13. Q332022 · 27 Jun 2022 · Shift 1

    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.$7\%$
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    Answer: A

14. Q312022 · 27 Jun 2022 · Shift 2

    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.$[{\text{M}}^{-1}{\text{L}}^3{\text{T}}^0]$
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    Answer: A

15. Q442022 · 27 Jun 2022 · Shift 2

    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.$\frac{L}{C}$
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    Answer: D

16. Q312022 · 26 Jun 2022 · Shift 1

    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.$[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-2}]$
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    Answer: C

17. Q542022 · 26 Jun 2022 · Shift 1· Numerical answer

    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 $=10{\text{ ms}}^{-2}$)

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    Answer: 6

18. Q572022 · 26 Jun 2022 · Shift 1· Numerical answer

    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 ________ $\times 10^{-2}\text{ mm}$.

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    Answer: 5

19. Q312022 · 26 Jun 2022 · Shift 2

    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.$[M{L}^2{T}^{-3}{A}^{-2}]$
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    Answer: C

20. Q582022 · 26 Jun 2022 · Shift 2· Numerical answer

    A travelling microscope is used to determine the refractive index of a glass slab. If 40 divisions are there in 1 cm on main scale and 50 Vernier scale divisions are equal to 49 main scale divisions, then least count of the travelling microscope is ________ $\times 10^{-6}\text{ m}$.

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    Answer: 5

21. Q312022 · 25 Jun 2022 · Shift 1

    If $Z=\frac{A^2{B}^3}{C^4}$, then the relative error in Z will be :

    1. A.$\frac{\Delta A}{A}+\frac{\Delta B}{B}+\frac{\Delta C}{C}$
    2. B.$\frac{2\Delta A}{A}+\frac{3\Delta B}{B}-\frac{4\Delta C}{C}$
    3. C.$\frac{2\Delta A}{A}+\frac{3\Delta B}{B}+\frac{4\Delta C}{C}$
    4. D.$\frac{\Delta A}{A}+\frac{\Delta B}{B}-\frac{\Delta C}{C}$
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    Answer: C

22. Q512022 · 25 Jun 2022 · Shift 2· Numerical answer

    For $z=a^2{x}^3{y}^{\frac{1}{2}}$, where 'a' is a constant. If percentage error in measurement of 'x' and 'y' are 4% and 12% respectively, then the percentage error for 'z' will be ____%.

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    Answer: 18

23. Q312022 · 24 Jun 2022 · Shift 2

    Identify the pair of physical quantities that have same dimensions:

    1. A.velocity gradient and decay constant
    2. B.Wien's constant and Stefan constant
    3. C.angular frequency and angular momentum
    4. D.wave number and Avogadro number
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    Answer: A