Units, Dimensions & Errors · JEE Main previous year questions

If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is

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

A quantity $f$ is given by $f=\sqrt{\frac{h{c}^5}{G}}$ where $c$ is speed of light, $G$ universal gravitational constant and $h$ is the Planck's constant. Dimension of $f$ is that of :

1. A.Energy
2. B.Momentum
3. C.Area
4. D.Volume

If the screw on a screw-gauge is given six rotations, it moves by 3 mm on the main scale. If there are 50 divisions on the circular scale the least count of the screw gauge is :

1. A.0.001 mm
2. B.0.01 cm
3. C.0.02 mm
4. D.0.001 cm

The distance x covered by a particle in one dimensional motion varies with time t as $x^2=a{t}^2+2bt+c$. If the acceleration of the particle depends on x as $x^{-n}$, where n is an integer, the value of n is _______.

The dimension of stopping potential ${\mathrm{V}}_0$ in photoelectric effect in units of Planck's constant 'h', speed of light 'c' and Gravitational constant 'G' and ampere A is :

1. A.${\mathrm{h}}^{1/3}{\mathrm{G}}^{2/3}{\mathrm{c}}^{1/3}{\mathrm{A}}^{-1}$
2. B.${\mathrm{h}}^0{\mathrm{c}}^5{\mathrm{G}}^{-1}{\mathrm{A}}^{-1}$
3. C.${\mathrm{h}}^{2/3}{\mathrm{c}}^{5/3}{\mathrm{G}}^{1/3}{\mathrm{A}}^{-1}$
4. D.${\mathrm{h}}^2{\mathrm{G}}^{3/2}{\mathrm{c}}^{1/3}{\mathrm{A}}^{-1}$

A particle of mass m is dropped from a height h above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of $\sqrt{2gh}$. If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of $\sqrt{\frac{h}{g}}$ is :

1. A.$\sqrt{\frac{1}{2}}$
2. B.$\frac{1}{2}$
3. C.$\sqrt{\frac{3}{2}}$
4. D.$\sqrt{\frac{3}{4}}$

The dimension of $\frac{B^2}{2{\mu }_0}$, where B is magnetic field and ${\mu }_0$ is the magnetic permeability of vacuum, is :

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

Which of the following combinations has the dimension of electrical resistance (${\in }_0$ is the permittivity of vacuum and ${\mu }_0$ is the permeability of vacuum)?

1. A.$\sqrt{\frac{{\in }_0}{{\mu }_0}}$
2. B.$\frac{{\in }_0}{{\mu }_0}$
3. C.$\sqrt{\frac{{\mu }_0}{{\in }_0}}$
4. D.$\frac{{\mu }_0}{{\in }_0}$

In the formula $\mathrm{X}=5Y{\mathrm{Z}}^2$ , X and Z have dimensions of capacitance and magnetic field, respectively. What are the dimensions of Y in SI units?

1. A.$[{\mathrm{M}}^{-3}{\mathrm{L}}^{-2}{\mathrm{T}}^8{\mathrm{A}}^4]$
2. B.$[{\mathrm{M}}^{-2}{\mathrm{L}}^{-2}{\mathrm{T}}^6{\mathrm{A}}^3]$
3. C.$[{\mathrm{M}}^{-1}{\mathrm{L}}^{-2}{\mathrm{T}}^4{\mathrm{A}}^2]$
4. D.$[{\mathrm{M}}^{-2}{\mathrm{L}}^0{\mathrm{T}}^{-4}{\mathrm{A}}^{-2}]$

In the density measurement of a cube, the mass and edge length are measured as $(10.00\pm 0.10)\mathrm{kg}$ and $(0.10\pm 0.01)\mathrm{m}$, respectively. The error in the measurement of density is :

1. A.$0.01kg/{\mathrm{m}}^3$
2. B.$0.10kg/{\mathrm{m}}^3$
3. C.$0.31kg/{\mathrm{m}}^3$
4. D.$0.07kg/{\mathrm{m}}^3$

The area of a square is $5.29\mathrm{c}{\mathrm{m}}^2$. The area of 7 such squares taking into account the significant figures is :-

1. A.$37.0\mathrm{c}{\mathrm{m}}^2$
2. B.$37\mathrm{c}{\mathrm{m}}^2$
3. C.$37.030\mathrm{c}{\mathrm{m}}^2$
4. D.$37.03\mathrm{c}{\mathrm{m}}^2$

In a simple pendulum experiment for determination of acceleration due to gravity (g), time taken for 20 oscillations is measured by using a watch of 1 second least count. The mean value of time taken comes out to be 30 s. The length of pendulum is measured by using a meter scale of least count 1 mm and the value obtained is 55.0 cm. The percentage error in the determination of g is close to :-

1. A.0.2%
2. B.3.5%
3. C.0.7%
4. D.6.8%

If surface tension (S), Moment of inertia (I) and Planck's constant (h), were to be taken as the fundamental units, the dimensional formula for linear momentum would be :-

1. A.$S^{1/2}{I}^{1/2}{h}^0$
2. B.$S^{3/2}{I}^{1/2}{h}^0$
3. C.$S^{1/2}{I}^{1/2}{h}^{-1}$
4. D.$S^{1/2}{I}^{3/2}{h}^{-1}$

The least count of the main scale of a screw gauge is $1\mathrm{mm}$. The minimum number of divisions on its circular scale required to measure $5\mu \mathrm{m}$ diameter of a wire is :

1. A.500
2. B.100
3. C.200
4. D.50

Let $\ell$, r, C and V represent inductance, resistance, capacitance and voltage, respectively. The dimension of $\frac{\ell }{rCV}$ in SI units will be :

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

An alpha-particle of mass m suffers 1-dimensional elastic collision with a nucleus at rest of unknown mass. It is scattered directly backwards losing, 64% of its initial kinetic energy. The mass of the nucleus is

1. A.2m
2. B.4m
3. C.1.5m
4. D.3.5m

Two rods A and B of identical dimensions are at temperature $30^{\circ }\mathrm{C}$. If A is heated upto $180^{\circ }\mathrm{C}$ and B upto $T^{\circ }\mathrm{C}$, then the new lengths are the same. If the ratio of the coefficients of linear expansion of A and B is $4:3$, then the value of T is

1. A.$200^{\circ }\mathrm{C}$
2. B.$270^{\circ }\mathrm{C}$
3. C.$230^{\circ }\mathrm{C}$
4. D.$250^{\circ }\mathrm{C}$

If speed (V), acceleration (A) and force (F) are considered as fundamental units, the dimension of Young's modulus will be:

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

A copper wire is wound on a wooden frame, whose shape is that of an equilateral triangle. If the linear dimension of each side of the frame is increased by a factor of 3, keeping the number of turns of the coil per unit length of the frame the same, then the self inductance of the coil:

1. A.decreases by a factor of $9\sqrt{3}$
2. B.increases by a factor of 27
3. C.decreases by a factor of 9
4. D.increases by a factor of 3

The percentage errors in quantities P, Q, R and S are 0.5%, 1%, 3% and 1.5% respectively in the measurement of a physical quantity $\text{A}=\frac{P^3{Q}^2}{\sqrt{R}S}$. The maximum percentage error in the value of A will be :

1. A.6.0%
2. B.7.5%
3. C.8.5%
4. D.6.5%

The relative error in the determination of the surface area of sphere is $\alpha$. Then the relative error in the determination of its volume is :

1. A.$\frac{3}{2}\alpha$
2. B.$\frac{2}{3}\alpha$
3. C.$\frac{5}{2}\alpha$
4. D.$\alpha$

In a screw gauge, 5 complete rotations of the screw cause it to move a linear distance of $0.25cm$. There are 100 circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of 4 main scale divisions and 30 circular scale divisions. Assuming negligible zero error, the thickness of the wire is :

1. A.$0.4300cm$
2. B.$0.2150cm$
3. C.$0.3150cm$
4. D.$0.0430cm$

The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively 1.5% and 1%, the maximum error in determining the density is:

1. A.6%
2. B.2.5%
3. C.3.5%
4. D.4.5%

A physical quantity P is described by the relation $\mathrm{P}={\mathrm{a}}^{1/2}{\mathrm{b}}^2{\mathrm{c}}^3{\mathrm{d}}^{-4}$ If the relative errors in the measurement of a, b, c and d respectively, are 2%, 1%, 3% and 5%, then the relative error in P will be :

1. A.$8\%$
2. B.$12\%$
3. C.$32\%$
4. D.$25\%$

Time (T), velocity (C) and angular momentum (h) are chosen as fundamentalquantities instead of mass, length and time. In terms of these, the dimensions of mass would be :

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