Units & Measurement — JEE Main PYQs

Match List I with List II.

List I:
(a) Capacitance, C
(b) Permittivity of free space, ${\epsilon }_0$
(c) Permeability of free space, ${\mu }_0$
(d) Electric field, E

List II:
(i) ${\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-3}{\mathrm{A}}^{-1}$
(ii) …

A physical '$y$' is represented by the formula $y=m^2{r}^{-4}{g}^x{\ell }^{-\frac{3}{2}}$ if the percentage errors found in $y,m$, $\mathrm{r},\ell$ and g are 18, $1,0.5,4$ and p respectively, then find the value of x and p .

A student determined Young's Modulus of elasticity using the formula $Y=\frac{{\mathrm{MgL}}^3}{4{\mathrm{bd}}^3\delta }$. The value of g is taken to be $9.8\mathrm{m}/{\mathrm{s}}^2$, without any significant error, his observation are as follows:

| Physical Quantity | Leas …

Two resistors ${\mathrm{R}}_1=(4\pm 0.8)\Omega$ and ${\mathrm{R}}_2=(4\pm 0.4)\Omega$ are connected in parallel. The equivalent resistance of their parallel combination will be:

One main scale division of a vernier callipers is ' $a$ ' $cm$ and $n^{\text{th }}$ division of the vernier scale coincide with $(n-1{)}^{\text{th }}$ division of the main scale. The least count of the callipers in mm is :

The resistance $R=\frac{V}{I}$, where $V=(50\pm 2)V$ and $I=(20\pm 0.2)A$. The percentage error in $R$ is ' $x$ ' $\%$. The value of ' $x$ ' to the nearest integer is ____

In order to determine the young's modulus of a wire of radius 0.2 cm (measured using a scale of least count $=0.001\mathrm{cm}$) and length 1 m (measured using a scale of least count $=1\mathrm{mm}$), a weight of mass 1 kg (measured using a scale of least count …

The radius of a sphere is measured to be $(7.50\pm 0.85)\mathrm{cm}$. Suppose the percentage error in its volume is $x$. The value of $x$, to the nearest integer, is ____

Suppose you have taken a dilute solution of oleic acid in such a way that its concentration becomes $0.01{\mathrm{cm}}^3$ of oleic acid per ${\mathrm{cm}}^3$ of the solution. Then you make a thin film of this solution (monomolecular thickness) of area $4{\mathrm{cm}}^2$ by co …

In the experiment of Ohm's law, a potential difference of 5.0 V is applied across the end of a conductor of length 10.0 cm and diameter of 5.00 mm. The measured current in the conductor is 2.00 A. The maximum permissible percentage error in the resistivity of the conductor is:

The time period of a simple pendulum is given by $\mathrm{T}=2\pi \sqrt{\frac{\ell }{g}}$. The measured value of the length of pendulum is 10 cm known to a 1 mm accuracy. The time for 200 oscillations of the pendulum is found to be 100 second using a clock of 1 s resolution. Th …

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

A simple pendulum is being used to determine the vale of gravitational acceleration $g$ at a certain place. The length of the pendulum is 25.0 cm and a stop watch with 1 s resolution measures the time taken for 40 oscillations to be 50 s. The accuracy in $g$ is:

For the four sets of three measured physical quantifies as given below. Which of the following options is correct?
(i) ${\mathrm{A}}_1=24.36,{\mathrm{B}}_1=0.0724,{\mathrm{C}}_1=256.2$
(ii) ${\mathrm{A}}_2=24.44,{\mathrm{B}}_2=16.082,{\mathrm{C}}_2=240.2$
(iii) …

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:

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

The quantities $x=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}},y=\frac{E}{B}$ and $z=\frac{l}{CR}$ are defined where $C$-capacitance, $R$ Resistance, $l$-length, $E$-Electric field, $B$-magnetic field and ${ϵ}_0$, ${\mu }_0$, - free space permittivity and permeability respective …

Using screw gauge of pitch 0.1 cm and 50 divisions on its circular scale, the thickness of an object is measured. It should correctly be recorded as

Amount of solar energy received on the earth's surface per unit area per unit time is defined a solar constant. Dimension of solar constant is:

A student measuring the diameter of a pencil of circular cross-section with the help of a vernier scale records the following four readings $5.50\mathrm{mm},5.55\mathrm{mm},5.45\mathrm{mm};5.65$ mm . The average of these four readings is 5.5375 mm and the standard devia …

A physical quantity $z$ depends on four observables $a,c,c$ and $d$, as $z=\frac{a^2{b}^{\frac{2}{3}}}{\sqrt{c}{d}^3}$. The percentages of error in the measurement of $\mathrm{a},\mathrm{b},\mathrm{c}$ and d are $2\%,1.5\%,4\%$ and $2.5\%$ respectively. …

Assume that the displacement (s) of air is proportional to the pressure difference ( $\Lambda \mathrm{p}$ ) created by a sound wave. Displacement (s) further depends on the speed of sound (v), density of air ( $\rho$ ) and the frequency (f). If …

A screw gauge has 50 divisions on its circular scale. The circular scale is 4 units ahead of the pitch scale marking, prior to use. Upon one complete rotation of the circular scale, a displacement of 0.5 mm is noticed on the pitch scale. The nature of zero error involved, and the …

The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is $\left(\frac{x}{100}\right)\%$. If the relative errors in measuring the mass and the diameter are $6.0\%$ and $1.5\%$ respectively, …

Dimensional formula for thermal conductivity is (here $\mathbf{K}$ denotes the temperature):

A quantity X is given by $IF{v}^2/W{L}^4$ in terms of moment of inertia I, force F, velocity v, work W and Length L. The dimensional formula for x is same as that of:

In SI units, the dimensions of $\sqrt{\frac{{ϵ}_0}{{\mu }_0}}$ is:

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:

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 …

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:

The density of a material in SI units is $128\mathrm{kg}{\mathrm{m}}^{-3}$. In certain units in which the unit of length is 25 cm and the unit of mass 50 g, the numerical value of density of the material is:

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

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

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

The force of interaction between two atoms is given by $F=\alpha \beta \exp \left(-\frac{x^2}{\alpha \mathrm{kt}}\right)$; where $x$ is the distance, k is the Boltzmann constant and T is temperature and $\alpha$ and $\beta$ are two constants. The dimension of $\beta$ …

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

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

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

Expression for time in terms of G (universal gravitational constant), h (Planck constant) and c (speed of light) is proportional to:

A carbon resistance has a following colour code. What is the value of the resistance?

The pitch and the number of divisions, on the circular scale, for a given screw gauge are 0.5 mm and 100 respectively. When the screw gauge is fully tightened without any object, the zero of its circular scale lies 3 divisions below the mean line.
The readings of the main scale a …

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:

The following observations were taken for determining surface tension T of water by capillary method :
diameter of capillary, $D=1.25\times 10^{-2}\mathrm{m}$
rise of water, $\mathrm{h}=1.45\times 10^{-2}\mathrm{m}$.

Using $\mathrm{g}=9.80\mathrm{m}/{\mathrm{s}}^2$ and …

A student measures the time period of 100 oscillations of a simple pendulum four times. The data set is 90 s, 91s, 95s and 92s. If the minimum division in the measuring clock is 1s, then the reported mean time should be :

A screw gauge with a pitch of 0.5 mm and a circular scale with 50 divisions is used to measure the thickness of a thin sheet of Aluminium. Before starting the measurement, it is found that when the two jaws of the screw gauge are brought in contact, the $45^{\text{th}}$ division …

The period of oscillation of a simple pendulum is $T=2\pi \sqrt{\frac{L}{g}}$. Measured value of $L$ is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using wrist watch of 1 s resolution. The accuracy in the determination of $g$ …

The current voltage relation of diode is given by $I=\left(e^{1000\mathrm{V}/\mathrm{T}}-1\right)\mathrm{mA}$, where the applied voltage $V$ is in volts and the temperature T is in degree Kelvin. If a student makes an error measuring $\pm 0.01\mathrm{V}$ while measuring th …

A student measured the length of a rod and wrote it as 3.50 cm. Which instrument did he use to measure it?

Let $[{\epsilon }_0]$ denote the dimensional formula of the permittivity of vacuum. If $\mathrm{M}=$ mass, $\mathrm{L}=$ length, $\mathrm{T}=$ time and $\mathrm{A}=$ electric current, then:

Resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference applied across it. If the percentage errors in the measurement of the current and the voltage difference are $3\%$ each, then error in the value of resistance of the wire is