Units, Dimensions & Errors · JEE Main previous year questions

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 ________ $\times 10^{-6}\text{ m}$.

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.$\frac{\Delta A}{A}+\frac{\Delta B}{B}-\frac{\Delta C}{C}$

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 ____%.

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

The temperature of an ideal gas in 3-dimensions is $300\mathrm{K}$. The corresponding de-Broglie wavelength of the electron approximately at $300\mathrm{K}$, is : [$m_{\mathrm{e}}$ = mass of electron $=9\times 10^{-31}\mathrm{kg}$, $h$ = Planck constant $=6.6\times 10^{-34}\mathrm{Js}$, $k_{\mathrm{B}}$ = Boltzmann constant $=1.38\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$]

1. A.$6.26\mathrm{nm}$
2. B.$8.46\mathrm{nm}$
3. C.$2.26\mathrm{nm}$
4. D.$3.25\mathrm{nm}$

A student determined Young's Modulus of elasticity using the formula $Y=\frac{Mg{L}^3}{4b{d}^3\delta }$. The value of g is taken to be $9.8m/{\mathrm{s}}^2$, without any significant error, his observations are as follows. $$\begin{array}{lll}\text{Physical Quantity}& \text{Least count of the Equipment used for measurement}& \text{Observed value}\\ \text{Mass (M)}& 1\mathrm{g}& 2\mathrm{kg}\\ \text{Length of bar (L)}& 1\mathrm{mm}& 1\mathrm{m}\\ \text{Breadth of bar (b)}& 0.1\mathrm{mm}& 4\mathrm{cm}\\ \text{Thickness of bar (d)}& 0.01\mathrm{mm}& 0.4\mathrm{cm}\\ \text{Depression }(\delta )& 0.01\mathrm{mm}& 5\mathrm{mm}\end{array}$$ Then the fractional error in the measurement of Y is :

1. A.0.0083
2. B.0.0155
3. C.0.155
4. D.0.083

1. A.at x > $x_4$ K.E. is constant throughout the region.
2. B.at x < $x_1$, K.E. is smallest and the particle is moving at the slowest speed.
3. C.at x = $x_2$, K.E. is greatest and the particle is moving at the fastest speed.
4. D.at x = $x_3$, K.E. = 4 J.

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

1. A.5 and $\pm 2$
2. B.4 and $\pm 3$
3. C.$\frac{16}{3}$ and $\pm \frac{3}{2}$
4. D.8 and $\pm 2$

A particle of mass 'm' is moving in time 't' on a trajectory given by $$\vec{r}=10\alpha t^2\hat{i}+5\beta (t-5)\hat{j}$$ Where $\alpha$ and $\beta$ are dimensional constants. The angular momentum of the particle becomes the same as it was for t = 0 at time t = ____ seconds.

The force is given in terms of time t and displacement x by the equation F = A cos Bx + C sin Dt The dimensional formula of $\frac{AD}{B}$ is :

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

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 ____.

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 :

1. A.$\frac{10a}{n}$
2. B.$\frac{10na}{(n-1)}$
3. C.$\left(\frac{n-1}{10n}\right)a$
4. D.$\frac{10a}{(n-1)}$

In a typical combustion engine the workdone by a gas molecule is given by $W={\alpha }^2\beta e^{-\frac{\beta x^2}{kT}}$, where x is the displacement, k is the Boltzmann constant and T is the temperature. If $\alpha$ and $\beta$ are constants, dimensions of $\alpha$ will be :

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

If 'C' and 'V' represent capacitance and voltage respectively then what are the dimensions of $\lambda$ where $C/V=\lambda$ ?

1. A.$[M^{-3}{L}^{-4}{I}^3{T}^7]$
2. B.$[M^{-2}{L}^{-3}{I}^2{T}^6]$
3. C.$[M^{-2}{L}^{-4}{I}^3{T}^7]$
4. D.$[M^{-1}{L}^{-3}{I}^{-2}{T}^{-7}]$

If e is the electronic charge, c is the speed of light in free space and h is Planck's constant, the quantity $\frac{1}{4\pi {\epsilon }_0}\frac{|e{|}^2}{hc}$ has dimensions of :

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

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 0.5mm is noticed on the pitch scale. The nature of zero error involved and the least count of the screw gauge, are respectively :

1. A.Positive, $0.1\text{ mm}$
2. B.Positive, $0.1\mu \mathrm{m}$
3. C.Positive, $10\mu \mathrm{m}$
4. D.Negative, $2\mu \mathrm{m}$

A clock has a continuously moving second's hand of $0.1\text{ m}$ length. The average acceleration of the tip of the hand (in units of $\mathrm{m}{\mathrm{s}}^{-2}$) is of the order of :

1. A.$10^{-3}$
2. B.$10^{-1}$
3. C.$10^{-2}$
4. D.$10^{-4}$

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, the value of x 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 $5.50\text{ mm}$, $5.55\text{ mm}$, $5.45\text{ mm}$, $5.65\text{ mm}$. The average of these four readings is $5.5375\text{ mm}$ and the standard deviation of the data is $0.07395\text{ mm}$. The average diameter of the pencil should therefore be recorded as :

1. A.$(5.54\pm 0.07)\text{ mm}$
2. B.$(5.5375\pm 0.0740)\text{ mm}$
3. C.$(5.5375\pm 0.0739)\text{ mm}$
4. D.$(5.538\pm 0.074)\text{ mm}$

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

1. A.2.123 cm
2. B.2.124 cm
3. C.2.125 cm
4. D.2.121 cm

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

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

The least count of the main scale of a vernier callipers is 1 mm. Its vernier scale is divided into 10 divisions and coincide with 9 divisions of the main scale. When jaws are touching each other, the $7^{\mathrm{th}}$ division of vernier scale coincides with a division of main scale and the zero of vernier scale is lying right side of the zero of main scale. When this vernier is used to measure length of a cylinder the zero of the vernier scale between 3.1 cm and 3.2 cm and $4^{\mathrm{th}}$ VSD coincides with a main scale division. The length of the cylinder is : (VSD is vernier scale division)

1. A.3.21 cm
2. B.2.99 cm
3. C.3.07 cm
4. D.3.2 cm

If speed V, area A and force F are chosen as fundamental units, then the dimension of Young’s modulus will be

1. A.$\mathrm{F}{\mathrm{A}}^{-1}{\mathrm{V}}^0$
2. B.$\mathrm{F}{\mathrm{A}}^2{\mathrm{V}}^{-1}$
3. C.$\mathrm{F}{\mathrm{A}}^2{\mathrm{V}}^{-2}$
4. D.$\mathrm{F}{\mathrm{A}}^2{\mathrm{V}}^{-3}$

A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature T. Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy (in units of RT) of the mixture is :

1. A.11
2. B.20
3. C.15
4. D.13