Units, Dimensions & Errors · JEE Advanced previous year questions

1. A.0.12 cm
2. B.0.11 cm
3. C.0.13 cm
4. D.0.14 cm

Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm, 0.05 mm, and 6.0 $\mu$m, respectively. Which of the following option(s) give(s) the volume of the strip in $\mathrm{c}{\mathrm{m}}^3$ with correct significant figures:

1. A.$3.2\times 10^{-5}$
2. B.$32.0\times 10^{-6}$
3. C.$3.0\times 10^{-5}$
4. D.$3\times 10^{-5}$

Consider a system of three connected strings, $S_1$, $S_2$ and $S_3$ with uniform linear mass densities $\mu$ kg/m, $4\mu$ kg/m and $16\mu$ kg/m, respectively, as shown in the figure. $S_1$ and $S_2$ are connected at the point $P$, whereas $S_2$ and $S_3$ are connected at the point $Q$, and the other end of $S_3$ is connected to a wall. A wave generator O is connected to the free end of $S_1$. The wave from the generator is represented by $y=y_0\cos (\omega t-kx)$ cm, where $y_0$, $\omega$ and $k$ are constants of appropriate dimensions. Which of the following statements is/are correct:

1. A.When the wave reflects from $P$ for the first time, the reflected wave is represented by $y={\alpha }_1{y}_0\cos (\omega t+kx+\pi )$ cm, where ${\alpha }_1$ is a positive constant.
2. B.When the wave transmits through $P$ for the first time, the transmitted wave is represented by $y={\alpha }_2{y}_0\cos (\omega t-kx)$ cm, where ${\alpha }_2$ is a positive constant.
3. C.When the wave reflects from $Q$ for the first time, the reflected wave is represented by $y={\alpha }_3{y}_0\cos (\omega t-kx+\pi )$ cm, where ${\alpha }_3$ is a positive constant.
4. D.When the wave transmits through $Q$ for the first time, the transmitted wave is represented by $y={\alpha }_4{y}_0\cos (\omega t-4kx)$ cm, where ${\alpha }_4$ is a positive constant.

A temperature difference can generate e.m.f. in some materials. Let $S$ be the e.m.f. produced per unit temperature difference between the ends of a wire, $\sigma$ the electrical conductivity and $\kappa$ the thermal conductivity of the material of the wire. Taking $M,L,T,I$ and $K$ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $Z=\frac{S^2\sigma }{\kappa }$ is:

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

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space ${\epsilon }_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^{\alpha }{\epsilon }_0^{\beta }{h}^{\gamma }{c}^{\delta }$ and $n$ is a non-zero integer, then $(\alpha ,\beta ,\gamma ,\delta )$ is given by

1. A.$(2n,-n,-n,-n)$
2. B.$(n,-n,-2n,-n)$
3. C.$(n,-n,-n,-2n)$
4. D.$(2n,-n,-2n,-2n)$

The specific heat capacity of a substance is temperature dependent and is given by the formula $C=kT$, where $k$ is a constant of suitable dimensions in SI units, and $T$ is the absolute temperature. If the heat required to raise the temperature of $1\text{ kg}$ of the substance from $-73{}^{\circ }\mathrm{C}$ to $27{}^{\circ }\mathrm{C}$ is $nk$, the value of $n$ is _____. [Given: $0\text{ K}=-273{}^{\circ }\mathrm{C}$.]

A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M(\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_c(r)=m\alpha /r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_c(r)$, but with a new time period $T_1$, then $(T_1^2-T_0^2)/T_1^2$ is given by [$G$ is the gravitational constant.]

1. A.$\frac{3\alpha }{GM{r}_0^2}$
2. B.$\frac{\alpha }{2GM{r}_0^2}$
3. C.$\frac{\alpha }{GM{r}_0^2}$
4. D.$\frac{2\alpha }{GM{r}_0^2}$

The dimensions of a cone are measured using a scale with a least count of 2 mm. The diameter of the base and the height are both measured to be 20.0 cm. The maximum percentage error in the determination of the volume is ____.