Let $f:[0,\infty )\to \mathbb{R}$ be a differentiable function such that $f(x)=1-2x+{\int }_0^x{e}^{x-t}f(t)dt$ for all $x\in [0,\infty )$. Then the area of the region bounded by $y=f(x)$ and the coordinate axes is

The area of the region $\{(x,y):|x-y|\le y\le 4\sqrt{x}\}$ is

The pair of physical quantities not having the same dimensions is :

The expression given below shows the variation of velocity (v) with time (t), $v={\mathrm{At}}^2+\frac{\mathrm{Bt}}{\mathrm{C}+\mathrm{t}}$. The dimension of ABC is :

Match List - I with List - II. List - I List - II (A) Young’s Modulus (I) M L-1 T-1 (B) Torque (II) M L-1 T-2 (C) Coefficient of Viscosity (III) M-1 L3 T-2 (D) Gravitational Constant (IV) M L2 T-2 Choose the correct answer from the options given below :

If $B$ is magnetic field and ${\mu }_0$ is permeability of free space, then the dimensions of $\left(B/{\mu }_0\right)$ is

Given below are two statements : Statement I: In a vernier callipers, one vernier scale division is always smaller than one main scale division. Statement II : The vernier constant is given by one main scale division multiplied by the number of vernier scale divisions. In the light of the above statements, choose the correct answer from the options given below.

The maximum percentage error in the measurment of density of a wire is [Given, mass of wire $=(0.60\pm 0.003)\mathrm{g}$ radius of wire $=(0.50\pm 0.01)\mathrm{cm}$ length of wire $=(10.00\pm 0.05)\mathrm{cm}]$

Match List - I with List - II. List - I List - II (A) Angular Impulse (I) M0 L2 T-2 (B) Latent Heat (II) M L2 T-3 A-1 (C) Electrical resistivity (III) M L2 T-1 (D) Electromotive force (IV) M L3 T-3 A-2 Choose the correct answer from the options given below:

The position of a particle moving on $x$-axis is given by $x(t)=A\sin t+B{\cos }^{2}t+C{t}^2+D$, where $t$ is time. The dimension of $\frac{ABC}{D}$ is

The energy of a system is given as $\mathrm{E}(\mathrm{t})={\alpha }^3{\mathrm{e}}^{-\beta t}$, where t is the time and $\beta =0.3{\mathrm{s}}^{-1}$. The errors in the measurement of $\alpha$ and $t$ are $1.2 \%$ and $1.6 \%$, respectively. At $t=5\mathrm{s}$, maximum percentage error in the energy is :

Match List - I with List - II .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List - I List - II (A) Permeability of free space (I) $\left[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-2}\right]$ (B) Magnetic field (II) $\left[\mathrm{M}{\mathrm{T}}^{-2}{\mathrm{A}}^{-1}\right]$ (C) Magnetic moment (III) $\left[\mathrm{ML}{\mathrm{T}}^{-2}{\mathrm{A}}^{-2}\right]$ (D) Torsional constant (IV) $\left[{\mathrm{L}}^2\mathrm{A}\right]$ Choose the correct answer from the options given below :

For an experimental expression $y=\frac{32.3\times 1125}{27.4}$, where all the digits are significant. Then to report the value of $y$ we should write

A quantity Q is formulated as $X^{-2}{Y}^{+\frac{3}{2}}{Z}^{-\frac{2}{5}}$. X, Y, and Z are independent parameters which have fractional errors of 0.1, 0.2, and 0.5, respectively in measurement. The maximum fractional error of Q is

Match List I with List II. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} List - I List - II (A) Coefficient of viscosity (I) $\left[{\mathrm{ML}}^0{\mathrm{T}}^{-3}\right]$ (B) Intensity of wave (II) $\left[{\mathrm{ML}}^{-2}{\mathrm{T}}^{-2}\right]$ (C) Pressure gradient (III) $\left[{\mathrm{M}}^{-1}{\mathrm{LT}}^2\right]$ (D) Compressibility (IV) $\left[{\mathrm{ML}}^{-1}{\mathrm{T}}^{-1}\right]$ Choose the correct answer from the options given below:

The dimension of $\sqrt{\frac{{\mu }_0}{{ϵ}_0}}$ is equal to that of: (${\mu }_0$ = Vacuum permeability and ${ϵ}_0$ = Vacuum permittivity)

If ${\mu }_0$ and ${ϵ}_0$ are the permeability and permittivity of free space, respectively, then the dimension of $\left(\frac{1}{{\mu }_0{ϵ}_0}\right)$ is :

Given a charge q , current I and permeability of vacuum ${\mu }_{{\mathrm{o}}^{\ast }}$. Which of the following quantity has the dimension of momentum ?

Match List - I with List - II. List - I List - II (A) Mass density (I) [ML2T−3] (B) Impulse (II) [MLT−1] (C) Power (III) [ML2T0] (D) Moment of inertia (IV) [ML−3T0] Choose the correct answer from the options given below :

A person measures mass of 3 different particles as $435.42\mathrm{g},226.3\mathrm{g}$ and 0.125 g . According to the rules for arithmetic operations with significant figures, the addition of the masses of 3 particles will be.