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   ........ \(\%\)

Heat is supplied to a diatomic gas at constant pressure. Then the ratio of \(\Delta Q : \Delta U : \Delta W\) is _______.

A simple harmonic motion is represented by \(y\, = 5\,(\sin \,3\pi t\, + \,\sqrt 3 \,\cos \,3\pi t)\,cm\) The amplitude and time period of the motion are

The relative permittivity of distilled water is \(81 .\) The velocity of light in it will be \(....\,\times 10^{7}\,{m} / {s}\) \(\left(\right.\) Given \(\left.\mu_{r}=1\right)\)

The temperature of \(3.00\, {mol}\) of an ideal diatomic gas is increased by \(40.0^{\circ} {C}\) without changing the pressure of the gas. The molecules in the gas rotate but do not oscillate. If the ratio of change in internal energy of the gas to the amount of workdone by the gas is \(\frac{{x}}{10} .\) Then the value of \({x}\) (round off to the nearest integer) is ..... . \(\left(\right.\) Given \(\left.{R}=8.31\, {J} {mol}^{-1} {K}^{-1}\right)\)

A parallel plate capacitor was made with two rectangular plates, each with a length of \(l=3 \mathrm{~cm}\) and breath of \(\mathrm{b}=1 \mathrm{~cm}\). The distance between the plates is \(3 \mu \mathrm{~m}\). Out of the following, which are the ways to increase the capacitance by a factor of 10 ?
A. \(l=30 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
B. \(l=3 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=30 \mu \mathrm{~m}\)
C. \(l=6 \mathrm{~cm}, \mathrm{~b}=5 \mathrm{~cm}, \mathrm{~d}=3 \mu \mathrm{~m}\)
D. \(l=1 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=10 \mu \mathrm{~m}\)
E. \(l=5 \mathrm{~cm}, \mathrm{~b}=2 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}\)
Choose the correct answer from the options given below:

A source of sound \(S\) is moving with the velocity of \(50\,m/s\) towards a stationary observer. The observer measures the frequency of the sound as \(1000\,Hz.\) What will be the apparent frequency of the source when it is moving away from the observer after crossing him ... \(Hz\) ? (Take velocity of sound in air is \(350\,m/s\) )

Number of \(4-\)digit numbers that are less than or equal to \(2800\) and either divisible by \(3\) or by \(11\) , is equal to \(............\).

A Carnot's engine working between \(400\, K\) and \(800\, K\) has a work output of \(1200\, J\) per cycle. The amount of heat energy supplied to the engine from the source in each cycle is ........... \(J\)

Four identical rectangular plates with length, \(l=2\, cm\) and breadth, \(b =\frac{3}{2}\, cm\) are arranged as shown in figure. The equivalent capacitance between \(A\) and \(C\) is \(\frac{ x \varepsilon_{0}}{ d } .\) The value of \(x\) is (Round off to the Nearest Integer)

A thin uniform tube is bent into a circle of radius \(r\) in the virtical plane. Equal volumes of two immiscible liquids, whose densities are \({\rho _1}\) and \({\rho _2}\left( {{\rho _1} > {\rho _2}} \right)\) fill half the circle. The angle \(\theta\) between the radius vector passing through the common interface and the vertical is

If \(\mathrm{S}=\{\mathrm{a} \in \mathrm{R}:|2 \mathrm{a}-1|=3[\mathrm{a}]+2\{\mathrm{a}\}\}\), where \([\mathrm{t}]\) denotes the greatest integer less than or equal to \(t\) and \(\{t\}\) represents the fractional part of \(t\), then \(72 \sum_{\mathrm{a} \in \mathrm{S}} \mathrm{a}\) is equal to ....................

Let \(\left(2 x ^{2}+3 x +4\right)^{10}=\sum \limits_{ r =0}^{20} a _{ r } x ^{ r } \cdot\) Then \(\frac{ a _{7}}{ a _{13}}\) is equal to