The displacement equations of two interfering waves are given by \(y_1  =10 \sin \left(\omega t+\frac{\pi}{3}\right) cm\) \(y_2 =5[\sin (\omega t)+\sqrt{3} \cos \omega t] \;cm\) respectively. The amplitude of the resultant wave is \(.............cm\).

Select the correct statement from the following

Water of mass \(m\) gram is slowly heated to increase the temperature from \(T_1\) to \(T_2\) The change in entropy of the water, given specific heat of water is \(1 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}\), is :

Two long parallel conductors \(S_{1}\) and \(S_{2}\) are separated by a distance \(10 \,cm\) and carrying currents of \(4\, A\) and \(2 \,A\) respectively. The conductors are placed along \(x\)-axis in \(X - Y\) plane. There is a point \(P\) located between the conductors (as shown in figure). A charge particle of \(3 \pi\) coulomb is passing through the point \(P\) with velocity \(\overrightarrow{ v }=(2 \hat{ i }+3 \hat{ j }) \,m / s\); where \(\hat{i}\) and \(\hat{j} \quad\) represents unit vector along \(x\) and \(y\) axis respectively. The force acting on the charge particle is \(4 \pi \times 10^{-5}(-x \hat{i}+2 \hat{j}) \,N\). The value of \(x\) is

The percentage decrease in the weight of a rocket, when taken to a height of \(32 km\) above the surface of earth will, be\(.....\%\) (Radius of earth \(=6400\,km\) )

The disintegration rate of a certain radioactive sample at any instant is \(4250\) disintegrations per minute.\(10\) minutes later, the rate becomes \(2250\) disintegrations per minute. The approximate decay cons \(.........\min^{-1}\)

List - I	List - II
(A) Heat capacity of body	\(\text{(I) } \mathrm{J} \mathrm{kg}^{-1}\)
(B) Specific heat capacity of body	\(\text{(II) } \mathrm{JK}^{-1}\)
(C) Latent heat	\(\text{(III) } \mathrm{J} \mathrm{kg}^{-1} \mathrm{~K}^{-1}\)
(D) Thermal conductivity	\(\text{(IV) } \mathrm{Jm}^{-1} \mathrm{~K}^{-1} \mathrm{~s}^{-1}\)

Choose the correct answer from the options given below:

If the function \(f\) given by \(f\,(x)\, = \,{x^3} - 3(a - 2){x^2} + 3ax\, + 7,\) for some \(a\in R\) is increasing in \((0, 1]\) and decreasing in \([1, 5),\) then a root of the equation, \(\frac{{f(x) - 14}}{{{{(x - 1)}^2}}} = 0\,(x\, \ne 1)\) is

By what percentage will the illumination of the lamp decrease if the current drops by \(20 \%\) ?

If \(\int\left(\frac{1}{x}+\frac{1}{x^3}\right) \left(\sqrt[23]{3 x^{-24}+x^{-26}}\right) d x \) \( =-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^\gamma\right)^{\frac{\alpha+1}{\alpha}}+C, x \gt 0,\) \((\alpha, \beta, \gamma \in Z)\), where \(C\) is the constant of integration, then \(\alpha+\beta+\gamma\) is equal to ________ .

Two wires A and B are made of same material having ratio of lengths \(\frac{L_A}{L_B}=\frac{1}{3}\) and their diameters ratio \(\frac{d_A}{d_B}=2\). If both the wires are stretched using same force, what would be the ratio of their respective elongations?

The area of the region enclosed by the curves \(y=\mathrm{e}^x, y=\left|\mathrm{e}^x-1\right|\) and \(y\)-axis is:

The truth of the given circuit diagram is _______.