The equations of two waves are given by \(y_{1}=5 \sin 2 \pi(x-v t) \,c m\,\) \(y_{2}=3 \sin 2 \pi(x-v t+1.5) \,c m\) These waves are simultaneously passing through a string. The amplitude of the resulting wave is.........\(cm\)

A small block of mass \(100\,g\) is tied to a spring of spring constant \(7.5\,N / m\) and length \(20\, cm\). The other end of spring is fixed at a particular point \(A\). If the block moves in a circular path on a smooth horizontal surface with constant angular velocity \(5\,rad / s\) about point \(A\), then tension in the spring is \(.........\,N\)

A potentiometer wire \(PQ\) of \(1\,m\) length is connected to a standard cell \(E _{1}\). Another cell \(E _{2}\) of emf \(1.02\, V\) is connected with a resistance \('r'\) and switch \(S\) (as shown in figure). With switch \(S\) open, the null position is obtained at a distance of \(49\, cm\) from \(Q\). The potential gradient in the potentiometer wire is.......\(V/cm\)

A nucleus disintegrates into two nuclear parts, in such a way that ratio of their nuclear sizes is \(1: 2^{1 / 3}\).Their respective speed have a ratio of \(n: 1\). The value of \(n\) is \(.....\).

A massless square loop, of wire of resistance \(10\,\Omega\). supporting a mass of \(1\,g\). hangs vertically with one of its sides in a uniform magnetic field of \(10^3\, G\), directed outwards in the shaded region. A dc voltage \(V\) is applied to the loop. For what value of V. the magnetic force will exactly balance the weight of the supporting mass of \(1\,g\) ? (If sides of the loop \(=10\,cm , g =10\,ms ^{-2}\) )

Three conductors of same length having thermal conductivity \(\mathrm{k}_1, \mathrm{k}_2\) and \(\mathrm{k}_3\) are connected as shown in figure.

Area of cross sections of \(1^{\text {st }}\) and \(2^{\text {nd }}\) conductor are same and for \(3^{\text {rd }}\) conductor it is double of the \(1^{\text {st }}\) conductor. The temperatures are given in the figure. In steady state condition, the value of \(\theta\) is _______ \({ }^{\circ} \mathrm{C}\).
(Given : \(\mathrm{k}_1=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_2=120 \mathrm{Js}^{-1}\mathrm{~m}^{-1}\) \( \mathrm{~K}^{-1}, \mathrm{k}_3=135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) )

A spherical body of mass \(2\,kg\) starting from rest acquires a kinetic energy of \(10000\,J\) at the end of \(5^{\text {th }}\) second. The force acted on the body is \(.....N\)

A monochromatic light wave with wavelength \(\lambda_1\) and frequency \(v_1\) in air enters another medium. If the angle of incidence and angle of refraction at the interface are \(45^{\circ}\) and \(30^{\circ}\) respectively, then the wavelength \(\lambda_2\) and frequency \(v_2\) of the refracted wave are :

If the sum and product of the first three term in an \(A.P\). are \(33\) and \(1155\), respectively, then a value of its \(11^{th}\) tern is

If the area of the bounded region \(R=\left\{(x, y): \max \left\{0, \log _{e} x\right\} \leq y \leq 2^{x}, \frac{1}{2} \leq x \leq 2\right\}\) is, \(\alpha\left(\log _{e} 2\right)^{-1}+\beta\left(\log _{e} 2\right)+\gamma\), then the value of \((\alpha+\beta-2 \gamma)^{2}\) is equal to:

\(f(x)=\left\{\begin{array}{cc}\frac{\sin (x-[x])}{x-[x]} & , \quad x \in(-2,-1) \\ \max \{2 x, 3[|x|]\} & , \quad|x|<1 \\ 1 & , \quad \text { otherwise }\end{array}\right.\) where \([t]\) denotes greatest integer \(\leq t\). If \(m\) is the number of points where \(f\) is not continuous and \(n\) is the number of points where \(f\) is not differentiable, then the ordered pair \(( m , n )\) is

Let \(f : [-1,3] \to  R\) be defined as \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  {\left| x \right| + \left[ x \right],}&{ - 1 \leq x < 1} \\ 
  {x + \left| x \right|,}&{1 \leq x < 2} \\ 
  {x + \left| x \right|,}&{2 \leq x \leq 3} 
\end{array}} \right.\)  where \([t]\) denotes the greatest integer less than or equal to \(t\). Then, \(f\) is discontinuous at:

The value of the definite integral \(\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{d x}{\left(1+e^{x \cos x}\right)\left(\sin ^{4} x+\cos ^{4} x\right)}\) is equal to: