A proton, a neutron, an electron and an \(\alpha\,-\)particle have same energy. If \(\lambda_{ p }, \lambda_{ n }, \lambda_{ e }\) and \(\lambda_{\alpha}\) are the de Broglie's wavelengths of proton, neutron, electron and \(\alpha\) particle respectively, then choose the correct relation from the following

A current through a wire depends on time as \(i =\alpha_{0} t +\beta t ^{2}\) where \(\alpha_{0}=20 A / s\) and \(\beta=8 As ^{-2} .\) Find the charge crossed through a section of the wire in \(15 \,s\) (in \(C\))

White light is passed through a double slit and interference is observed on a screen \(1.5 \,{m}\) away. The separation between the slits is \(0.3 \,{mm}\). The first violet and red fringes are formed \(2.0 \,{mm}\) and \(3.5\, {mm}\) away from the central white fringes. The difference in wavelengths of red and voilet light is \(.... \,{nm} .\)

Two closed vessels of same volume are joined through a narrow tube and both vessels are filled with air of pressure \(90\) kPa and temperature \(400\) K. Keeping the temperature of one vessel constant at \(400\) K the second vessel temperature is raised to \(500\) K. The final pressure in the vessels is _______ kPa.

A monochromatic source of light operating at \(15\) kW emits \(2.5\times 10^{22}\) photons/s. The region of an electromagnetic spectrum to which the emitted electromagnetic radiation belongs to ________. (Take \(h=6.6\times 10^{-34}\) J·s and \(c=3\times 10^8\) m/s).

The \(rms\) speeds of the molecules of Hydrogen, Oxygen and Carbondioxide at the same temperature are \({V}_{{H}}, {V}_{0}\) and \({V}_{{C}}\) respectively then

The position-time graphs for two students \(A\) and \(B\) returning from the school to their homes are shown in figure \((A)\) \(A\) lives closer to the school \((B)\) \(B\) lives closer to the school \((C)\) \(A\) takes lesser time to reach home \((D)\) \(A\) travels faster than \(B\) \((E)\) \(B\) travels faster than \(A\) Choose the correct answer from the options given below :

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

If the variable line \(3 x+4 y=\alpha\) lies between the two circles \((x-1)^{2}+(y-1)^{2}=1\) and \((x-9)^{2}+(y-1)^{2}=4\) without intercepting a chord on either circle, then the sum of all the integral values of \(\alpha\) is .... .

Let the foot of perpendicular from a point \(P(1,2,-1)\) to the straight line  \(L: \frac{x}{1}=\frac{y}{0}=\frac{z}{-1}\) be \(N\). Let a line be drawn from \(P\) parallel to the plane \(x+y+2 z=0\) which meets \(L\) at point \(Q\). If \(\alpha\) is the acute angle between the lines \(\mathrm{PN}\) and \(\mathrm{PQ}\), then \(\cos \alpha\) is equal to \(.....\)

For some \(\theta\in(0,\frac{\pi}{2}),\) let the eccentricity and the length of the latus rectum of the hyperbola \(x^{2}-y^{2}sec^{2}\theta=8\) be \(e_{1}\) and \(l_{1}\), respectively, and let the eccentricity and the length of the latus rectum of the ellipse \(x^{2}sec^{2}\theta+y^{2}=6\) be \(e_{2}\) and \(l_{2},\) respectively. If \(e_{1}^{2}=e_{2}^{2}(sec^{2}\theta+1)\), then \((\frac{l_{1}l_{2}}{e_{1}e_{2}})tan^{2}\theta\) is equal to ___ .

A boy pushes a box of mass \(2\, kg\) with a force \(\overrightarrow{ F }=(20 \hat{ i }+10 \hat{ j }) N\) on a frictionless surface. If the box was initially at rest, then ...........  \(m\) is displacement along the \(x-\)axis after \(10\, s\).

If \(\alpha\) and \(\beta\) are the roots of the equation \(2 z^2-3 z-2 \mathrm{i}=0\), where \(\mathrm{i}=\sqrt{-1}\), then \(16 \cdot \operatorname{Re}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \operatorname{lm}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)\) is equal to