A deuteron and a proton moving with equal kinetic energy enter into to a uniform magnetic field at right angle to the field. If \(r_{d}\) and \(r_{p}\) are the radii of their circular paths respectively, then the ratio \(\frac{r_{d}}{r_{p}}\) will be \(\sqrt{ x }: 1\) where \(x\) is ..........

Water boils in an electric kettle in \(20\) minutes after being switched on. Using the same main supply, the length of the heating element should be _______ to _______ times of its initial length if the water is to be boiled in \(15\) minutes.

One mole of an ideal monoatomic gas is taken along the path \(ABCA\)  as shown in the \(PV\) diagram. The maximum temperature attained by the gas along the path \(BC\)  is given by

Two beams, \(A\) and \(B\), of plane polarized light with mutually perpendicular planes of polarization are seen through a polaroid. From the position when the beam \(A\) has maximum intensity ( and beam \(B\) has zero intensity), a rotation of polaroid through \(30 ^o \) makes the two beams appear equally bright. If the initial intensities of the two beams are \(I_A\) and \(I_B\)  respectively, then \(\frac{{{I_A}}}{{{I_B}}}\)=

An object with mass 500 g moves along x -axis with speed \(v=4 \sqrt{x} \mathrm{~m} / \mathrm{s}\). The force acting on the object is :

A constant magnetic field of \(1 \,{T}\) is applied in the \({x}\,>\,0\) region. A metallic circular ring of radius \(1 \,{m}\) is moving with a constant velocity of \(1 \,{m} / {s}\) along the \(x\)-axis. At \(t=0\) \(s,\) the centre of \(O\) of the ring is at \({x}=-1 \,{m}\). What will be the value of the induced \(emf\) in the ring at \(t=1\, s\) ? (Assume the velocity of the ring does not change.) (In \({V}\))

A poly-atomic molecule ( \(C_V=3 R, C_P=4 R\), where \(R\) is gas constant) goes from phase space point \(\mathrm{A}\left(\mathrm{P}_{\mathrm{A}}=10^5 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{A}}=4 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{B}\left(\mathrm{P}_{\mathrm{B}}=5 \times 10^4 \mathrm{~Pa}, \mathrm{~V}_{\mathrm{B}}=6 \times 10^{-6} \mathrm{~m}^3\right)\) to point \(\mathrm{C}\left(\mathrm{P}_{\mathrm{C}}=10^4\right.\) \(\left.\mathrm{Pa}, \mathrm{V}_{\mathrm{C}}=8 \times 10^{-6} \mathrm{~m}^3\right)\). A to B is an adiabatic path and \(B\) to \(C\) is an isothermal path. The net heat absorbed per unit mole by the system is :

Ship \(A\) is sailing towards north -east with velocity \(\vec v = 30\,\hat i + 50\hat j\,km/hr\) where \(\hat i\) points east and \(\hat j\) , north. Ship \(B\) is at a distance of \(80\, km\) east and \(150\, km\) north of Ship \(A\) and is sailing towards west at \(10\, km/hr\). \(A\) will be at minimum distance from \(B\) is.........\(hrs\)

The number of integral values of \(k\), for which one root of the equation \(2 x^2-8 x+k=0\) lies in the interval \((1,2)\) and its other root lies in the interval \((2,3)\), is :

Let \(A , B , C\) be \(3 \times 3\) matrices such that \(A\) is symmetric and \(B\) and \(C\) are skew-symmetric.Consider the statements \((S1): A ^{13} B ^{26}- B ^{26} A ^{13}\) is symmetric \((S2):A ^{26} C ^{13}- C ^{13} A ^{26}\) is symmetric Then,

Let \(f : R \to R\) be a function such that \(f(2 - x)\, = f(2 + x)\) and \(f(4 -x)\, = f(4 + x)\), for all \(x \in  R\) and \(\int\limits_0^2 {f\left( x \right)\,dx = 5} \) . Then the value of \(\int\limits_{10}^{50} {f\left( x \right)\,\,dx} \) is

If \(\overrightarrow{ a }=\hat{ i }+2 \hat{ k }, \overrightarrow{ b }=\hat{ i }+\hat{ j }+\hat{ k }, \overrightarrow{ c }=7 \hat{ i }-3 \hat{ k }+4 \hat{ k }\) \(\overrightarrow{ r } \times \overrightarrow{ b }+\overrightarrow{ b } \times \overrightarrow{ c }=\overrightarrow{0}\) and \(\overrightarrow{ r } \cdot \overrightarrow{ a }=0\) then \(\overrightarrow{ r } \cdot \overrightarrow{ c }\) is equal to :

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly \(7\) solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in N\) then \(\sum_{\mathrm{k}=1}^{\mathrm{n}} \frac{\mathrm{k}}{2^{\mathrm{k}}}\) is equal to :