Surface tension of two liquids (having same densities), \(T_1\) and \(T_2\), are measured using capillary rise method utilizing two tubes with inner radii of \(r_1\) and \(r_2\) where \(r_1>r_2\). The measured liquid heights in these tubes are \(h_1\) and \(h_2\) respectively. [Ignore the weight of the liquid about the lowest point of miniscus]. The heights \(h _1\) and \(h _2\) and surface tensions \(T _1\) and \(T _2\) satisfy the relation :

In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by \(\hat{ k }\) and \(2 \hat{ i }-2 \hat{ j },\) respectively What is the unit vector along direction of propagation of the wave.

A mass \(m\) attached to free end of a spring executes SHM with a period of \(1\; s\). If the mass is increased by \(3\; kg\) the period of oscillation increases by one second, the value of mass \(m\) is \(..............kg\).

A circular loop of radius \(20\text{ cm}\) and resistance \(2\ \Omega\) is placed in a time varying magnetic field \(\vec{B} = (2t^2 + 2t + 3)\text{ T}\). At \(t = 0\), for the plane of the loop being perpendicular to the magnetic field and, the induced current in the loop at \(t = 3\text{ s}\) is \(\dfrac{\alpha}{50}\text{ A}\). The value of \(\alpha\) is _______.
(Take \(\pi = 22/7\))

The magnitudes of power of a biconvex lens (refractive index 1.5) and that of a plano-concave lens (refractive index \(=1.7\)) are same. If the curvature of plano-concave lens exactly matches with the curvature of back surface of the biconvex lens, then ratio of radius of curvature of front and back surface of the biconvex lens is ___________ .

If \(917 \mathring A\) be the lowest wavelength of Lyman series then the lowest wavelength of Balmer series will be .......\(\mathring A\).

The electric field of a plane electromagnetic wave propagating along the \(x\) direction in vacuum is \(\overrightarrow{ E }= E _{0} \hat{ j } \cos (\omega t - kx )\). The magnetic field \(\overrightarrow{ B },\) at the moment \(t =0\) is :

If the set of all values of a, for which the equation \(5 x^3-15 x-a=0\) has three distinct real roots, is the interval \((\alpha, \beta)\), then \(\beta-2 \alpha\) is equal to ______

The value \(\sum \limits_{ r =0}^{22}{ }^{22} C _{ r }{ }^{23} C _{ r }\) is \(.......\)

The term independent of \(x\) in the expansion of \(\left( {\frac{1}{{60}} - \frac{{{x^8}}}{{81}}} \right).{\left( {2{x^2} - \frac{3}{{{x^2}}}} \right)^6}\) is equal to

If the system of equations  \(x+2 y+3 z=3\)  ; \(4 x+3 y-4 z=4\) ; \(8 x+4 y-\lambda z=9+\mu\) has infinitely many solutions, then the ordered pair \((\lambda, \mu)\) is equal to

If \(y=y(x)\) is the solution of the differential equation \(x \frac{d y}{d x}+2 y=x e^{x}, y(1)=0\) then the local maximum value of the function \(z(x)=x^{2} y(x)-e^{x}\), \(x \in R\) is

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to