Two coherent point sources \(S_1\) and \(S_2\) vibrating in phase emit light of wavelength \(\lambda\). The separation between them is \(2 \lambda\) as shown in figure. The first bright fringe is formed at \(P\) due to interference on a screen placed at a distance \(D\) from \(S_1(D>>\lambda)\), then \(O P\) is

The difference between the radii of $n^{\text{th }}$ and $(n+1{)}^{th}$ orbits of hydrogen atom is equal to the radius of $(\mathrm{n}-1{)}^{\mathrm{th}}$ orbit of hydrogen. The angular momentum of the electron in the ${\mathrm{n}}^{\mathrm{th}}$ orbit is$\_\_\_\_\_\_$($h$ is Planck's constant)

A block of mass \(90 \mathrm{~kg}\) is suspended by three strings \(A, B\) and \(C\) as shown in figure. Tensions in the strings \(A, B\) and \(C\) respectively are \(\left(g=10 \mathrm{~ms}^{-2}, \sin 37^{\circ}=0.6, \cos 37^{\circ}=0.8\right)\)

A plane electromagnetic wave travels in free space along $z$- axis. At a particular point in space, the electric field along $x$-axis is $8.7 \mathrm{V} {\mathrm{m}}^{-1}$. The magnetic field along $\mathrm{y}$- axis is

The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would:

When a rectangular coil is rotated in a uniform magnetic field about an axis passing through its centre and perpendicular to the field, the emf induced in the coil varies

A gas at \(37^{\circ} \mathrm{C}\) is compressed adiabatically to half of its volume, then the final temperature of the gas is
(Ratio of specific heat capacities of the gas is 1.5)

\(\int_0^\pi \frac{1}{1+\sin x} d x\) is equal to

The pair of molecules / ions with the same bond order value is

Three charged particles of each mass \(0.1 \mathrm{~g}\) and charge ' \(\mathrm{q}\) ' are suspended from a common rigid point by insulated massless threads of each \(1 \mathrm{~m}\) long. If the three particles are in equilibrium and are located at the corners of an equilateral triangle of side \(3 \mathrm{~cm}\), the charge ' \(q\) ' on each particle is ____ \(\mathrm{nC}\). (The angle made by the line joining the centroid of the triangle and the point of suspension with the vertical is very small).
(Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) and \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\) )

The line \(x-2 y-3=0\) cuts the parabola \(y^2=4 \mathrm{ax}\) at the points P and Q . If the focus of this parabola is \(\left(\frac{1}{4}, k\right)\), then \(P Q=\)

\((\mathrm{r}, \theta)\) denotes \(r(\cos \theta+i \sin \theta)\). If \(x=(1, \alpha), y=(1, \beta)\), \(\mathrm{z}=(1, \gamma)\) and \(x+y+z=0\) then \(\sum \cos (2 \alpha-\beta-\gamma)=\)