Two coherent point sources \(S_1\) and \(S_2\) are separated by a small distance '\(d\)' as shown. The fringes obtained on the screen will be

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

The displacement and the increase in the velocity of a moving particle in the time interval of \(t\) to \((t+1) \mathrm{s}\) are \(125 \mathrm{~m}\) and \(50 \mathrm{~m} / \mathrm{s}\), respectively. The distance travelled by the particle in \((\mathrm{t}+2)^{\mathrm{th}} \mathrm{s}\) is __________ \(\mathrm{m}\).

For the determination of refractive index of glass slab, a travelling microscope is used whose main scale contains 300 equal divisions equals to 15 cm. The vernier scale attached to the microscope has 25 divisions equals to 24 divisions of main scale. The least count (LC) of the travelling microscope is (in cm) :

The time period of revolution of electron in its ground state orbit in a hydrogen atom is \(1.6 \times 10^{-16} \;\mathrm{s}\). The frequency of revolution of the electron in its first excited state (in \(s^{-1}\) ) is:

If \(\rho\) is the density and \(\eta\) is coefficient of viscosity of fluid which flows with a speed \(v\) in the pipe of diameter \(d\), the correct formula for Reynolds number \(R _{ e }\) is ..............

A capacitor of capacitance \(C\) is charged to a potential \(V\). The flux of the electric field through a closed surface enclosing the positive plate of the capacitor is \(......\)

A data consists of \(20\) observations \(x_1, x_2, \ldots, x_{20}\). If \(\sum_{i=1}^{20}(x_i + 5)^2 = 2500\) and \(\sum_{i=1}^{20}(x_i - 5)^2 = 100\), then the ratio of mean to standard deviation of this data is:

If \(\alpha \), \(\beta \) and \(\gamma \) are three consecutive terms of a non-constant \(G.P.\) such that the equations \(\alpha x^2 + 2\beta x + \gamma  = 0\) and \(x^2 + x -1 = 0\) have a common root, then \(\alpha(\beta  + \gamma )\) is equal to

In the given figure the blocks \(A , B\) and C weigh \(4 kg, 6 kg\) and 8 kg respectively. The co-efficient of sliding friction between any two surfaces is 0.5 . The force \(\overrightarrow{ F }\) required to slide the block C with constant speed is _________ N . (Used \(g =10 m / s ^2\) )

If an equation of a tangent to the curve, \(y = \cos \,\left( {x + f} \right),\, - 1\, - \pi \le x \le 1 + \pi ,\) is \(x + 2y = k\) then \(k\) is equal to

Let \(\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}\), and \(\overrightarrow{ u }\) be a vector such that \(|\vec{u}|=\alpha > 0\). If the minimum value of the scalar triple product \([\vec{u} \vec{v} \vec{w}]\) is \(-\alpha \sqrt{3401}\), and \(|\vec{u} . \hat{i}|^2=\frac{m}{n}\) where \(m\) and \(n\) are coprime natural numbers, then \(m + n\) is equal to \(.........\).

If \(z_1, z_2, z_3 \in C\) are the vertices of an equilateral triangle, whose centroid is \(\mathrm{z}_0\), then \(\sum_{\mathrm{k}=1}^3\left(\mathrm{z}_{\mathrm{k}}-\mathrm{z}_0\right)^2\) is equal to