Class 12th

If ${\text{∮}\mathrm{}}_{\mathrm{s}}\vec{\mathrm{E}}\cdot \overrightarrow{\mathrm{d}\mathrm{S}}=0$  over a surface, then:

An electric dipole is placed at an angle of ${30}^{?}$ with an electric field of intensity $2*{10}^5{\mathrm{N}\mathrm{C}}^{-1}$ . It experiences a torque equal to $4\mathrm{N}\mathrm{m}$ . Calculate the magnitude of charge on the dipole, if the dipole length is $2\mathrm{c}\mathrm{m}$ .

When two monochromatic lights of frequency, $v$ $$ and $\frac{v}{2}$ $\frac{}{}$ are incident on a photoelectric metal, their stopping potential becomes $\frac{V_S}{2}$ $\frac{{}_{}}{}$ and $V_S$ ${}_{}$ respectively. The threshold frequency for this metal is

When light propagates through a material medium of relative permittivity ${\epsilon }_{\mathrm{r}}$ and relative permeability ${\mu }_{\mathrm{r}}$ , the velocity of light, $\mathrm{v}$ is given by : ( $\mathrm{c}$  - velocity of light in vacuum)

A square loop of side $1\mathrm{m}$ $$ and resistance $1\mathrm{\Omega }$ $$ is placed in a magnetic field of $0.5\mathrm{T}$ $$ . If the plane of loop is perpendicular to the direction of a magnetic field, the magnetic flux through the loop is

Let P = [[-30, 20, 56],,] and A = [[2, 7, ω²], [-1, -ω, 1], [0, -ω, -ω+1]] where ω = (-1 + i√3)/2, and I be the identity matrix of order 3. If the determinant of the matrix (P?¹AP - I)² is αω², then the value of α is equal to.

The total number of 3 x 3 matrices A having entries from the set {0,1,2,3} such that the sum of all the diagonal entries of AA? is 9, is equal to.

Let f: (0, 2) → R be defined as f(x) = log?(1 + tan(πx/4)). Then, lim (n→∞) (2/n) [f(1/n) + f(2/n) + . + f(1)] is equal to.

Let A(1,0), B(6,2) and C(3/2, 6) be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point (-7/6, -1/3) is

Let the curve y = y(x) be the solution of the differential equation, dy/dx = 2(x + 1). If the numerical value of area bounded by the curve y = y(x) and x-axis is 4√8/3, then the value of y(1) is equal to.