Class 12th

The value of the integral ${\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{dx}{\left(1+e^x\right)\left(si{n}^{6}x+co{s}^{6}x\right)}isequalto}$

Let f(x) $\{\begin{array}{ccc}\hfill \frac{sin\left(x-\left[x\right]\right)}{x-\left[x\right]}\hfill & \hfill ,\hfill & \hfill x\in \left(-2,-1\right)\hfill \\ \hfill max\left\{2x,3\left[\left|x\right|\right]\right\}\hfill & \hfill ,\hfill & \hfill \left|x\right|<1\hfill \\ \hfill 1\hfill & \hfill ,\hfill & \hfill otherwise\hfill \end{array}$

where [t] denotes greatest integer $\le t.$ If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, then the ordered pair (m, n) is

Let the system of linear equations

x + y + $\alpha z=2$

3x + y + z = 4

x + 2z = 1

have a unique solution $\left(x^{\ast },y^{\ast },z^{\ast }\right).$ If $\left(\alpha ,x^{\ast }\right),\left(y^{\ast },\alpha \right)and\left(x^{\ast },-y^{\ast }\right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

The sum of all the real roots of the equation $\left(e^{2x}-4\right)\left(6e^{2x}-5e^x+1\right)=0is$

A circular coil of 1000 turns each with area 1m² is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of 0.07 T. The maximum voltage generation will be ________V.

A ray of light is incident at an angle of incidence $60°$ on the glass slab of refractive index $\sqrt[]{3}.$ After refraction, the light ray emerges out from other parallel faces and lateral shift between incident ray and emergent ray is $4\sqrt[]{3}$ cm. The thickness of the glass slab is _______cm.

A sample contains 10^-2kg each of two substances A and B with half lives 4s and 8s respectively. The ration of their atomic weights is 1 : 2. The ratio of the amounts of A and B after16s is $\frac{x}{100}$ . The value of x is ______.

In the given circuit, the value of current I_L will be _________mA. (When R_L = 1 k $\Omega$ )

A potentiometer wire of length 10m and resistance 20 $\Omega$ is connected in series with a 25 V battery and an external resistance $30\Omega$ . A cell of emf E in secondary circuit is balanced by 250cm long potentiometer wire. The value of E (in volt) is $\frac{x}{10}.$ The value of x is ______.

An antenna is placed in a dielectric medium of dielectric constant 6.25. If the maximum size of that antenna is 5.0 mm, it can radiate a signal of minimum frequency of ________ GHz.

(Given µ_r = 1 for dielectric medium)