Class 12th

Le aa₁, a₂, a₃,…… be an A.P. If ${\displaystyle \sum _{r=1}^{\infty }\frac{a_r}{2^r}=4,}$ then 4a₂ is equal to……

Let a line with direction ratios a, 4a, 7 be perpendicular to the lines with direction ratios 3, 1, 2b and b, a, 2. If the point of intersection of the line $\frac{x+1}{a^2+b^2}=\frac{y-2}{a^2-b^2}=\frac{z}{1}$ and the plane x – y + z = 0 is $\left(\alpha ,\beta ,\gamma \right),then\alpha +\beta +\gamma$ is equal to………

Dimensional formula for thermal conductivity is (here $K$ denotes the temperature):

Let the mean and the variance of 20 observations x1, x2,…….x20 be 15 and 9, respectively. For $\in R$ , if the mean of ${\left(x_1+\alpha \right)}^2,{\left(x_2+\alpha \right)}^2,....,{\left(x_{20}+\alpha \right)}^2$ is 178, then the square of the maximum value of is equal to………….

Let $S=\left\{\theta \in \left(0,2\pi \right):7co{s}^2\theta -3si{n}^2\theta -2co{s}^{2}2\theta =2\right\}.$ Then, the sum of roots of all the equations $x^2-2\left(ta{n}^2\theta +co{t}^2\theta \right)x+6si{n}^2\theta =0,\theta \in S,is.........$

The position vector of 1kg object is $\overrightarrow{r}=\left(3\widehat{i}-\widehat{j}\right)m$ and its velocity $\overrightarrow{v}=\left(3\widehat{j}+\widehat{k}\right)m{s}^{-1}.$ The magnitude of is angular momentum is $\sqrt[]{x}Nm$ where x is________.

Let f(x) = $3^{{\left(x^2-2\right)}^3+4},x\in R.$ Then which of the following statements are true?

P : x = 0 is a point of local minima of f

Q : x = $\sqrt[]{2}$ is a point of inflection of f

R : $f\text{'}$ is increasing for x > $\sqrt[]{2}$

An AC source is connected to an inductance of 100 mH, a capacitance of 100 $\mu F$  and a resistance of 120 $\Omega$  as shown in figure. The time in which the resistance having a thermal capacity 2 J/°C will get heated by 16°C is________ s.

Let S = {1, 2, 3,…., 2022}. Then the probability, that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is:

Given figure shows few data points in a photo electric effect experiment for a certain metal. The minimum energy for ejection of electron from its surface is: (Plancks constant $\mathrm{h}=6.62*{10}^{-34}\mathrm{J}.\mathrm{s}$ )