Class 12th

If y = tan^-1 (sec x³ tanx³), $\frac{\pi }{2}<x^3<\frac{3\pi }{2},then$

Let $\widehat{a}$ and $\widehat{b}$ be two unit vectors such that $\left|\left(\widehat{a}+\widehat{b}\right)+2\left(\widehat{a}*\widehat{b}\right)\right|$ = 2. If $\theta \in (0,\pi )$ is the angle between $\widehat{a}and\widehat{b}$ , then among the statements:

$\left(S1\right):2\left|\widehat{a}*\widehat{b}\right|=\left|\widehat{a}-\widehat{b}\right|$

(S1) : The projection of $\widehat{a}on\left(\widehat{a}+\widehat{b}\right)is\frac{1}{2}$

Let the points on the plane P be equidistance from the points (–4, 2, 1) and (2, –2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is

If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda }$ and $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}is\frac{1}{\sqrt[]{3}},$ then the sum of all possible values of $\lambda$ is

A random variable X has the following probability distribution

The number of distinct real roots of the equation x⁷ – 7x – 2 = 0 is

$\underset{x\to \infty }{lim}\left(\frac{n^2}{\left(n^2+1\right)\left(n+1\right)}+\frac{n^2}{\left(n^2+4\right)\left(n+2\right)}+\frac{n^2}{\left(n^2+9\right)\left(n+3\right)}+...+\frac{n^2}{\left(n^2+n^2\right)\left(n+n\right)}\right)isequalto$

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