Class 12th

Let S = $\left\{z\in C:\left|z-3\right|\le 1andz\left(4+3i\right)\le 24\right\}.$ If $\alpha +i\beta$ is the point in S which in closest to 4i, then 25 $\left(\alpha +\beta \right)$ is equal to

Let $\lambda$ be the largest value of $\lambda$ for which the function $f_{\lambda }\left(x\right)=4\lambda x^3-36\lambda x^2+36x+48$ is increasing for all $x\in R.$ Then $f_{\lambda }.(1)+f_{\lambda }.(-1)$ is equal to

The slope of normal at any point (x,y), X > 0, y > 0 on the curve y = y(x) is given by $\frac{x^2}{xy-x^2{y}^2-1}.$ If the curve passes through the point (1, 1), then e.y (e) is equal to

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

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