Ncert Solutions Maths class 12th

The value of <!-- [if gte mso 9]><xml> </xml><![endif]-->  $\underset{n\to \infty }{lim}\frac{1}{n}{\displaystyle \sum _{r=0}^{2n-1}\frac{n^2}{n^2+4r^2}is:}$

Let $\overrightarrow{a}=\widehat{i}+\widehat{j}+\widehat{k}and\overrightarrow{b}=\widehat{j}-\widehat{k}.If\overrightarrow{c}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is a vector such that $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}and\overrightarrow{a}.\overrightarrow{c}=3,$ <!-- [if gte mso 9]><xml> </xml><![endif]--> then $\overrightarrow{a}.\left(\overrightarrow{b}\times \overrightarrow{c}\right)is$ <!-- [if gte mso 9]><xml> </xml><![endif]-->equal to

Let A and B be independent events such that P (A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = $\frac{5}{9},is:$ <!-- [if gte mso 9]><xml> </xml><![endif]--> 

The value of ${\displaystyle \underset{\frac{-1}{\sqrt[]{2}}}{\overset{\frac{1}{\sqrt[]{2}}}{\int }}{\left({\left(\frac{x+1}{x-1}\right)}^2+{\left(\frac{x-1}{x+1}\right)}^2-2\right)}^{1/2}}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> dx is:

A plane P contains the line x + 2y + 3z + 1 = 0 x – y – z – 6, and is perpendicular to the plane -2x + y + z + 8 = 0. Then which of the following points lies on P?

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set:

Let a,b $\in R,b\ne 0.$  Define a function f(x) =   $\{\begin{array}{cc}\hfill asin\frac{\pi }{2}\left(x-1\right),\hfill & \hfill forx\le 0\hfill \\ \hfill \frac{tan2x-sin2x}{b{x}^3},\hfill & \hfill forx>0\hfill \end{array}$

If f is continuous at x = 0, then 10 – ab is equal to__________

Let $z=\frac{1-i\sqrt[]{3}}{2},i=\sqrt[]{-1}.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> Then the value of

$21+{\left(z+\frac{1}{z}\right)}^3+{\left(z^2+\frac{1}{z^2}\right)}^3+{\left(z^3+\frac{1}{z^3}\right)}^3+.....+{\left(z^{21}+\frac{1}{z^{21}}\right)}^{3}is\_\_\_\_\_\_\_\_\_.$

Let the line L be the projection of the line $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> in the plane x – 2y – z = 3. If d is that distance of the point (0, 0, 6) from L, then d² is equal to____________.

The area of the region S = <!-- [if gte mso 9]><xml> </xml><![endif]-->  $\left\{\left(x,y\right):3x^2\le 4y\le 6x+24\right\}is\_\_\_\_\_\_\_\_\_\_\_\_$