Class 12th

The value of   $\underset{n\to \infty }{lim}\frac{1}{n}{\displaystyle \sum _{r=0}^{2n-1}\frac{n^2}{n^2+4r^2}is:}$

Let $\overrightarrow{a}=\widehat{i}+\widehat{j}+\widehat{k}and\overrightarrow{b}=\widehat{j}-\widehat{k}.If\overrightarrow{c}$  is a vector such that $\overrightarrow{a}*\overrightarrow{c}=\overrightarrow{b}and\overrightarrow{a}.\overrightarrow{c}=3,$  then $\overrightarrow{a}.\left(\overrightarrow{b}*\overrightarrow{c}\right)is$ equal to

Let y = y(x) be a solution curve of the differential equation $\left(y+1\right)ta{n}^{2}xdx+tanxdy+ydx=0,$ $x\in \left(0,\frac{\pi }{2}\right).$  If $\underset{x\to 0^{+}}{lim}xy\left(x\right)=1,$  then the value of y   $\left(\frac{\pi }{4}\right)is:$

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) = $\frac{5}{9},is:$  

The value of ${\displaystyle \underset{\frac{-1}{\sqrt[]{2}}}{\overset{\frac{1}{\sqrt[]{2}}}{\int }}{\left({\left(\frac{x+1}{x-1}\right)}^2+{\left(\frac{x-1}{x+1}\right)}^2-2\right)}^{1/2}}$  dx is:

If A = $\left(\begin{array}{cc}\hfill \frac{1}{\sqrt[]{5}}\hfill & \hfill \frac{2}{\sqrt[]{5}}\hfill \\ \hfill \frac{-2}{\sqrt[]{5}}\hfill & \hfill \frac{1}{\sqrt[]{5}}\hfill \end{array}\right),$  B = $\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill i\hfill & \hfill 1\hfill \end{array}\right),i=\sqrt[]{-1},$  and Q = A^TBA, then the inverse of the matrix AQ²⁰²¹A^T is equal to

The line graph given below shows the marked price, selling price and cost of manufacturing of an aircraft  (in crore).

What is the maximum discount the company has given to a customer in any year?

During halogens test, sodium fusion extract is boiled with concentrated HNO₃ to

Given below are two statements: one is labeled as Assertion A  and the other is labeled as Reason R.

           Assertion A:  A mixture contains benzoic acid  and naphthalene. THE pure benzoic acid can be separated out by the use of benzene.

           Reason R: Benzoic acid is soluble in hot water.

           In the light of the above statements, choose the most appropriate answer form the options given below:

The metal ion (in gaseous state) with lowest spin-only magnetic moment value is