Class 12th

Among the following, basic oxide is:

Hydrogen has three isotopes : protium (¹H), deuterium (²H or D) and tritium (³H or T). They have nearly same chemical properties but different physical properties. They differ in

Let $M=\left[\begin{array}{cc}\hfill 0\hfill & \hfill -\alpha \hfill \\ \hfill \alpha \hfill & \hfill 0\hfill \end{array}\right],$ where is a non-zero real number an $N={\displaystyle \sum _{k=1}^{49}{M}^{2k}.}$ If (I – M²)N = 2I, then the positive integral value of is

Let f and g be twice differentiable even functions on (2, 2) such that $f\left(\frac{1}{4}\right)=0,f\left(\frac{1}{2}\right)=0,f\left(1\right)=1$ and $f\left(\frac{3}{4}\right)=0,g\left(1\right)=2$

Then, the minimum number of solutions of $f\left(x\right)g"\left(x\right)+f\text{'}\left(x\right)g\text{'}\left(x\right)=0in\left(-2,2\right)$ is equal to

For real numbers a, b(a > b > 0), let

$Area\left\{\left(x,y\right):x^2+y^2\le a^{2}and\frac{x^2}{a^2}+\frac{y^2}{b^2}\ge 1\right\}=30\pi and$

$Area\left\{\left(x,y\right):x^2+y^2\ge b^{2}and\frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1\right\}=18\pi$

Then the value of (a – b)² is equal to

Let y = y(x), x > 1, be the solution of the differential equation

$\left(x-1\right)\frac{dy}{dx}+2xy=\frac{1}{x-1},withy\left(2\right)=\frac{1+e^4}{2e^4}.$ If y(3) = $\frac{e^{\alpha }+1}{\beta e^{\alpha }}$ , then the value of α+β is equal to

Let $\overrightarrow{a}=\widehat{i}-2\widehat{j}+3\widehat{k},\overrightarrow{b}=\widehat{i}+\widehat{j}+\widehat{k}and\overrightarrow{c}$ be a vector such that $\overrightarrow{a}+\left(\overrightarrow{b}*\overrightarrow{c}\right)=\overrightarrow{0}and\overrightarrow{b}.\overrightarrow{c}=5.$ Then, the value of $3\left(\overrightarrow{c}.\overrightarrow{a}\right)$ is equal to

Let A, B, C be three points whose position vectors respectively are

$\overrightarrow{a}=\widehat{i}+4\widehat{j}+3\widehat{k}$

$\overrightarrow{b}=2\widehat{i}+\alpha \widehat{j}+4\widehat{k},\alpha \in R$

$\overrightarrow{c}=3\widehat{i}-2\widehat{j}+5\widehat{k}$

If is the smallest positive integer for which $\overrightarrow{a},\overleftarrow{b},\overrightarrow{c}$ are noncollinear, then the length of the median, in $\Delta ABC,$ through A is

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y +2z = 16. Let T be a plane passing through the point Q and contains the line $\overrightarrow{r}=-\widehat{k}+\lambda \left(\widehat{i}+\widehat{j}+2\widehat{k}\right),\lambda \in R.$ Then, which of the following points lies on T?

The distance of the origin from the centroid of the triangle whose two sides have the equations x – 2y + 1 = 0 and 2x – y – 1 = 0 and whose orthocenter is $\left(\frac{7}{3},\frac{7}{3}\right)is:$