Class 12th

If $(a+\sqrt[]{2}b\mathrm{c}\mathrm{o}\mathrm{s}x)(a-\sqrt[]{2}b\mathrm{c}\mathrm{o}\mathrm{s}y)=a^2-b^2$ , where $a>b>0$ , then $\frac{dx}{dy}$ at $\left(\frac{\pi }{4},\frac{\pi }{4}\right)$ is:

Let $x_0$ be the point of local maxima of $f\left(x\right)=\vec{a}\cdot (\vec{b}*\vec{c})$ where $\vec{a}=x\hat{i}-2\hat{j}+3\hat{k},\vec{b}=-2\hat{i}+x\hat{j}-\hat{k}$ and $\vec{c}=7\hat{i}-2\hat{j}+x\hat{k}$ . Then the value of $\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ at $x=x_0$ is

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are $\mathrm{5,7},\mathrm{10,12,14,15}$ , then the absolute difference of the remaining two observations is:

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(ab)$ be given ellipse, length of whose latus rectum is 10 . If its eccentricity is the maximum value of the function, $?\left(t\right)=\frac{5}{12}+t-t^2$ , then $a^2+b^2$ is equal to:

Among the following the number of state variable is____________.

Internal energy (U)

Volume (V)

Heat (q)

Enthalpy (H)

For a reaction, given below is the graph of ink vs $\frac{1}{T}.$  The activation energy for the reaction is equal to_____________ cal mol^-1 (Nearest integer)

(Given: R = 2 cal K^-1 mol^-1)

Let $f$ be a twice differentiable function on $\left(\mathrm{1,6}\right)$ . If $f\left(2\right)=8,f^{\mathrm{\text{'}}}\left(2\right)=5,f^{\mathrm{\text{'}}}\left(x\right)\ge 1$ and $f$ " $\left(x\right)\ge 4$ , for all $x\in \left(\mathrm{1,6}\right)$ , then:

On reaction with stronger oxidizing agent like KlO₄,hydrogen peroxide oxidizes with the evolution of O₂. The oxidation number of I in KlO₄, changes to__________.

A sample of 0.125g of an organic compound when analyzed by Duma's method yields 22.78 mL of nitrogen gas collected over KOH solution at 280 K and 759 mm Hg. The percentage of nitrogen in the given organic compound is__________. (Nearest integer)

[Given: (a) The vapour pressure of water of 280K is 14.2mm Hg

(b) R = 0.082 L atm K^-1 mol^-1]

Let $f\left(x\right)=|x-2|$ and $g\left(x\right)=f\left(f\right(x\left)\right),x\in \left[\mathrm{0,4}\right]$ . Then ${\int }_0^3?\left(g\right(x)-f(x\left)\right)dx$ is equal to: