Class 12th

dy/dx = (e? – x)? ¹
dx/dy = e? - x
dx/dy + x = e? ⇒ I.F. = e^ (∫dy) = e?
∴ xe? = ∫e? dy + c
xe? = e²? /2 + c
y (0) = 0 ⇒ c = -1/2
∴ x = e? /2 - (1/2)e? ⇒ e? – e? = 2x
e²? – 2xe? - 1 = 0
e? = x ± √ (x² + 1) ⇒ e? = x + √ (x² + 1)
y = ln (x + √ (x² + 1)

Kindly consider the following Image 

${\mathrm{C}\mathrm{r}\mathrm{O}}_4^{2-}+2{\mathrm{H}}^{+}+2{\mathrm{H}}_2{\mathrm{O}}_2?\underset{\text{Chromium Pentoxide}}{{\mathrm{C}\mathrm{r}\mathrm{O}}_5+3{\mathrm{H}}_2\mathrm{O}}$

Molarity $=\frac{\text{moles of}\text{solute}}{\text{volume of solution in}L}=\frac{1*1.15}{1342}*1000=0.86\mathrm{M}$

f' (x) = cosx + sinx − k ≤ 0∀x ∈ R
k ≥ √2

The number of >C = 0 groups in tripeptide is 0 .

Kindly consider the following figure

Charge (q)   $=it=2*50*60=6000\mathrm{C}$

$\Rightarrow \mathrm{}6000C=\left(\frac{6000}{96500}F\right)$

${\mathrm{M}\mathrm{n}\mathrm{O}}_4^{-}+3{\mathrm{e}}^{(-)}+2{\mathrm{H}}_2\mathrm{O}\stackrel{\text{weakly alkaline medium}}{?}{\mathrm{M}\mathrm{n}\mathrm{O}}_2+4{\mathrm{O}\mathrm{H}}^{-}$

Mass of ${\mathrm{M}\mathrm{n}\mathrm{O}}_2$    deposited $=\frac{1}{3}*\frac{6000}{96500}*87=1.8$

$\to$ ${\pi }_{{\mathrm{K}\mathrm{N}\mathrm{O}}_3}={\pi }_{\text{glucose.}}$

$(\mathrm{i}\mathrm{C}\mathrm{R}\mathrm{T}{)}_{{\mathrm{K}\mathrm{N}\mathrm{O}}_3}=(\mathrm{C}\mathrm{R}\mathrm{T}{)}_{\text{glucose}}$

 $\frac{}{}\frac{}{}$ $\frac{\mathrm{i}*1.34}{101.1}*\mathrm{R}\mathrm{T}=\frac{4.77}{180}*\mathrm{R}*\mathrm{T}$

$\mathrm{i}=\frac{4.77}{180}*\frac{101.1}{1.34}=2$

Hence, extent of dissociation is $100\mathrm{\%}$

Kindly consider the following figure