Class 12th

$\begin{array}{l}\frac{d}{dx}\left(e^{2}x\right)=2e^{2x}\\ e^{2x}=\frac{1}{2}-\frac{d}{dx}\left(e^{2x}\right)\\ e^{2x}=\frac{d}{dx}\left(\frac{1}{2}{e}^{2x}\right)\end{array}$

Therefore, an anti-derivative of $e^{2x}is\frac{1}{2}{e}^{2x}$

$\begin{array}{l}\frac{dx}{}sin3x=3cos3x \\ cos3x=\frac{1}{3}dx\left(sin3x\right) \\ cos3x= \frac{d}{dx}\left(\frac{1}{3}sin3x\right)\end{array}$

Therefore, an anti-derivative of $cos3xis\frac{1}{3}sin3x$

$\begin{array}{l}\frac{d}{dx}cos2x=–2sin2x \\ sin2x=–\frac{1}{2}\frac{d}{dx}\left(cos2x\right) \\ sin2x =\frac{d}{dx}\left(\frac{1}{2}cos2x\right)\end{array}$

Therefore, an anti-derivative of $sin2xis-\frac{1}{2}cos2x$

This is a assertion and reason answer type question as classified in NCERT Exemplar

(ii) Assertion and reason both are correct statements but reason is not correct

Explanation for assertion.

Explanation:  Assertion is correct statement but reason is a  wrong statement because a semipermeable membrane allows solvent molecules to move through a lower concentration solution to a higher concentration solution, osmosis is the flow of solvent molecules from the solvent side to the solution side through a semipermeable membrane.

This is a assertion and reason answer type question as classified in NCERT Exemplar

(i) Assertion and reason both are correct statements and reason is correct explanation for assertion.

Explanation:  When a nonvolatile solute (such as NaCl) is added to water, a NaCl solution is created. When compared to pure water, the solution has a lower vapour pressure because there are less water molecules at the liquid's surface. A dip, or drop in the freezing point of water, is noticed as a result of this decrease in vapour pressure.

(i) Assertion and reason both are correct statements and reason is correct explanation for assertion.

Explanation: (i) In terms of 'Molarity (M), ' the number of moles of the solute dissolved per litre of solution is the strength of a solution.

(ii) Because the volume of a liquid changes with temperature, the volume of a solution changes as well, while the number of moles of solute in it remains constant (or unchanged). As a result, the molarity of the solution would be affected.

This is a matching answer type question as classified in NCERT Exemplar

(i)- (d);    (ii)- (c);    (iii)- (b);   (iv)- (e);    (v)- (a)