Question

In the section each question has some statements $(\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D},...)$given in Column-I and some statements $(\mathrm{p},\mathrm{q},\mathrm{r},\mathrm{s},\mathrm{t},...)$in Column -II. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column - II for example, if for a given question, statement B matches with the statements given in q and r, then for that particular question against statement B, darken the bubbles corresponding to q and r in the ORS. i.e., answer r will be q and r

Let the functions defined in Column-I have domain $\left(\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$. Match the statements in column-I with the values in column-II

Column-I	Column-II
(a) $\mathrm{x}+\sin \mathrm{x}$ (b) $\sec \mathrm{x}$	(p) Increasing (q) Decreasing (r) Neither Increasing nor Decreasing

• A. $\mathrm{A}\to \mathrm{p};\mathrm{B}\to \mathrm{r}$
• B. $\mathrm{A}\to \mathrm{r};\mathrm{B}\to \mathrm{q}$
• C. $\mathrm{A}\to \mathrm{p};\mathrm{B}\to \mathrm{q}$
• D. $\mathrm{A}\to \mathrm{q};\mathrm{B}\to \mathrm{r}$

Answer 1

The correct option is A

$\mathrm{A}\to \mathrm{p};\mathrm{B}\to \mathrm{r}$

Step 1: Applying differentiation formula for the given equation in Column-I

$\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{x}=1;\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{sinx}=\mathrm{cosx};\frac{d}{dx}secx=secx\tan x$

Step 2: Solving the given equations in column-I

now, solving (a) in column-I,

(a):

$$\begin{gathered} \mathrm{let},\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}+\mathrm{sinx} \\ \therefore \mathrm{f}\text{'}\left(\mathrm{x}\right)=1+\mathrm{cosx}>0,\mathrm{for}\mathrm{all}\mathrm{x}\in \left(\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right) \end{gathered}$$

now, solving (b) in column-I

(b):

$$\begin{gathered} \mathrm{let}\mathrm{g}\left(\mathrm{x}\right)=\mathrm{secx} \\ \therefore g\text{'}\left(\mathrm{x}\right)=\mathrm{secx}\mathrm{tanx} \end{gathered}$$

which is positive if $0<x<\frac{\mathrm{\pi }}{2}$ and negative if $\frac{-\mathrm{\pi }}{2}<x<0$

therefore, g(x) is neither increasing nor decreasing.

Here, statement A in column-I matches with statement p in column-II

$\therefore \mathrm{A}\to \mathrm{p}$

and statement B in column-I matches with statement r in column-II

$\therefore \mathrm{B}\to \mathrm{r}$

Hence, the correct option is option (A).