IIT-JEE 1992Question 1
In this question there are entries in columns $I$ and $II$. Each entry in column $I$ is related to exactly one entry in column $II$. Write the correct letter from column $II$ against the entry number in column $I$ in your answer book.
Let the functions defined in column $I$ have domain $\left(-\dfrac\pi2,\dfrac\pi2\right)$.
Column I: (A) $x+\sin x$; (B) $\sec x$.
Column II: (p) increasing; (q) decreasing; (r) neither increasing nor decreasing.
Differentiation and its Application
Answer$(A)-(p),\ (B)-(r)$
Solution
Concept. Check the sign of each derivative on $\left(-\frac\pi2,\frac\pi2\right)$. Formulas used. (1) $(\sec x)'=\sec x\tan x$Step 1 - (A). $(x+\sin x)'=1+\cos x>0$: increasing. (A) $\to$ (p). Step 2 - (B). By (1), $\sec x>0$ and $\tan x$ changes sign at $0$: $\sec x$ decreases on $\left(-\frac\pi2,0\right)$ and increases on $\left(0,\frac\pi2\right)$. Neither. (B) $\to$ (r). Answer: $(A)-(p),\ (B)-(r)$