NIMCET 2025 — Mathematics PYQ

To find where the function attains a maximum, we first find its derivative and set it to zero.

Step 1: Simplify the function

The given function is:

Multiplying the numerator and denominator by $\cos x$:

Step 2: Differentiate with respect to $x$

Using the quotient rule $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v{u}^{\prime }-u{v}^{\prime }}{v^2}$:

Let $u=x\cos x⟹u^{\prime }=\cos x-x\sin x$

Let $v=\cos x+x\sin x⟹v^{\prime }=-\sin x+\sin x+x\cos x=x\cos x$

Now, calculate $y^{\prime }$:

Since ${\sin }^{2}x+{\cos }^{2}x=1$:

Step 3: Set the derivative to zero for stationary points

For maxima or minima, $y^{\prime }=0$:

Taking the square root:

Final Answer:

The curve attains a maximum when $x=\cos x$. The correct option is (A).