Zadanie 6a, Paper 2, maj 2025, TZ1, Matematyka AA IB HL

(i) $f(x)=3\cos x/\sin x+2\sin x$. (ii) The curve decreases from +∞ as x → 0⁺ to −∞ as x → π⁻. Its horizontal intercept is x = 2π/3 ≈ 2.09: setting the numerator to zero gives $3\cos x+2(1-\cos^2x)=0$, with admissible root $\cos x=-1/2$. There are vertical asymptotes at 0 and π. For shape, $f\prime=-3\csc^2x+2\cos x<0$; the curve changes from convex to concave once, before π/2.

A1: correct sine/cosine expression. A1: a single decreasing branch with correct endpoint behaviour and curvature. A1: correct intercept.