Trigonometry & Inverse Trigonometry
Domain of cos⁻¹ + sin⁻¹ — Intersection
nta_pyq_2026_jan
Grade 12
Question:
If the domain of the function $f(x)=\cos^{-1}\!\left(\dfrac{2x-5}{11-3x}\right)+\sin^{-1}(2x^2-3x+1)$ is the interval $[\alpha,\beta]$, then $\alpha+2\beta$ is equal to:
Step-by-Step Solution
Key Concept: For $\cos^{-1}$: $-1\leq\tfrac{2x-5}{11-3x}\leq1\Rightarrow x\leq\tfrac{16}{5}$ or $x>6$ (from $11-3x>0$... solving). For $\sin^{-1}$: $-1\leq2x^2-3x+1\leq1\Rightarrow x(2x-3)\leq0\Rightarrow0\leq x\leq\tfrac{3}{2}$.
Domain $=[0,\tfrac{3}{2}]$. $\alpha+2\beta=3$.
Correct Answer: 1