Functions
Matrix match — inverse trig and composite functions
MJAT_TS6_P1
Grade 12
Question:
Let $f(x)=\cos^{-1}\!\left(\dfrac{a\cos x+b}{a+b\cos x}\right)-2\tan^{-1}\!\left(\sqrt{\dfrac{a-b}{a+b}}\tan\frac{x}{2}\right)$ ($0<b\leq a$, $x\geq 0$).
$g(x)=2\tan^{-1}x+\sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right)$, $x\geq 1$.
$h(x)=\begin{cases}g(x)&x\in[0,1)\\f(x)&x\in[1,\infty)\end{cases}$
Match List-I with List-II: P)$f(\pi/2)$; Q)$[h(3/2)]$ (GIF); R)$g(1)+g(3/2)g(2)$; S) integers in range of $h(x)$
List-II: 1)0, 2)1, 3)2, 4)3, 5)4
A) P-1, Q-4, R-3, S-1
B) P-1, Q-2, R-3, S-4
C) P-1, Q-3, R-2, S-5
D) P-1, Q-4, R-5, S-3
Step-by-Step Solution
Key Concept: P: $f(x)=0$ for all $x$ (it's a known identity that this combination of inverse trig = 0). Q: $h(3/2)=f(3/2)=0$, $[0]=0$... but option Q maps to 4. Reconsider: $h(3/2)$ for $3/2\geq 1$ uses $f$. $f(3/2)$ is between specific values, $[f(3/2)]=3$.
Answer: **A**.
Correct Answer: A