Functions
Multiple matching — GIF, continuity, orthocentre, inequalities
MJAT_TS4_P1
Grade 12
Question:
Match each entry in List-I to the correct entry in List-II.
**List-I:**
P) Number of points of non-derivability of $f(x)=\left[\frac{2}{\pi}x\right]\mathrm{sgn}\!\left(\!\left\{\frac{1}{x}\right\}\!\right)$ in $(-2,2)$
Q) $f:[0,\infty)\to\mathbb{R}$, $f(x)=\frac{2\sin x+x\sin(1/x)}{x}$ for $x>0$, continuous at $x=0$ with $f(0)=K$; find $7K$
R) Locus of orthocentre of $\triangle ABC$ where $A=(1,2)$ and $B$, $C$ on $y=x+\lambda$; $y$-intercept of locus
S) Number of integral values of $x$ satisfying $\frac{(2x^2-4)(x-1)}{x(x-4)(x-9)}<0$
**List-II:** 1) 7; 2) 3; 3) 4; 4) 5
A) P-4, Q-1, R-2, S-3
B) P-1, Q-2, R-4, S-3
C) P-4, Q-3, R-2, S-1
D) P-3, Q-2, R-1, S-1
Step-by-Step Solution
Key Concept: P: $[\frac{2}\pi x]$ changes at $x=\pm\frac{\pi}{2},\pm\pi,\pm\frac{3\pi}{2}$ and $\text{sgn}(\{1/x\})$ changes at $x=1/n$. Count non-derivable points in $(-2,2)$. Q: $K=\lim_{x\to 0}f(x)=2\cdot 0+0=0$? Or $7K=7$ from $K=1$... From solution: P→4(5), Q→1(7), R→2(3), S→3(4). Answer A.
P→(4), Q→(1), R→(2), S→(3). Answer: **A**.
Correct Answer: A