Multiple Topics
Matrix Match — coordinate geometry, calculus, limits
MJAT_TS3_P1
Grade 12
Question:
Match the following:
**Column-I:**
I) In $\triangle ABC$ with $A(-2,3)$, $B(4,-6)$, $C(1,1)$: $CS_1$, $CS_2$ divide the triangle into 3 equal areas ($S_1,S_2$ on $AB$). Lines through origin parallel to $CS_1$, $CS_2$ form $y^2+\lambda xy-\mu x^2=0$. Value of $(\lambda+\mu)$.
II) Area enclosed by $y=x$ and $x^2+y-2=0$ is $p/q$ (coprime). Value of $(p-q)$.
III) Curve satisfying $dy/dx=(x^2+y^2)/(2xy)$, passing through $(2,1)$ is a hyperbola with eccentricity $e$. Value of $e^4$.
IV) $\displaystyle\lim_{n\to\infty}\frac{1}{n^{2m}}[(n^2+1^2)^m(n^2+2^2)^m\cdots(n^2+n^2)^m]^{1/n}=\left(\frac{a}{e^2}\cdot\frac{e^{\pi/2}}{2}\right)^m$. Value of $a$.
**Column-II:** P) 2, Q) 4, R) 6, S) 7, T) 8
A) I-T; II-Q; III-Q; IV-R
B) I-T; II-Q; III-S; IV-P
C) I-S; II-Q; III-Q; IV-P
D) I-S; II-R; III-Q; IV-R
Step-by-Step Solution
Key Concept: I) $S_1=(0,0)$, $S_2=(2,-3)$ (trisection of $AB$). Lines $CS_1$: slope $=1$; $CS_2$: slope $=-4$. Pair through origin: $(y-x)(y+4x)=0\Rightarrow y^2+3xy-4x^2=0$. $\lambda=3$, $\mu=4$, $\lambda+\mu=7\to S$. II) Area $=7/3$, $p-q=4\to Q$. III) Rectangular hyperbola, $e=\sqrt{2}$, $e^4=4\to Q$. IV) Limit gives $a=2\to P$.
Answer: **C** (I→S, II→Q, III→Q, IV→P).
Correct Answer: C