Multiple Topics
Matrix Match — planes, integrals, functions, geometry
MJAT_TS3_P1
Grade 12
Question:
Match the following:
**Column-I:**
I) Distance from origin to plane through $(1,1,1)$ perpendicular to $x/3=y/0=z/4$ is $p/q$ (coprime). Value of $(p-q)$.
II) $\displaystyle\lim_{\delta x\to 0}\sum_{i=1}^n x_i\,\delta x$ over $[0,4]$ divided into $n$ equal parts.
III) $f(x)\geq M$ for all $x$, where $xf(x)+(1-x)f(-x)=x^2+x+1$. $M=p/q$ (coprime). Value of $(q-p)$.
IV) Trapezium $ABCD$ with $AB\parallel CD$, $AD\perp AB$, $AB=3CD=4$, inscribed circle radius $r$. Value of $4r^2$.
**Column-II:** P) 1, Q) 2, R) 3, S) 6, T) 8
A) I-R; II-T; III-P; IV-Q
B) I-Q; II-P; III-T; IV-R
C) I-R; II-T; III-S; IV-Q
D) I-Q; II-T; III-P; IV-R
Step-by-Step Solution
Key Concept: I) Plane: $3(x-1)+4(z-1)=0\Rightarrow 3x+4z=7$. Distance $=7/5$, $p-q=2\to Q$. II) Riemann sum $=\int_0^4 x\,dx=8\to T$. III) $f(x)=3x^2-x+1\geq 11/12$, $M=11/12$, $q-p=12-11=1\to P$. IV) $r^2=3/4$ (from solution $4r^2=3\to R$).
Answer: **D** (I→Q, II→T, III→P, IV→R).
Correct Answer: D