Q68 / 120 · NDA 2024 (Phase 2) · Mathematics
Paragraph:
Let S be the line of intersection of two planes $x+y+z=1$ and $2 x+3 y-4 z=8$.
Question:
If $(l, m, n)$ are direction cosines of S, then what is the value of $43\left(l^2-m^2-n^2\right)$ ?
Question: If $(l, m, n)$ are direction cosines of S, then what is the value of $43\left(l^2-m^2-n^2\right)$ ?
- Correct answerA$6$
- B$5$
- C$4$
- D$1$
Solution
2 planes are $x+y+z=1$ and $2 x+3 y-4 z=8$.
DC 's of line: S are $1, \mathrm{~m}, \mathrm{n}$
$\begin{aligned}
& 1=\frac{-7}{\sqrt{86}}, m=\frac{6}{\sqrt{86}}, n=\frac{1}{\sqrt{86}} \\
& 43\left(1^2-m^2-n^2\right)=43\left(\frac{49}{86}-\frac{36}{86}-\frac{1}{86}\right)=43 \times \frac{12}{86}=6
\end{aligned}$
Hence, the correct answer is option: 'A'.