Question
Passage: P(-2,-3,5), Q(4,-1,5), R(6,-4,8) and S(2,-6,10) are four points. Question: Consider the following statements : I. PQ is parallel to RS. II. PR is perpendicular to QS. Which of the statements given above is/are correct ?
NDA
Mathematics
Three Dimensional Geometry
2026
NDA 2026 (Phase 1)
NDA Mathematics Question (2026) — Solution
Options
- A. I only
- B. II only
- C. Both I and II
- D. Neither I nor II
Answer
D. Neither I nor II
Step-by-step solution
The direction ratios of the line segment PQ are 4 - (-2), -1 - (-3), 5 - 5, which gives 6, 2, 0 . The direction ratios of the line segment RS are 2 - 6, -6 - (-4), 10 - 8, which gives -4, -2, 2 . Since the ratios of their corresponding direction ratios are not equal ( 6 -4 2 -2 0 2 ), PQ is not parallel to RS. Thus, Statement I is incorrect. The direction ratios of the line segment PR are 6 - (-2), -4 - (-3), 8 - 5, which gives 8, -1, 3 . The direction ratios of the line segment QS are 2 - 4, -6 - (-1), 10 - 5, which gives -2, -5, 5 . For PR to be perpendicular to QS, the dot product of their direction vectors must be zero. Calculating the dot product: PR QS = (8)(-2) + (-1)(-5) + (3)(5) = -16 + 5 + 15 = 4 Since the dot product is not zero, PR is not perpendicular to QS. Thus, Statement II is incorrect. Therefore, neither Statement I nor Statement II is correct. Answer: Neither I nor II
Related: Mathematics — Three Dimensional Geometry · All PYQ Banks