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UPSC NDA I 2026 Mathematics Paper

Question Numbers: 53-54

For the next two (02) items that follow:

\mathrm{P}(-2,-3,5), \mathrm{Q}(4,-1,5), \mathrm{R}(6,-4,8)

and S(2, -6, 10) are four points.

Question : 53 of 120

Marks: +1, -0

Which of the statements given above is/are correct ?

Solution:

Concept:

Use vectors and dot/cross products to check parallel and perpendicular conditions in 3D.

Explanation:

\overrightarrow{PQ} = Q - P = (6, 2, 0)

\overrightarrow{RS} = S - R = (-4, -2, 2)

Since

(6, 2, 0)

is not a scalar multiple of

(-4, -2, 2)

PQ

is not parallel to

RS

So statement I is false.

\overrightarrow{PR} = R - P = (8, -1, 3)

\overrightarrow{QS} = S - Q = (-2, -5, 5)

Dot product:

(8)(-2) + (-1)(-5) + (3)(5) = -16 + 5 + 15 = 4 \neq 0

PR

is not perpendicular to

QS

Statement II is false.

Answer:

Neither I nor II

Option D

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