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UPSC NDA I 2026 Mathematics Paper

Question Numbers: 57-58

For the next two (02) items that follow:

Let

\mathrm{X}(\mathrm{a}, \mathrm{p}), \mathrm{Y}(\mathrm{b}, \mathrm{q})

and

\mathrm{Z}(\mathrm{c}, \mathrm{r})

be the points such that a, b and c are in AP.

Question : 57 of 120

Marks: +1, -0

If p, q and r are in AP, then the points X, Y and Z are

Solution:

Concept:

Points with both

x

- and

y

-coordinates in Arithmetic Progression lie on a straight line.

Explanation:

Let

a, b, c

be in AP, so

b - a = c - b = d

Let

p, q, r

be in AP, so

q - p = r - q = e

Thus the points are:

X(a, p),\quad Y(a+d, p+e),\quad Z(a+2d, p+2e)

This is of the form

(a+td, p+te)

, which represents a straight line as

t

varies.

Hence

X

Y

and

Z

are collinear.

Answer:

on a straight line

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