Test Index

UPSC NDA I 2026 Mathematics Paper

Question Numbers: 51-52

For the next two (02) items that follow:

The equation of a straight line which passes through the origin and makes an angle

\alpha\left(0 \leqslant \alpha \leqslant 90^{\circ}\right)

with the line

\mathrm{L}: \mathrm{x}+\sqrt{3} y+3 \sqrt{3}=0

\mathrm{x}-\sqrt{3} y=0

Question : 52 of 120

Marks: +1, -0

What is the angle made by the line L with positive direction of y-axis?

Solution:

Concept:

The angle between a line and the positive y-axis is found using the line's inclination with the positive x-axis.

Explanation:

Given

L: x+\sqrt{3}y+3\sqrt{3}=0

Rewrite as

y=-\frac{1}{\sqrt{3}}x-3

So slope

m=-\frac{1}{\sqrt{3}}=\tan 150^\circ

Thus,

L

makes an angle

150^\circ

with the positive x-axis.

The positive y-axis makes

90^\circ

with the positive x-axis.

Therefore, the angle between

L

and the positive y-axis is:

150^\circ-90^\circ=60^\circ

Answer:

C. 60°

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