UPSC NDA I 2025 Mathematics Paper

The sides of the triangle are:
‌a=5cm
‌b=7cm
‌c=3cm
Statement I: The triangle is obtuse-angled.
Using the Cosine Rule:
cos‌B=‌

a2+c2−b2

2ac

Substitute the values for a,b=7 and c
cos‌B=‌

52+32−72

2×5×3

=‌

25+9−49

30

=‌

−15

30

=−‌

1

2

Now, calculate angle B:
B=cos−1(−‌

1

2

)=120∘
Thus, angle B is 120∘, which confirms that the triangle is an obtuse-angled triangle,
Statement II: The sum of the acute angles of the triangle is also acute.
Since the sum of the angles in a triangle is 180∘, the sum of the acute angles is:
A+C=180∘−120∘=60∘
This confirms that the sum of the acute angles is acute,
∴ Both statements are correct.