UPSC CDS I 2025 Mathematics Paper

Angle at the vertex of the conical body =120∘
Sum of slant height, height, and radius =9+3√‌3cm
Formula used:
Volume of the cone =(1∕3)πr2h
Relationship between slant height, radius, and height =l2=r2+h2
Let the height of the cone be h , radius be r , and slant height be 1 .
We know that angle at the vertex is 120∘ and tan(60∘)=r∕h, so r=√‌3×h.
We are given that r+h+1=9+3√3, substituting r=√3×h and l=√‌(r2+h2)=2h :
‌(√‌3h)+h+2h=9+3√‌3
‌⇒(√‌3+3)h=9+3√‌3
‌⇒h=(9+3√‌3)∕(√‌3+3)
Rationalize the denominator:
‌h=(9+3√‌3)(√‌3−3)∕(3−9)
‌⇒h=−18∕−6
‌⇒h=3cm
Now, r=√‌3×h=√‌3×3=3√‌3cm.
Now, volume of the cone =(1∕3)πr2h=(1∕3)π(3√‌3)2×3
⇒ Volume =(1∕3)π×27×3=27π cubic cm.