UPSC NDA I 2023 Mathematics Solved Paper

Concept:Use the cosine rule to relate sides to angles and compare coefficients with the given equation.Explanation:For a triangle with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$, the cosine rule gives:$b^2 + c^2 - a^2 = 2bc \cos A$$a^2 + c^2 - b^2 = 2ac \cos B$$a^2 + b^2 - c^2 = 2ab \cos C$Adding these three equations: $a^2 + b^2 + c^2 = 2ab \cos C + 2bc \cos A + 2ca \cos B$The given condition is $a^2 + b^2 + c^2 = ac + \sqrt{3}bc$Comparing coefficients of corresponding terms:Coefficient of $ab$: $2\cos C = 0 \Rightarrow \cos C = 0 \Rightarrow C = 90^\circ$Coefficient of $bc$: $2\cos A = 1 \Rightarrow \cos A = \frac{1}{2} \Rightarrow A = 60^\circ$Coefficient of $ca$: $2\cos B = \sqrt{3} \Rightarrow \cos B = \frac{\sqrt{3}}{2} \Rightarrow B = 30^\circ$Thus, the angles are $30^\circ$, $60^\circ$, $90^\circ$ — a right‑angled triangle.Answer:Right angled triangle.