UPSC NDA II 2023 Mathematics Solved Paper

Concept:The general equation $\frac{x^{2}}{A} + \frac{y^{2}}{B} = 2$ can be rewritten as $\frac{x^{2}}{2A} + \frac{y^{2}}{2B} = 1$. The conic type depends on the signs of $A$ and $B$: both positive gives an ellipse, one positive and one negative gives a hyperbola, and equal positive values give a circle.Explanation:For $k = 19$: $A = 24 - 19 = 5 > 0$, $B = 19 - 16 = 3 > 0$. Both denominators are positive, so the equation $\frac{x^{2}}{5} + \frac{y^{2}}{3} = 2$ represents an ellipse. Statement 1 is correct.For $k = 12$: $A = 24 - 12 = 12 > 0$, $B = 12 - 16 = -4 < 0$. The denominators have opposite signs, so $\frac{x^{2}}{12} + \frac{y^{2}}{-4} = 2$ simplifies to $\frac{x^{2}}{12} - \frac{y^{2}}{4} = 2$, which is a hyperbola. Statement 2 is correct.For $k = 20$: $A = 24 - 20 = 4 > 0$, $B = 20 - 16 = 4 > 0$. Both denominators are equal and positive, giving $\frac{x^{2}}{4} + \frac{y^{2}}{4} = 2$, or $x^{2} + y^{2} = 8$, which is a circle. Statement 3 is correct.All three statements are true.Answer:C. All three