UPSC CDS 1 2022 Math Paper

Concept:Relate side length of a square to its diagonal, and side length of an equilateral triangle to its area.Explanation:Let the side of the square be $x$ cm.Diagonal of square = $\sqrt{2} \cdot x = 12\sqrt{2}$ cm.$\Rightarrow x = 12$ cm.Perimeter of square = $4x = 48$ cm.Area of square = $x^2 = 144$ sq cm.Let the side of the equilateral triangle be $a$ cm.Area of equilateral triangle = $\frac{\sqrt{3}}{4}a^2 = 64\sqrt{3}$ sq cm.$\Rightarrow a^2 = 256 \Rightarrow a = 16$ cm.Perimeter of triangle = $3a = 48$ cm.Statement 1: Both perimeters equal 48 cm. So statement 1 is correct.Statement 2: Four times area of square = $4 \times 144 = 576$.$3\sqrt{3}$ times area of triangle = $3\sqrt{3} \times 64\sqrt{3} = 3 \times 64 \times 3 = 576$.Thus $4 \times \text{Area}_{\text{square}} = 3\sqrt{3} \times \text{Area}_{\text{triangle}}$. So statement 2 is also correct.Answer:Both statements 1 and 2 are correct. Hence, option C.