UPSC CDS 1 2022 Math Paper

Concept:The total cost is constant at ₹10,000. When the length increases by 2 m, the cost per metre decreases by ₹250. This relationship gives an equation in the original length.Explanation:Let the original length be $x$ metres.Original cost per metre is $\frac{10000}{x}$.New length is $(x+2)$ metres, so new cost per metre is $\frac{10000}{x+2}$.The difference in cost per metre is ₹250:$\frac{10000}{x} - \frac{10000}{x+2} = 250$.Combine the fractions: $\frac{10000(x+2) - 10000x}{x(x+2)} = 250$.Simplify numerator: $10000x + 20000 - 10000x = 20000$.Thus $\frac{20000}{x(x+2)} = 250$.Multiply both sides by $x(x+2)$: $20000 = 250x(x+2)$.Divide by 250: $80 = x(x+2)$ → $x^2 + 2x - 80 = 0$.Factorize: $(x+10)(x-8)=0$.So $x = -10$ or $x = 8$. Length cannot be negative, so $x = 8$ m.Answer:The original length of the piece of cloth is 8 m.