UPSC CDS 2 2019 Math Paper

Concept:This question tests the understanding of rational numbers, irrational numbers, inequalities, and the representation of repeating decimals as fractions.Explanation:We check each statement one by one.Statement 1: $\sqrt{75}=\sqrt{25\times 3}=5\sqrt{3}$.Since $\sqrt{3}$ is irrational, $\sqrt{75}$ is also irrational.Hence statement 1 is false.Statement 2: Solve $-4x/5 < -7/8$ for positive integer $x$.Multiply both sides by $40$ (positive, inequality direction unchanged): $-32x < -35$.Multiply by $-1$ (flips inequality): $32x > 35$ $\Rightarrow$ $x > 35/32 \approx 1.09375$.So $x=2$ is a positive integer satisfying the inequality.Thus statement 2 is true.Statement 3: Consider $\frac{x-2}{x} < 1$.For $x>0$, multiply by $x$ (positive): $x-2 < x$ $\Rightarrow$ $-2<0$, true.For $x<0$, multiply by $x$ (negative flips inequality): $x-2 > x$ $\Rightarrow$ $-2>0$, false.Example: $x=-1$ gives $\frac{-3}{-1}=3 < 1$ is false.Therefore statement 3 is false for all real $x$.Statement 4: $4.\overline{23}$ is a repeating decimal.Any repeating decimal can be expressed in the form $\frac{p}{q}$ with integers $p,q$ ($q\neq 0$).Hence statement 4 is true.Thus only statements 2 and 4 are correct.Answer:Option D: 2 and 4.