UPSC CDS 2 2021 Math Paper

Concept:Check each statement for a negative real number $x$ and identify which ones are false.Explanation:Given $x < 0$, $x$ is a negative real number.Statement 1: Natural number $k > 0$. Since $x < 0$, $kx < 0$ always. Hence $kx > 0$ is never true. So statement 1 is not correct.Statement 2: Take $x = -1$. Then $x^2 + x = 1 + (-1) = 0$, which is not $> 0$. So the inequality $x^2 + x > 0$ does not hold always. Thus statement 2 is not correct.Statement 3: For $x < 0$, we have $2x < x$ because adding $x$ gives $x < 0$ which is true. Also $x < -x$ because $-x > 0$. Therefore $2x < x < -x$ holds for all negative $x$. So statement 3 is correct.Statement 4: $x^2$ may be irrational. Example: $x = -\sqrt[4]{3}$, then $x^2 = \sqrt{3}$, which is irrational. Hence $x^2$ is not always rational. So statement 4 is not correct.Thus the statements that are not correct are 1, 2, and 4.Answer:Option B: 1, 2 and 4.