Which of the following statements are true? Justify your answers. (Thi

Question

Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) Eliminating z from $x + 2y + 3z = 2$ , $3x + 2y + 3z = 6$ and $2x + 3y = 5$ gives $x + 2y = 2$ .
ii) The roots of $x^3 - 8x - 3 = 0$ are given by $x = \frac{8 \pm \sqrt{64 + 12}}{2}$ .
iii) $(\sqrt{2}, 1, \frac{3}{5}) \in \mathbb{Q} \times \mathbb{Z} \times \mathbb{R}$ .
iv) Given any n positive numbers in $\mathbb{R}$ , the product of their harmonic mean and their arithmetic mean is 1.
v) If A and B are two sets such that $(A \cup B)^c$ is empty, then either $A = \emptyset$ or $B = \emptyset$ .
vi) For any $x, y \in \mathbb{R}, |x - y| \geq ||x| - |y||$ .
vii) The geometrical representation of the set $\{ix \mid x \in \mathbb{R}\}$ is a point.
viii) Any finite set is a subset of $\mathbb{Z}$ .
ix) Every biquadratic equation has at least one real root.
x) The converse of the statement, ‘Every student of MTE-04 has completed FST-01’, is ‘Every student of FST-01 has completed MTE-04’.

09 Jan 2026

Answer :

Word Count : 307

(i) Subtracting the first equation from the second gives ((3x+2y+3z)-(x+2y+3z)=6-2), so (2x=4) and hence (x=2). Substituting (x=2) into (2x+3y=5) gives (4+3y=5), so (y=\frac13). Then (x+2y=2+\frac23=\frac83\neq 2). Hence the statement is false. (ii) The given expression (\frac{8\pm\sqrt{64+12}}{2}) is the quadratic formula, but the equation is cubic. Substituting (x=\frac{8\pm\sqrt{76}}{2}) into (x^3-8x-3) does not satisfy the equation. Hence the statement is false. (iii) ((\sqrt2,1,\frac35)\in\mathbb Q\times\mathbb Z\times\mathbb R) would require (\sqrt2\in\mathbb Q), which is false since (\sqrt2) is irrational. Hence ____ ____ ______ ____ __________ _______.
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