Question

Please read the following sentences carefully: I — 103 and 7 are the only prime factors of 1000027 II — $\sqrt[6]{6!} > \sqrt[7]{7!}$ III — If I travel one half of my journey at an average speed of x km/h, it will be impossible for me to attain an average speed of 2x km/h for the entire journey.

Accepted Answer

C. Statement I: $1000027 = 7 	imes 103 	imes 1387 = 7 	imes 103 	imes 19 	imes 73$, so 19 is also a prime factor (False). Statement II: Raising to power 42 gives $(6!)^7 > (7!)^6 \implies (6!)^7 > 7^6 (6!)^6 \implies 6! > 7^6 = 117649$, which is False ($720 < 117649$). Statement III: Average speed $V = \frac{2 v_1 v_2}{v_1 + v_2} = \frac{2 x y}{x + y}$. For $V = 2x$, $\frac{2 x y}{x + y} = 2x \implies y = x + y \implies 0 = x$, impossible. Thus Statement III is True.

Evaluating Mathematical Statements

Explanation

Practise this under exam conditions