Question

A number $x$ is chosen at random from the first 100 natural numbers. The probability that it satisfies

• A. $x^2-25x\le 150$ is $0.3$
• B. $x^2-17x+30\le 0$ is $0.88$
• C. $30x-x^2$ is a perfect square of a natural number is $0.07$
• D. $30x-x^2<0$ is $0.7$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $x^2-25x\le 150$ is $0.3$
B $x^2-17x+30\le 0$ is $0.88$
C $30x-x^2$ is a perfect square of a natural number is $0.07$
D $30x-x^2<0$ is $0.7$

$x^2-25x-150\le 0\iff (x-30)(x+5)\le 0\iff x\le 30$
$x^2-17x+30\ge 0\iff (x-2)(x-15)\ge 0\iff x\le 2$ or $x\ge 15$
$30x-x^2$ is a perfect square of a natural number if $x$ $=3,6,12,15,18,21$ or $24$.
Lastly $30x-x^2<0\iff x(30-x)<0$ or $x>30$.