Let X denote the number of hours a student studies on any day. If

$P(X=x)=\left\{\begin{matrix}0,1, & x=0\\ kx, & x=1 & or\, 2\\k(5-x), & if & x=3\, or \, 4\\0, & & otherwise\end{matrix}\right.$

where k > 0 is a constant, then $P(X≥2)$ is equal to :

0.75

Given the probability distribution:

$P(X=0)=0.1$

$P(X=x)=kx \text{ for } x=1,2$

$P(X=x)=k(5-x) \text{ for } x=3,4$

Using $\sum P(X=x)=1$:

$0.1 + k(1+2) + k(2+1) = 1$

$0.1 + 6k = 1$

$k = 0.15$

Now,

$P(X \ge 2) = P(2) + P(3) + P(4)$

$= 2k + 2k + k = 5k$

$= 5(0.15) = 0.75$

final answer: The probability that the student studies at least 2 hours is 0.75