Question

Attempt any two of the following subquestions :
(i) If $\alpha +\beta =5$ and ${\alpha }^2+{\beta }^2=35$ find the quadratic equation whose roots are $\alpha$ and $\beta$.
(ii) Two dice are thrown. Find the probability of getting :
(a) The sum of the number on their upper faces is at least 9 .
(b) The sum of the numbers on their upper faces is 15.
(c) The number on the upper face of the second die is greater than the number on the upper face of the first die.

Answer 1

$1)\alpha +\beta =5,{\alpha }^2+{\beta }^2=25⟹(\alpha +\beta {)}^2-2\alpha \beta =25⟹\alpha \beta =-5$

the quadratic equation is $x^2-5x-5=0$

$2\left)\right(a)$ number of ways in which the sum of the number on their upper faces is atleast $9$ is $4+3+2+1=10$

Probability for this is $\frac{10}{6\times 6}=\frac{5}{18}$

$\left(b\right)$number of ways in which the sum of the number on their upper faces is 15 is $0$

Probability for this is $0$

$\left(c\right)$Number of ways for which the number on the upper face of the second die is greather than the number on the upper face of the first die is $5+4+3+2+1=15$

Probability for this is $\frac{15}{36}=\frac{5}{12}$