Question

the probality that the root of the equation x2+2nx,+4n+5÷n=0are not real numbers when nEN such that n less than equal to 5​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA REQUIRED

The roots of the quadratic equation

$a {x}^{2} + bx +c = 0$

are not real if

$\sf{ \:Discriminant = D = {b}^{2} - 4ac < 0 \: }$

TO DETERMINE

The probability that the roots of the equation

$\displaystyle \: {x}^{2} + 2nx +( 4n + \frac{5}{n} ) = 0 \: \: ....(1)$

are not real with the condition

$\sf{Here \: n \in \mathbb{N} \: and \: n \leqslant 5}$

CALCULATION

$\sf{Here \: n \in \mathbb{N} \: and \: n \leqslant 5}$

$\sf{ So \: n = 1, 2, 3, 4, 5\: }$

So the total number of possible outcomes = 5

Let A be the event that the roots of the equation (1) are not real with given conditions

Now Comparing the equation (1) with

$a {x}^{2} + bx +c = 0 \: \: we \: get$

$\displaystyle \: a = 1 \: , b = 2n \:, \: c = ( 4n + \frac{5}{n} )$

So

$\sf{ \:Discriminant = D }$

$\sf{ = {b}^{2} - 4ac \: }$

$\displaystyle \: = {(2n)}^{2} - 4 \times 1 \times (4n + \frac{5}{n} )$

$\displaystyle \: = {4n}^{2} - 4 (4n + \frac{5}{n} )$

By the given condition

$\sf{ \:Discriminant = D = {b}^{2} - 4ac < 0 \: }$

$\implies \: {n}^{3} - 4 {n}^{2} - 5 < 0$

Which is true for n = 1 , 2, 3, 4

So the number event points for the event A is 4

So the total number of possible outcomes for the event A is 4

RESULT

The required probability is

$P(A)\displaystyle \: = \frac{4}{5}$

Answer 2

Answer:

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