If p,;q,;r are in A.P. and are positive, the | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(a) $p,\;q,\;r$ are positive and are in $A.P.$

$	herefore \;q = \frac{{p + r}}{2}$ ......(i)

The roots of $p{x^2} + qx + r = 0$ are real

$ \

Vedclass · Accepted Answer

$\left| {\,\frac{r}{p} - 7\;} \right|\; \ge 4\sqrt 3 $

Vedclass · Answer

(a) $p,\;q,\;r$ are positive and are in $A.P.$

$	herefore \;q = \frac{{p + r}}{2}$ ......(i)

The roots of $p{x^2} + qx + r = 0$ are real

$ \Rightarrow $ ${q^2} \ge 4pr$

$ \Rightarrow $${\left[ {\frac{{p + r}}{2}} ight]^2} \ge 4pr$ [using (i)]

$ \Rightarrow $ ${p^2} + {r^2} - 14pr \ge 0$

$ \Rightarrow $ ${\left( {\frac{r}{p}} ight)^2} - 14\left( {\frac{r}{p}} ight) + 1 \ge 0$

$(\because \;p > 0\;{	ext{and}}\;p 
e 0)$

$ \Rightarrow $ ${\left( {\frac{r}{p} - 7} ight)^2} - 48 \ge 0$ 

$ \Rightarrow $${\left( {\frac{r}{p} - 7} ight)^2} - {(4\sqrt 3 )^2} \ge 0$

$ \Rightarrow $ $\left| {\;\frac{r}{p} - 7\;} ight|\; \ge 4\sqrt 3 $.

If $p,\;q,\;r$ are in $A.P.$ and are positive, the roots of the quadratic equation $p{x^2} + qx + r = 0$ are all real for

Similar Questions

There are $15$ terms in an arithmetic progression. Its first term is $5$ and their sum is $390$. The middle term is

Write the first five terms of the following sequence and obtain the corresponding series :

$a_{1}=-1, a_{n}=\frac{a_{n-1}}{n}, n\, \geq\, 2$

The sum of all the elements of the set $\{\alpha \in\{1,2, \ldots, 100\}: \operatorname{HCF}(\alpha, 24)=1\}$ is

If $^n{C_4},{\,^n}{C_5},$ and ${\,^n}{C_6},$ are in $A.P.,$ then $n$ can be

The first term of an $A.P. $ is $2$ and common difference is $4$. The sum of its $40$ terms will be