For p, q in R,consider the real-valued functi | vedclass

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

Question

(A) The roots of $f(x) = (x - p)^2 - q = 0$ are $x = p \pm \sqrt{q}$.The absolute difference between the roots is $|(p + \sqrt{q}) - (p - \sqrt{q})| =

Vedclass · Accepted Answer

$50$

Vedclass · Answer

(A) The roots of $f(x) = (x - p)^2 - q = 0$ are $x = p \pm \sqrt{q}$.The absolute difference between the roots is $|(p + \sqrt{q}) - (p - \sqrt{q})| = 2\sqrt{q}$.Given $a_1, a_2, a_3, a_4$ are in arithmetic progression with mean $p$ and common difference $d > 0$,we have $a_1 = p - \frac{3d}{2}, a_2 = p - \frac{d}{2}, a_3 = p + \frac{d}{2}, a_4 = p + \frac{3d}{2}$.Since $|f(a_i)| = 500$,we have $|(a_i - p)^2 - q| = 500$.For $i=4$,$|(\frac{3d}{2})^2 - q| = 500 \Rightarrow |\frac{9d^2}{4} - q| = 500$.For $i=1$,$|(-\frac{3d}{2})^2 - q| = 500 \Rightarrow |\frac{9d^2}{4} - q| = 500$.For $i=2$,$|(-\frac{d}{2})^2 - q| = 500 \Rightarrow |\frac{d^2}{4} - q| = 500$.Since $q > 0$ and $d > 0$,$\frac{9d^2}{4} - q = 500$ and $q - \frac{d^2}{4} = 500$ (as $\frac{d^2}{4} Adding these equations: $(\frac{9d^2}{4} - q) + (q - \frac{d^2}{4}) = 500 + 500$ $\Rightarrow 2d^2 = 1000$ $\Rightarrow d^2 = 500$.Substituting $d^2 = 500$ into $q - \frac{d^2}{4} = 500$,we get $q - 125 = 500 \Rightarrow q = 625$.The absolute difference between the roots is $2\sqrt{q} = 2\sqrt{625} = 2 	imes 25 = 50$.

Similar Questions

The sum of the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \dots$ to $9$ terms is

If $A$ is an arithmetic mean between two numbers and $S$ is the sum of $n$ arithmetic means between the same two numbers,then:

The $20^{\text{th}}$ term from the end of the progression $20, 19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots, -129 \frac{1}{4}$ is:

If the sum of three numbers in an arithmetic progression is $33$ and their product is $792$,what is the smallest number among them?

The natural numbers are arranged in rows as follows:
$1$
$2, 3$
$4, 5, 6$
$7, 8, 9, 10$
$. . .$
What is the sum of the numbers in the $n^{th}$ row?

Vedclass Test Series

Exam Paper Generator

Online Exam Module