The number of values of k for which the equat | vedclass

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

Question

(A) Let $f(x) = x^2 - 3x + k$. For the roots $\alpha, \beta$ to be real,distinct,and lie in the interval $(0, 1)$,the following conditions must be sat

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) Let $f(x) = x^2 - 3x + k$. For the roots $\alpha, \beta$ to be real,distinct,and lie in the interval $(0, 1)$,the following conditions must be satisfied:$1$. Discriminant $D > 0$: $D = (-3)^2 - 4(1)(k) = 9 - 4k > 0 \implies k $2$. $f(0) > 0$: $0^2 - 3(0) + k > 0 \implies k > 0$.$3$. $f(1) > 0$: $1^2 - 3(1) + k > 0 \implies 1 - 3 + k > 0 \implies k > 2$.$4$. The vertex of the parabola $x = -b/(2a)$ must lie in $(0, 1)$: $x = 3/2 = 1.5$. Since $1.5$ is not in the interval $(0, 1)$,it is impossible for both roots to lie in $(0, 1)$.Alternatively,for roots to lie in $(0, 1)$,the sum of roots $\alpha + \beta$ must satisfy $0 < \alpha + \beta < 2$. Here,$\alpha + \beta = 3$,which is not less than $2$. Therefore,no such $k$ exists.

The number of values of $k$ for which the equation $x^2 - 3x + k = 0$ has two real and distinct roots lying in the interval $(0, 1)$ is:

Similar Questions

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$ and $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in $G.P.$,where $\Delta = b^2 - 4ac$,then:

Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-x-1=0$. If $p_{k}=(\alpha)^{k}+(\beta)^{k}, k \geq 1$,then which one of the following statements is not true?

Solve the given two equations and select the correct answer from the given options.
$I.$ $x = \sqrt{2304}$
$II.$ $y^2 = 2304$

The number of real roots of the equation ${e^{\sin x}} - {e^{-\sin x}} - 4 = 0$ is:

The number of real roots of the equation $e^{4x} + e^{3x} - 4e^{2x} + e^x + 1 = 0$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module