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 $\alpha$ and $\beta$ to lie in the interval $(0, 1)$ with $\alpha 
eq \beta$,the following conditions mu

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) Let $f(x) = x^2 - 3x + k$. For the roots $\alpha$ and $\beta$ to lie in the interval $(0, 1)$ with $\alpha 
eq \beta$,the following conditions must be satisfied:$1$. Discriminant $\Delta > 0$: $\Delta = (-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 $x = -\frac{b}{2a}$ must lie in $(0, 1)$: $x = -\frac{-3}{2(1)} = \frac{3}{2} = 1.5$.Since the vertex $1.5$ does not lie in the interval $(0, 1)$,it is impossible for both roots to lie in $(0, 1)$.Thus,there are no such values of $k$.

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 $b > a$,then in which interval do the roots of the equation $(x - a)(x - b) - 1 = 0$ lie?

If the maximum value of $2x - 7 - ax^2$ cannot exceed $20$,then the minimum value of $a$ is

Number of integral values of $a$ for which the smaller root of the quadratic equation $x^2 - 2ax + a^2 - 4 = 0$ is smaller than $1$ and the bigger root is greater than $6$ is:

The smallest value of $k$,for which both the roots of the equation $x^2-8kx+16(k^2-k+1)=0$ are real,distinct and have values at least $4$,is

For what condition is the value of the quadratic expression $x^2 + 2bx + c$ always positive?

Vedclass Test Series

Exam Paper Generator

Online Exam Module