The set of values of a such that x^2 - 2ax + | vedclass

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

Question

(B) Let $f(x) = x^2 - 2ax + a^2 - 6a$. For $f(x) \leqslant 0$ to hold for all $x \in [1, 2]$,we must have $f(1) \leqslant 0$ and $f(2) \leqslant 0$.St

Vedclass · Accepted Answer

$[5 - \sqrt{21}, 4 + \sqrt{15}]$

Vedclass · Answer

(B) Let $f(x) = x^2 - 2ax + a^2 - 6a$. For $f(x) \leqslant 0$ to hold for all $x \in [1, 2]$,we must have $f(1) \leqslant 0$ and $f(2) \leqslant 0$.Step $1$: Solve $f(1) \leqslant 0$.$f(1) = 1 - 2a + a^2 - 6a = a^2 - 8a + 1 \leqslant 0$.The roots of $a^2 - 8a + 1 = 0$ are $a = \frac{8 \pm \sqrt{64 - 4}}{2} = 4 \pm \sqrt{15}$.So,$a \in [4 - \sqrt{15}, 4 + \sqrt{15}]$  $(i)$.Step $2$: Solve $f(2) \leqslant 0$.$f(2) = 4 - 4a + a^2 - 6a = a^2 - 10a + 4 \leqslant 0$.The roots of $a^2 - 10a + 4 = 0$ are $a = \frac{10 \pm \sqrt{100 - 16}}{2} = 5 \pm \sqrt{21}$.So,$a \in [5 - \sqrt{21}, 5 + \sqrt{21}]$  $(ii)$.Step $3$: Find the intersection of $(i)$ and $(ii)$.$a \in [4 - \sqrt{15}, 4 + \sqrt{15}] \cap [5 - \sqrt{21}, 5 + \sqrt{21}] = [5 - \sqrt{21}, 4 + \sqrt{15}]$.

The set of values of $a$ such that $x^2 - 2ax + a^2 - 6a \leqslant 0$ for all $x \in [1, 2]$ is

Similar Questions

The range of $a$ for which the roots of $x^2 - 2x - a^2 + 1 = 0$ lie between the roots (exclusive) of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ 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

If the graph of $y = ax^2 + bx + c$ is as follows,where $\Delta ABC$ is a right-angled isosceles triangle with hypotenuse $AC = 4\sqrt{2} \text{ units}$,then the minimum value of $ax^2 + bx + c$ is:

Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

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

Vedclass Test Series

Exam Paper Generator

Online Exam Module