The largest integral value of a for which every solution of the equation x - x -5-2 x -6-0 satisfies the inequality a -3 x 2-2 a - 3 x -8 a - 0- where - y - and y denote the greatest integer and fractional part functions respectively- is

X[x] 5x+2x 2[x]+6=0 ⇒ [x] (x 2) 3(x 2)=0 ⇒ ([x] 3)(x 2)=0 ⇒ x=2 or [x]=3 ⇒ x∈ [3,4)∪{2} (a 3)x^2+2(a+3)x 8a ≤ 0 ⇒ ax^2 3x^2+2ax+ 6x 8a ≤ 0 ⇒ ax^2 2ax 3x^2+6x ...

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Byju's Answer

Standard XII

Mathematics

Roots of a Quadratic Equation

The largest i...

Question

The largest integral value of $a$ for which every solution of the equation $x ([x] - 5) + 2 {x} + 6 = 0$ satisfies the inequality $(a - 3) x^{2} + 2 (a + 3) x - 8 a \leq 0,$ where $[y]$ and ${y}$ denote the greatest integer and fractional part functions respectively, is

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Solution

$x [x] - 5 x + 2 x - 2 [x] + 6 = 0$
$\Rightarrow [x] (x - 2) - 3 (x - 2) = 0$
$\Rightarrow ([x] - 3) (x - 2) = 0$
$\Rightarrow x = 2 or [x] = 3$
$\Rightarrow x \in [3, 4) \cup {2}$

$(a - 3) x^{2} + 2 (a + 3) x - 8 a \leq 0$
$\Rightarrow a x^{2} - 3 x^{2} + 2 a x + 6 x - 8 a \leq 0$
$\Rightarrow a x^{2} - 2 a x - 3 x^{2} + 6 x + 4 a x - 8 a \leq 0$
$\Rightarrow a x (x - 2) - 3 x (x - 2) + 4 a (x - 2) \leq 0$
$\Rightarrow (x - 2) [x (a - 3) + 4 a] \leq 0$

Case $I :$ $a - 3 > 0 \Rightarrow a > 3$
$\frac{- 4 a}{a - 3} \geq 4$
$\Rightarrow - 4 a \geq 4 a - 12$
$\Rightarrow 8 a \leq 12$
$\Rightarrow a \leq \frac{3}{2}$ No solution.

Case $II :$ $a - 3 < 0$
$\frac{- 4 a}{a - 3} \leq 2$
$\Rightarrow - 4 a \geq 2 a - 6$
$\Rightarrow 6 \geq 6 a$
$\Rightarrow a \leq 1$
$∴$ Greatest integral value of $a$ is $1$ .

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The largest integral value of a for which every solution of the equation x([x]−5)+2{x}+6=0 satisfies the inequality (a−3)x2+2(a+3)x−8a≤0, where [y] and {y} denote the greatest integer and fractional part functions respectively, is

The largest integral value of $a$ for which every solution of the equation $x ([x] - 5) + 2 {x} + 6 = 0$ satisfies the inequality $(a - 3) x^{2} + 2 (a + 3) x - 8 a \leq 0,$ where $[y]$ and ${y}$ denote the greatest integer and fractional part functions respectively, is