CAT 2018

Slot 2

QA

Question & Solution

Let f(x) = $2x^2−ax+2$. We can see that f(x) is a quadratic function. 

For, f(x) > 0, Discriminant (D) < 0

$\Rightarrow$ $(-a)^2-4*2*2<0$

$\Rightarrow$ (a-4)(a+4)<0

$\Rightarrow$ a $\epsilon$ (-4, 4)

Therefore, integer values that 'a' can take = {-3, -2, -1, 0, 1, 2, 3}

Let g(x) = $x^2−bx+8$. We can see that g(x) is also a quadratic function. 

For, g(x)≥0, Discriminant (D) $\leq$ 0

$\Rightarrow$ $(-b)^2-4*8*1<0$

$\Rightarrow$ $(b-\sqrt{32})(b+\sqrt{32})<0$

$\Rightarrow$ b $\epsilon$ (-$\sqrt{32}$, $\sqrt{32}$)

Therefore, integer values that 'b' can take = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

We have to find out the largest possible value of $2a−6b$. The largest possible value will occur when 'a' is maximum and 'b' is minimum. 

a$_{max}$ = 3, b$_{min}$ = -5

Therefore, the largest possible value of $2a−6b$ = 2*3 - 6*(-5) = 36.