If a and b are integers such that 2 {x}^{2}- {ax}+2>0 and {x}^{2}- {bx}+8 \geq 0 for all real numbers {x}, then the largest possible value of 2 {a}-6 {b} is

CAT 2018 Question Paper Shift 2 Solved Paper

Question : 95 of 100

Marks: +1, -0

If a and b are integers such that

2 {x}^{2}- {ax}+2>0

and

{x}^{2}- {bx}+8 ≥ 0

for all real numbers

{x}

, then the largest possible value of

2 {a}-6 {b}

Solution:

2 x^{2}-a x+2>0 ∀ x ∈ R

⇒ ∆<0

⇒ a^{2}-4 × 2 × 2<0

⇒ a^{2} < 16

⇒ \;-4 < a x^{2}-b x+8 ≥ 0

∀ x ∈ R

⇒ b^{2}-4(8) ≤ 0

⇒ -4 √{2} ≤ b ≤ 4 √{2}

As b is integer

-5 ≤ b ≤ 5

Therefore, maximum possible value of

2 a-6 b

2(3)-6(-5)=36

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