KVPY SX SB Exam 2013 Question Paper

Question : 81 of 120

Marks: +1, -0

Part-II

MATHEMATICS

Let f(x) be a non-constant polynomial with real coefficients such that

f(1/2)=100

and

f(x )< 1 0 0

for all real x. Which of the following statements is NOT necessarily true?

The coefficient of the highest degree term in $f( x )$ is negative.
$f(x)$ has at least two real roots.
If $x ≠1/2$ then $f(x ) < 100$ .
At least one of the coefficients of $f( x )$ is bigger than 50.

Validate

Solution:

Coefficient of highest degree term must be negative becuase if it is positive, then x → ∞, y → ∞ and it is not possible, since f(x) ≤ 100.
Now, graph will be like

at least two real roots will be there, & if

x ≠ 1/2

, then f(x) < 100, it is not always true, as the graph can be like this also

Now, let the highest coefficients, it can have is 49
then,

f(1/2)=49+49/2+49/2^2+...

But the sum cannot be equal to 100.

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