NIMCET 2008 — Mathematics PYQ

Step 1: Identify the Standard Polynomial Form

There is a well-known result in functional equations for polynomials. If a polynomial $f(x)$ satisfies the equation:

Then the polynomial must be of the form:

where $n$ is a non-negative integer.

Step 2: Determine the Sign and the Value of $n$

We are given that $f(3)=28$. Let's test both possible forms:

• Case 1: $f(x)=1-x^n$

  $$f(3)=1-3^n=28$$
  $$-3^n=27⟹3^n=-27$$

  This has no real integer solution for $n$.

• Case 2: $f(x)=1+x^n$

  $$f(3)=1+3^n=28$$
  $$3^n=27$$
  $$3^n=3^3$$

  Equating the powers, we get $n=3$.

Step 3: Define the Polynomial

Based on the result above, the specific polynomial is:

Step 4: Calculate $f(4)$

Substitute $x=4$ into our polynomial:

Final Answer:

The value of $f(4)$ is $65$.

The correct option is (b).