Let be a polynomial function of second degree and . If are in AP, then are in:

AP Explanation: Solution Step 1: Define the polynomial function Let the second-degree polynomial be: Step 2: Use the given condition So, the function is: Step 3: Find the derivative Differentiating with respect to : Step 4: Evaluate at Given are in Arithmetic Progression (AP), they satisfy: Now, find the values of the derivatives: Step 5: Check the progression of Find the difference between consecutive terms: Since the common difference is constant ( ), the terms are in AP. Final Answer: The correct option is 4 .

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NIMCET 2013 — Mathematics PYQ

NIMCET | Mathematics | 2013

Let $f(x)$ be a polynomial function of second degree and $f(1) = f(-1)$ . If $a, b, c$ are in AP, then $f'(a), f'(b), f'(c)$ are in:

Choose the correct answer:

Explanation

Solution

Step 1: Define the polynomial function

Let the second-degree polynomial be:

$f(x) = Ax^2 + Bx + C$

Step 2: Use the given condition $f(1) = f(-1)$

$A(1)^2 + B(1) + C = A(-1)^2 + B(-1) + C$

$A + B + C = A - B + C$

$2B = 0 \implies B = 0$

So, the function is:

$f(x) = Ax^2 + C$

Step 3: Find the derivative $f'(x)$

Differentiating with respect to $x$ :

$f'(x) = 2Ax$

Step 4: Evaluate at $a, b, c$

Given $a, b, c$ are in Arithmetic Progression (AP), they satisfy:

$b - a = c - b = d \text{ (common difference)}$

Now, find the values of the derivatives:

$f'(a) = 2Aa$

$f'(b) = 2Ab$

$f'(c) = 2Ac$

Step 5: Check the progression of $f'(a), f'(b), f'(c)$

Find the difference between consecutive terms:

$f'(b) - f'(a) = 2Ab - 2Aa = 2A(b - a) = 2Ad$

$f'(c) - f'(b) = 2Ac - 2Ab = 2A(c - b) = 2Ad$

Since the common difference is constant ( $2Ad$ ), the terms $f'(a), f'(b), f'(c)$ are in AP.

Final Answer:

The correct option is 4.

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