NOTE

Question 1 (Paper - 1)

Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in \{1, 2, 3\}$.

Define the functions $f: \mathbb{R} \to \mathbb{R}$, $g: \mathbb{R} \to \mathbb{R}$, and $h: \mathbb{R} \to \mathbb{R}$ by:

$$f(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4$$$$g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4$$$$h(x) = f(x + 1) - g(x + 2)$$

If $f(x) \ne g(x)$ for every $x \in \mathbb{R}$, then the coefficient of $x^3$ in $h(x)$ is:

$$\begin{array}{|c|c|} \hline \text{(A)} & 8 \\ \hline \text{(B)} & 2 \\ \hline \text{(C)} & -4 \\ \hline \text{(D)} & -6 \\ \hline \end{array}$$

Finding the Coefficient of $x^3$ in $h(x)$

Given

Let the functions be defined as:

$$f(x) = a_1 + 10x + a_2 x^2 + a_3 x^3 + x^4$$ $$g(x) = b_1 + 3x + b_2 x^2 + b_3 x^3 + x^4$$ $$h(x) = f(x + 1) - g(x + 2)$$

We are told that $f(x) = g(x)$ for all $x \in \mathbb{R}$, so:

• $a_1 = b_1$
• $a_2 = b_2$
• $a_3 = b_3$
• Constant $x^4$ terms are equal (both have coefficient 1)

Step 1: Expand $f(x + 1)$

Using binomial expansion:

$$(x + 1)^3 = x^3 + 3x^2 + 3x + 1$$ $$(x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1$$

So,

$$f(x + 1) = a_1 + 10(x + 1) + a_2(x + 1)^2 + a_3(x + 1)^3 + (x + 1)^4$$

We only need the coefficient of $x^3$:

• From $a_3(x + 1)^3$: contributes $a_3 \cdot x^3$
• From $(x + 1)^4$: contributes $4x^3$

So total contribution to $x^3$ in $f(x + 1)$ is:

$$a_3 + 4$$

Step 2: Expand $g(x + 2)$

Using binomial expansion:

$$(x + 2)^3 = x^3 + 6x^2 + 12x + 8$$ $$(x + 2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16$$

So,

$$g(x + 2) = b_1 + 3(x + 2) + b_2(x + 2)^2 + b_3(x + 2)^3 + (x + 2)^4$$

We only need the coefficient of $x^3$:

• From $b_3(x + 2)^3$: contributes $b_3 \cdot x^3$
• From $(x + 2)^4$: contributes $8x^3$

So total contribution to $x^3$ in $g(x + 2)$ is:

$$b_3 + 8$$

Step 3: Coefficient of $x^3$ in $h(x)$

Since:

$$h(x) = f(x + 1) - g(x + 2)$$

The coefficient of $x^3$ is:

$$(a_3 + 4) - (b_3 + 8) = a_3 - b_3 - 4$$

Given $a_3 = b_3$, we have:

$$a_3 - b_3 - 4 = -4$$

✅ Final Answer

TIP

Correct option: C

The coefficient of $x^3$ in $h(x)$ is $-4$.