Let ℝ denote the set of all real numbers. Let a_ i , b_ i ∈ ℝ for i ∈\ 1,2,3\ . Define the functions f: ℝ arrow ℝ, g: ℝ arrow ℝ , and h: ℝ arrow ℝ by aligned & f(x)=a_1+10 x+a_2 x^2+a_3 x^3+x^4, & g(x)=b_1+3 x+b_2

Correct answer: C. -4 aligned & h(x)=f(x+1)-g(x+2) & =a_1+10(x+1)+a_2(x+1)^2+a_3(x+1)^3+(x+1)^4-b_1-3(x+2)-b_2(x+2)^2-b_3(x+2)^3-(x+2)^4 aligned Coeff. of x^3 in h(x) aligned & =a_3-b_3-4 & f(x)-g(x) ≠ 0 x ∈ R & a_1+10 x+a_2 x^2+a_3 x^3+x^4-b_1-3 x-b_2 x^2-b_3 x^3-x^4 ≠ 0 & x^3(a_3-b_3)+x^2(a_2-b_2)+7 x+(a_1-b_1) ≠ 0 aligned Cubic Eq. will become zero at atleast are value of x So it will be quadratic a_3-b_3=0

JEE Advanced 2025Paper 1

Exam: JEE Advanced
Year: 2025
Paper: Paper 1
Subject: Mathematics
Chapter: Functions

JEE Advanced 2025 · Paper 1 · Mathematics · Functions

Let ℝ denote the set of all real numbers. Let a_ i , b_ i ∈ ℝ for i...

Let

\mathbb{R}

denote the set of all real numbers. Let

a_{\mathrm{i}}, b_{\mathrm{i}} \in \mathbb{R}

for

\mathrm{i} \in\{1,2,3\}

. Define the functions

f: \mathbb{R} \rightarrow \mathbb{R}, g: \mathbb{R} \rightarrow \mathbb{R}

, and

h: \mathbb{R} \rightarrow \mathbb{R}

\begin{aligned} & f(x)=a_1+10 x+a_2 x^2+a_3 x^3+x^4, \\ & g(x)=b_1+3 x+b_2 x^2+b_3 x^3+x^4, \\ & h(x)=f(x+1)-g(x+2) .\end{aligned}

f(x) \neq g(x)

for every

x \in \mathbb{R}

, then the coefficient of

x^3 in h(x)

Options

Correct answer

C. -4

Step-by-step solution

\begin{aligned} & h(x)=f(x+1)-g(x+2) \\ & =a_1+10(x+1)+a_2(x+1)^2+a_3(x+1)^3+(x+1)^4-b_1-3(x+2)-b_2(x+2)^2-b_3(x+2)^3-(x+2)^4 \end{aligned}

Coeff. of

x^3 in h(x)

\begin{aligned} & =a_3-b_3-4 \\ & f(x)-g(x) \neq 0 \quad \forall x \in R \\ & \Rightarrow a_1+10 x+a_2 x^2+a_3 x^3+x^4-b_1-3 x-b_2 x^2-b_3 x^3-x^4 \neq 0 \\ & \Rightarrow x^3\left(a_3-b_3\right)+x^2\left(a_2-b_2\right)+7 x+\left(a_1-b_1\right) \neq 0 \end{aligned}

Cubic Eq. will become zero at atleast are value of

So it will be quadratic

\Rightarrow \mathrm{a}_3-\mathrm{b}_3=0$

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