Let denote the set of all real numbers. Let a i , b i for i { 1 , 2 , 3 } . Define the functions f : , g : , and h : by [2025]

Question

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

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

$f (x) = a_{1} + 10 x + a_{2} x^{2} + a_{3} x^{3} + x^{4},$

$g (x) = b_{1} + 3 x + b_{2} x^{2} + b_{3} x^{3} + x^{4},$

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

If $f (x) \neq g (x)$ for every $x \in ℝ$ , then the coefficient of $x^{3}$ in $h (x)$ is [2025]

Smart Achievers · Accepted Answer

(3)

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

$= a_{1} + 10 (x + 1) + a_{2} {(x + 1)}^{2} + a_{3} {(x + 1)}^{3} + {(x + 1)}^{4} - b_{1} - 3 (x + 2) - b_{2} {(x + 2)}^{2} - b_{3} {(x + 2)}^{3} - {(x + 2)}^{4}$

$Coeff. of x^{3} in h (x) = a_{3} - b_{3} - 4; f (x) - g (x) \neq 0, \forall x \in ℝ$

$\Rightarrow a_{1} + 10 x + a_{2} x^{2} + a_{3} x^{3} + x^{4} - b_{1} - 3 x - b_{2} x^{2} - b_{3} x^{3} - x^{4} \neq 0$

$\Rightarrow x^{3} (a_{3} - b_{3}) + x^{2} (a_{2} - b_{2}) + 7 x + (a_{1} - b_{1}) \neq 0$

Cubic eq. will become zero at least one value of $x$

So, it will be quadratic

$\Rightarrow a_{3} - b_{3} = 0$

$∴$ $Coefficient of x^{3} in h (x) is - 4 .$

Solution

1 8

2 2

3 - 4

4 - 6

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Solution

Q. Let ℝ denote the set of all real numbers. Let ai,bi∈ℝ for i∈{1,2,3}. Define the functions f:ℝ→ℝ, g:ℝ→ℝ, and h:ℝ→ℝ by f(x)=a1+10x+a2x2+a3x3+x4, g(x)=b1+3x+b2x2+b3x3+x4, h(x)=f(x+1)-g(x+2). If f(x)≠g(x) for every x∈ℝ, then the coefficient of x3 in h(x) is [2025]

1 8

2 2

3 - 4

4 - 6

Want Us to Email you About Special Offers & Updates?