Question

If $f\left(x\right)=a{x}^2+bx+c$ satisfies the identity $f(x+1)-f\left(x\right)=8x+3$ for all $xϵR$ . Then $(a,b)=$

• A. $(2,1)$
• B. $(4,-1)$
• C. $(-1,4)$
• D. $(-1,1)$

Answer 1

Answer: (B)

$(4,-1)$

$\begin{array}{c}f\left(x\right)=a{x}^2+bx+c\\ f\left(x+1\right)-f\left(x\right)=8x+3\\ a{\left(x+1\right)}^2+b\left(x+1\right)+c-\left[a{x}^2+bx+c\right]=8x+3\\ a+b+2ax=8x+3\\ \therefore a=1\\ and,a+b=3\therefore b=-1\\ \left(a,b\right)\equiv \left(4,-1\right)\\ Hence,theoptionBisthecorrectanswer.\end{array}$