Q243 / 270 · NDA 2021 (Phase 1) · Mathematics

If the function $f(x)=\left\{\begin{array}{ll}a+b x, & x \lt 1 \\ 5, & x=1 \\ b-a x, & x \gt 1\end{array}\right.$ is continuous, then what is the value of $(a+b)$ ?

  • Correct answer
    A
    $5$
  • B
    $10$
  • C
    $15$
  • D
    $20$

Solution

$f(x)=\left\{\begin{array}{lc}a+b x, & x \lt 1 \\ 5, & x=1, \text { is continuous } \\ b-a x, & x \gt 1\end{array}\right.$ $\begin{aligned} & \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=f(1) \\ & \lim _{x \rightarrow 1^{-}}(a+b x)=\lim _{x \rightarrow 1^{+}}(b-a x)=5\end{aligned}$ $a+b=5\ldots$ (i) and $b-a=5$.\ldots(ii) From (i) and (ii) $b=5$ and $a=0$ $\therefore a+\mathrm{b}=0+5=5$

Hence, the correct answer is option: 'A'.

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