Q104 / 160 · TS EAMCET 2017 · Mathematics

If $f(x)=\left\{\begin{array}{cc}\sin x & \text { if } x \leq 0 \\ x^2+a^2 & \text { if } 0 < x < 1 \\ b x+2 & \text { if } 1 \leq x \leq 2 \\ 0 & \text { if } x>2\end{array}\right.$ is continuous on $\mathrm{IR}$, then $a+b+a b=$

  • A
    -2
  • B
    0
  • C
    2
  • Correct answer
    D
    -1

Solution

Given, $$ \begin{aligned} & f(x)=\left\{\begin{array}{cl} \sin x, & \text { if } x \leq 0 \\ x^2+a^2, & \text { if } 0 < x < 1 \\ b x+2, & \text { if } 1 \leq x \leq 2 \\ 0, & \text { if } x>2 \end{array}\right. \text { is continuous on IR, } \\ & \therefore \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x) \\ & \text { and } \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x) \\ & \Rightarrow \lim _{x \rightarrow 0^{-}} \sin x=\lim _{x \rightarrow 0^{-}} x^2+a^2 \\ & \text { and } \lim _{x \rightarrow 2^{-}} b x+2=\lim _{x \rightarrow 2^{+}} 0 \\ & \Rightarrow \quad 0=0+a^2 \text { and } 2 b+2=0 \\ & \Rightarrow \quad a=0 \text { and } b=-1 \\ & \text { Now, } a+b+a b=0+(-1)+0=-1 \\ & \end{aligned} $$

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

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