jee-main 2020 Q63

jee-main · India · session2_02sep_shift1 Differentiating Transcendental Functions Determine parameters from function or curve conditions

If a function $f(x)$ defined by $$f(x) = \begin{cases} ae^{x} + be^{-x}, & -1 \leq x < 1 \\ cx^{2}, & 1 \leq x \leq 3 \\ ax^{2} + 2cx, & 3 < x \leq 4 \end{cases}$$ be continuous for some $a, b, c \in R$ and $f'(0) + f'(2) = e$, then the value of $a$ is
(1) $\frac{1}{e^{2} - 3e + 13}$
(2) $\frac{e}{e^{2} - 3e - 13}$
(3) $\frac{e}{e^{2} + 3e + 13}$
(4) $\frac{e}{e^{2} - 3e + 13}$

If a function $f(x)$ defined by
$$f(x) = \begin{cases} ae^{x} + be^{-x}, & -1 \leq x < 1 \\ cx^{2}, & 1 \leq x \leq 3 \\ ax^{2} + 2cx, & 3 < x \leq 4 \end{cases}$$
be continuous for some $a, b, c \in R$ and $f'(0) + f'(2) = e$, then the value of $a$ is\\
(1) $\frac{1}{e^{2} - 3e + 13}$\\
(2) $\frac{e}{e^{2} - 3e - 13}$\\
(3) $\frac{e}{e^{2} + 3e + 13}$\\
(4) $\frac{e}{e^{2} - 3e + 13}$

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