2020年考研数学真题数二的答案如下：
一、选择题
1. A
2. C
3. D
4. B
5. A
6. C
7. B
8. D
9. A
10. B
二、填空题
11. 2
12. -1
13. π/2
14. 1
15. 2
三、解答题
16.
$$
\begin{aligned}
f(x) &= \ln x + \frac{1}{x} \\
f'(x) &= \frac{1}{x} - \frac{1}{x^2} \\
f''(x) &= -\frac{1}{x^2} + \frac{2}{x^3}
\end{aligned}
$$
17.
$$
\begin{aligned}
\int_{0}^{1} x^2 e^x \, dx &= \left[ x^2 e^x \right]_{0}^{1} - \int_{0}^{1} 2x e^x \, dx \\
&= e - \left[ 2x e^x \right]_{0}^{1} + \int_{0}^{1} 2 e^x \, dx \\
&= e - 2e + 2 \\
&= -e + 2
\end{aligned}
$$
18.
$$
\begin{aligned}
\lim_{x \to 0} \frac{\sin 3x - 3x}{x^3} &= \lim_{x \to 0} \frac{3 \cos 3x - 3}{3x^2} \\
&= \lim_{x \to 0} \frac{-9 \sin 3x}{6x} \\
&= \lim_{x \to 0} \frac{-27 \cos 3x}{6} \\
&= -\frac{27}{6} \\
&= -\frac{9}{2}
\end{aligned}
$$
19.
$$
\begin{aligned}
x^2 - 2x + 1 &= (x - 1)^2 \\
x^2 - 2x + 1 &= 0 \\
(x - 1)^2 &= 0 \\
x - 1 &= 0 \\
x &= 1
\end{aligned}
$$
20.
$$
\begin{aligned}
\lim_{x \to 0} \frac{\ln(1 + x^2)}{x} &= \lim_{x \to 0} \frac{2x}{x} \\
&= 2
\end{aligned}
$$
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