QQ号码

我的QQ号码是一个10位数，下面共有十道试题，每道题的答案都是一个一位数字。假设第 i 道题的答案是 ni​，则我的QQ号码为：

QQ号码 = \overline{n_1 n_2 \ldots n_{10}}

只有您正确解答了所有题目，才可能获取我的联系方式。

  1. 设函数f(x) = x2e-x , 求函数f(x) 在区间 [0, +∞) 上的最大值所对应的x的值。
  2. Given a 3×3 real symmetric matrix A with eigenvalues λ1 = 1, λ2 = 3, and λ3 = 3, find the determinant of matrix A.
  3. Given that ai are positive integers for i = 1, 2, 3, … , 9. For any natural number k where 1< k < 9, exactly one of the following conditions holds:
    ak = ak-1 + 1, or
    ak = ak+1 – 1
    Additionally, a1 = 6 and a9 = 9.
    Let m = \sum_{i=1}^9 a_i, If the minimum value of m is mmin , then find the tens digit of mmin.
  4. 已知实数x, y满足:
    x ≤ 3
    2x – y -2 ≥ 0
    3x + y -8 ≥ 0
    求z = x – y 的最大值。
  5. Given that the domain of the function f(x) is [0, +∞), and it satisfies f(x) = f( 1\over {x+1} ), with the range of the function being A.
    If the set { y | y = f(x), x ∈ [0, a] } can take all values of A. Find the minimum value of a, denoted as a0 , and find the second decimal place of a0
  6. Given the parabola y2 = 2px where p > 0.
    Let points A and B lie on the parabola in the first quadrant, and let F be the focus of the parabola.
    It’s known that |AF| = 2, |BF| = 4, and |AB| = 3; the slope of line AB is k.
    Find 4k2

  7. As shown in the figure, a sufficiently long metal rod slides horizontally on rails with a distance L = 0.5m between them. The resistance of the rod is not negligible. On the left side, there is a power source with an electromotive force E = 3V, and the system is located in a uniform magnetic field directed perpendicularly downward. After the switch S is closed, the rod with mass m = 0.1 kg begins to move from rest. The length of the rod is equal to the distance between the rails. The current in the circuit gradually decreases over time, and eventually reaches a steady state. At this steady state, the induced electromotive force across the rod equals E, and the current in the circuit becomes zero. During the motion from the moment the switch is closed to when the rod reaches the steady state, the total energy supplied by the power source is Es =10J. The heat generated in the power source and the metal rod is Q1 = 0.5J and Q2 = 4.5J, respectively. Find the ratio between the resistance R of the metal rod and the internal resistance r of the power source, i.e. R/r
  8. In Iron pentacarbonyl, what is the coordination number of the central metal?
  9. 设Pn(xn, yn)是直线 2x – y = n\over {n+1} (n ∈ N*) 与圆 x2 + y2 = 2 在第一象限的交点，求 \lim_{{x \to \infty}} \frac{{ 1 - y_n}}{{x_n - 1}}
  10. Let V be a six-dimensional vector space, and let T: V→V be a linear transformation satisfying T3 = T2 and T2 ≠ T. Then find the number of distinct eigenvalues of T.