QQ號碼

我的QQ號碼是一個10位數，下面共有十道試題，每道題的答案都是一個一位數字。假設第 i 道題的答案是 ni​，則我的QQ號碼為：

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

只有您正確解答了所有題目，才可能獲取我的聯繫方式。

  1. 設函數f(x) = x2e-x , 求函數f(x) 在區間 [0, +∞) 上的最大值所對應的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. 已知實數x, y滿足:
    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/rCircuit diagram: rod on rails in a magnetic field
  8. In Iron pentacarbonyl, what is the coordination number of the central metal?
  9. 設Pn(xn, yn)是直線 2x – y = n\over {n+1} (n ∈ N*) 與圓 x2 + y2 = 2 在第一象限的交點，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.