8.f = , plot the curve. Hin. Through the quadratic form, transform it to principal axes, and express [x₁  x₂] in terms of the new coordinate vector [y₁ y₂]. Please try specifying each dimension of your plotted conic section. (10 分)

4. Use the Laplace transform to solve the following ODE (10 分)  y(0)=0, y'(0)=-1

5. Use the Laplace transform to solve the following ODE (10 分) y" +2y'+5y = 25t-1000(t-π),y(0)=-2,y'(0)=5

6. LetFind the resultant matrices of the following expressions or give reasons why they are undefined. Each calculation result should be given in a step-by-step way. (a) BTA；(b) (3A -2B)T (e)aBaT；(d)3AT-2BT (10分)

7.Find the det(A) = ?, det(A x B)=?, and =?.

9. Find the complex Fourier Series of f(x)=ex if-p＜x ＜pandf(x+2p)=T(x).(10分)

10. Find the trigonometric polynomial F(x) of the form:for which the square error with respect to the given f(x) = x2 on the interval -π ＜x ＜ π is minimum. Compute the minimum value for N = 1, 2, ..., 5. (10 分)

1.Suppose that X and Yare two continuous random variables with the joint probability density function /(x.y) - x + 2y for 0 ＜x＜2 and 0 ＜y ＜ 2. The covariance of Xand Y is________.

2.A random variable X has a mean p = 5 and variance σ2 = 36. Based on Chebyshev's inequality, a lower bound on Pr(-7＜X＜ 17)is____________.

3.Suppose that X is a father's height, Y is a mother's height, Z is a child's ultimate height (as an adult), and G is delined as: G = I if the child is a boy and G = -I ifthe child is a girl. Let M = (X+ Y)/2. Suppose that, conditional on known values of G, X, and Y, Z has a continuous uniform distribution with range [M -5 +3G, M + 5 + 3G]. If a girl's father is 4 inches taller than her mother, the probability that she will grow to be taller than her mother is__________.

4.A manufacturer of car tires claims that his tires will last, on the average, 4 years with a standard deviation of 1 year. Given that 5 of these tires have lifetimes of 2.1, 2.2, 3.0, 3.6, 4.1 years, if someone wants to test whether his tires have a standard deviation of1 year, the figure of the appropriate test statistic is equal to__________.