110年 - 110 國立中央大學_碩士班招生考試_光電類：工程數學#105768

1.For an equation given as , what is the relationship between x and y?  (AB) x⁴-4x-1 =y⁴-2y²+3log|y|(AC)x⁴-4x-1=y³-2y²+41og|y| (AD)x⁴-2x-1=y⁴-2y²+41oglyl (AE)x⁴-4x-2 =y⁴-2y²+4logly|(BC)x³-4x-1 =y⁴-2y²+4log|y|(BD)x⁴-4x-1 =y⁴-2y²+4loglyl (BE) x⁴-4x-1 =y⁴-2y³+4loglyl.

2. If = 2y,with y(0) =1, what is y(x)?(AB)ex(AC)(AD)(AE) 2e^x(BC)(BD) (BE) 1

3. lf and y=10 when x =5, what isywhen x = ?(AB)5 (AC)9 (AD)(AE)(BC)8 (BD)7 (BE)6

4. Ifandy= I when x = 2, what is y when x = 1?(AB)1.14(AC)-1.14 (AD) 2.14 (AE) -2.24 (BC) 3.14 (BD) -3.14 (BE) None of the above.

5. For the differential equation: (s2+4v)(dy/dx) + (2xy+1) = 0, what is the function P(x.y) that satisfies the relationship: ?(AB)P(x，y)=x²y+x+2y²(AC)P(x，y)=x²+x+2y²  (AD)P(x，y)=x²y+x+y² (AE)P(x，y)=xy+x+2y²(BC)P(x，y)=2x²y+x+2y²(BC)P(x，y)=xy²+x+2y² (BE) P(xy)=x²y+2y²

6. If P(f,t) is the general solution of the function:, what is P(2,1)?(AB)2.31(AC)3.31 (AD)0 (AE)1.31(BC)4.31 (BD)5.31 (BE)6.31

7.If with y(0) =2, what is y(π)? (AB)5.1 (AC) 1.1 (AD) 2.1 (AE)6.1(BC) 7.1 (BD) 3.1 (BE) 4.1

8. If = 2y, with y(0) = -1, what is y(0.5)?(AB) 0.368 (AC) 2.718 (AD) 5.436 (AE) 0.736 (BC) 8. 1 55 (BD) 1.104 (BE) 1.472

9. The amount, A(t), of carbon 14 in an archaeological sample decays exponentially with time, and A(t) can be estimated by the equation: , where C and k are constants. If the half-life of carbon 14 is 5570 years, what is k?(AB) (AC) (AD) (AE) (BC) (BD) (BE)

10.If= -4x+2y and y = 1 when x =0, what is x wheny= 3?(AB)-3(AC)2 (AD)1 (AE)0 (BC)-2 (BD)3(BE) None of the above

Part B.
Solving the following problems(50%)(每一大題10分,請務必寫出計算過程)
(11) (10pt) Find the rank of A:

(12) (10pt) Calculate the matrix function, where the matrix M is defined as

(13)(10pt )For a quadratic form Q = x^TAx, if for any nonzero we always have Q ＞0, then this quadratic form is positive definite.

 For n=3 and , is the quadratic form Q=x ^TAx positive definite? 

(b) (5pt) Find the unit normal vector of the surface at an arbitrary point.

15. (10pt) Find the Fourier transform of the function: