Đề thi môn Hàm phức và phép biến đổi laplace năm học 2014-2015 (Mã đề thi: 1001-060-209)

ĐỀ THI MÔN: HÀM PHỨC VÀ PHÉP BIẾN ĐỔI LAPLACE<br /> MÃ MÔN HỌC: 1001060<br /> THỜI GIAN: 75 PHÚT<br /> NGÀY THI: 04/06/2015<br /> Đề thi gồm 02 trang bao gồm 10 câu hỏi trắc nghiệm và 3 câu hỏi tự luận<br /> (Được phép sử dụng tài liệu)<br /> MÃ ĐỀ THI: 1001-060-209<br /> <br /> PHẦN TRẮC NGHIỆM LỰA CHỌN (5 ĐIỂM)<br /> Câu 1: Cho hàm số u (x , y ) = ax + e x cos (ay ). Xác định hằng số phức a sao cho u(x , y ) là phần thực<br /> của một hàm giải tích trên ℂ .<br /> A. a = 1 hoặc a = 2<br /> C. a = 0<br /> <br /> B. a = 1 hoặc a = −1<br /> D. Không tồn tại a<br /> Câu 2: Giả sử hàm gốc f (t ) có ảnh là F (p ) , L  f (t ) = F (p ) . Hãy chọn phát biểu ĐÚNG:<br /> <br /> <br /> F ( p − 3)<br /> <br /> p<br /> A. L e t f (3t ) = F  <br />  <br /> <br /> 3<br /> <br /> <br />  <br /> <br /> B. L e 3t * f (t ) =<br /> <br /> <br /> <br /> t<br />  F ( p − 3)<br /> C. L  ∫ e 3u f (u )dt  =<br /> p −3<br />  0<br /> <br /> <br /> D. L e 3t f (t ) = F (p − 3)<br /> <br /> <br /> <br /> Câu 3: Tìm biến đổi Laplace L te −2t sin (5t ) :<br /> <br /> <br /> A. L te −2t sin (5t ) =<br /> <br /> <br /> C. L te −2t sin (5t ) =<br /> <br /> <br /> <br /> 10 (p + 2)<br /> B. L te −2t sin (5t ) =<br /> 2<br /> <br />  <br /> 2<br /> <br /> (p − 2) + 25<br /> <br /> <br /> <br /> <br /> <br /> 10p − 20<br /> <br /> (p<br /> <br /> 2<br /> <br /> )<br /> <br /> 2<br /> <br /> − 4 p + 29<br /> 10p + 20<br /> <br /> (p<br /> <br /> 2<br /> <br /> 10p − 20<br /> D. L te −2t sin (5t ) =<br /> 2<br /> <br />  <br /> 2<br /> <br /> ( p + 2) + 25<br /> <br /> <br /> <br /> <br /> <br /> )<br /> <br /> + 4 p + 29<br /> <br /> p<br /> <br /> 2<br /> <br /> Câu 4: Cho hàm f (z ) có khai triển Laurent tại trong lân cận của điểm z = 0 là:<br /> <br /> <br /> <br /> <br /> 22n<br /> 1<br /> .<br /> <br /> +<br /> f (z ) = ∑ (−1) <br /> <br /> (2n )! z 2n +1 (2n )! z 2n <br /> <br /> <br /> <br /> n =0<br /> <br /> <br /> ∞<br /> <br /> Tính tích phân I =<br /> <br /> ∫<br /> <br /> n<br /> <br /> z 5 f (z )dz .<br /> <br /> |z |=2<br /> <br /> A.<br /> <br /> 8πi<br /> 5!<br /> <br /> B. −<br /> <br /> Câu 5: Cho hàm phức f (z ) =<br /> <br /> (<br /> <br /> 2πi<br /> 6!<br /> <br /> C.<br /> <br /> ( ) . Tìm phần thực Re<br /> <br /> z Re e z<br /> <br /> )<br /> <br /> (<br /> <br /> (<br /> <br /> )<br /> <br /> D. Re f (z )<br /> sin πz<br /> <br /> z (2z − 1)<br /> 2<br /> <br /> xe x cos y<br /> =−<br /> y<br /> <br /> . Hãy chọn phát biểu ĐÚNG:<br /> <br /> <br /> 1<br /> <br /> <br /> <br /> A. Res f (z ), 0 = −π và Res  f (z ),  = 2<br /> <br /> <br /> 2<br /> <br /> <br /> )<br /> <br /> )<br /> <br /> B. Re f (z ) = e x cos y<br /> <br /> xe x cos y<br /> C. Re ( f ( z ) ) =<br /> y<br /> <br /> (<br /> <br /> ( f ) với z = x + iy .<br /> <br /> Im (z )<br /> <br /> A. Re f (z ) = −e x cos y<br /> <br /> Câu 6: Cho hàm phức f (z ) =<br /> <br /> 4<br /> 1<br /> <br /> <br /> D. 2πi  − <br /> <br /> <br />  5! 6! <br /> <br /> <br /> 2πi<br /> 6!<br /> <br /> <br /> <br /> 1<br /> <br /> <br /> B. Res f (z ), 0 = −π và Res  f (z ),  = 4<br /> <br /> 2<br /> <br /> <br /> <br /> (<br /> <br /> )<br /> <br /> Trang 1/6 - Mã đề thi 1001-060-209<br /> <br /> <br /> 1<br /> <br /> C. Res f (z ), 0 = −πi và Res  f (z ),  = 2<br /> <br /> <br /> <br /> <br /> 2<br /> <br /> <br /> (<br /> <br /> <br /> 1<br /> <br /> D. Res f (z ), 0 = 2 và Res  f (z ),  = −π<br /> <br /> <br /> <br /> <br /> 2<br /> <br /> <br /> )<br /> <br /> (<br /> <br /> Câu 7: Biến đổi Laplace ngược nào sau đây là SAI?<br /> <br /> <br /> 1<br />  = e 2t − et<br /> A. L −1  2<br />  p − 3p + 2 <br /> <br /> <br /> <br /> <br /> <br /> <br />  2p − 1 <br /> 1<br /> <br />  = e t 2 cos (2t ) + sin (2t )<br /> <br /> C. L −1 <br /> <br /> 2<br /> <br /> <br /> <br /> <br /> <br /> 2<br /> <br /> <br />  (p − 1) + 4 <br /> <br /> B. L<br /> <br /> )<br /> <br /> −1<br /> <br /> 3<br />  2<br /> t<br /> 3 <br /> t<br /> 2<br /> <br /> −<br /> = 2e − 3e<br />  p − 1 2p + 3 <br /> <br /> <br /> <br />  3p − 2 <br />  = 3 cos (3t ) − 2 sin (3t )<br /> D. L −1  2<br />  p + 9<br /> 3<br /> <br /> <br /> <br /> 1<br /> Câu 8: Khai triển Laurent của hàm f (z ) = (2z + 1) cos   trong lân cận của điểm z = 0 là:<br />  <br /> <br /> z <br />  <br /> <br /> <br /> <br /> <br /> ∞<br /> ∞<br /> n<br /> n<br /> 2<br /> 1  1<br /> 2<br /> 1  1<br /> <br /> <br /> A. ∑ (−1) <br /> B. ∑ (−1) <br /> +<br />  2n<br /> +<br /> <br /> <br /> <br /> (2n + 2)! (2n )!  z<br /> (2n + 2)! (2n )!  z 2n<br /> <br /> <br /> <br /> <br /> <br /> <br /> n =0<br /> n =0<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> ∞<br /> ∞<br /> n<br /> n<br /> <br /> 2<br /> 1 <br /> 2<br /> 1<br /> <br /> <br /> <br /> <br /> C. ∑ (−1) <br /> D. ∑ (−1) <br /> +<br /> <br /> +<br /> <br /> (n + 1)! z 2n −1 n ! z 2n <br /> (2n )! z 2n −1 (2n )! z 2n <br /> <br /> <br /> <br /> <br /> <br /> <br /> n =0<br /> n =0<br /> <br /> <br /> <br /> <br /> t<br /> <br /> Câu 9: Tìm ảnh của hàm gốc e 2t * ∫ sin (3u )du :<br /> 0<br /> <br /> A.<br /> <br /> C.<br /> <br /> 3<br /> <br /> ( p − 2 )( p<br /> <br /> 2<br /> <br /> B.<br /> <br /> )<br /> <br /> +9<br /> <br /> 1<br /> 1<br /> 3<br /> +<br /> + 2<br /> p p −2 p + 9<br /> <br /> D.<br /> <br /> 3<br /> <br /> (<br /> <br /> )<br /> <br /> p ( p − 2) p 2 + 9<br /> <br /> 1<br /> 3<br /> +<br /> p − 2 p p2 + 9<br /> <br /> (<br /> <br /> )<br /> <br /> 3<br /> <br /> Câu 10: Cho hàm phức f (z ) =<br /> A.<br /> B.<br /> C.<br /> D.<br /> <br /> z<br /> z<br /> z<br /> z<br /> <br /> (<br /> <br /> ez<br /> <br /> z z 2 + 6z + 18<br /> <br /> )<br /> <br /> . Hãy chọn phát biểu SAI:<br /> <br /> = 0 là cực điểm cấp 2<br /> = −3 + 3i và z = −3 − 3i là các điểm bất thường cô lập<br /> = −3 − 3i là cực điểm cấp 1<br /> = −3 + 3i là cực điểm cấp 1<br /> <br /> PHẦN TỰ LUẬN (5 ĐIỂM)<br /> Câu 11 (1.5 điểm). Áp dụng phép biến đổi Laplace giải phương trình vi phân sau:<br /> y′′ + y = tet + 1 với điều kiện y ( 0 ) = y′ ( 0 ) = 0.<br /> Câu 12 (2.0 điểm). Áp dụng phép biến đổi Laplace giải phương trình tích phân:<br /> t<br /> <br /> y + e2 t * ∫ y ( u ) du = t + e 2t .<br /> 0<br /> <br /> Câu 13 (1.5 điểm). Cho hàm phức f ( z ) = ze<br /> <br /> 3<br /> z −1<br /> <br /> .<br /> <br /> a) Khai triển Laurent hàm f trong lân cận của điểm z = 1.<br /> b) Sử dụng kết quả này tính tích phân I =<br /> <br /> ∫<br /> <br /> f ( z ) dz.<br /> <br /> | z − i|= 3<br /> <br /> Trang 2/6 - Mã đề thi 1001-060-209<br /> <br /> .........................................................................................................................................................................<br /> .........................................................................................................................................................................<br /> .........................................................................................................................................................................<br /> 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