mathc192comp - Find the principal value of h e 2-1-√ 3 i...

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BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI II SEMESTER, 2005-2006 MATH C192: Mathematics II Comprehensive Examination Date: December 10, 2005 Time: 3 hrs. Day : Saturday Max. Marks: 40 Q1(a). Show that T : V 3 V 3 defined by T ( x 1 , x 2 , x 3 ) = ( x 1 + x 2 + x 3 , x 3 + x 2 , x 3 ) is nonsingular and find its inverse. [3] Q1(b). Let T : V 3 V 3 a linear map defined by T ( e 1 ) = e 1 - e 2 , T ( e 2 ) = 2 e 2 + e 3 , and T ( e 3 ) = e 1 + e 2 + e 3 where e 1 , e 2 , and e 3 are the standard bases for V 3 . Find Null Space N ( T ), rank r ( T ) and nullity n ( T ). [4] Q2(a). Let W 1 be a subspace of a vector space V. Let U ( 6 = 0) V be such that u 6∈ W 1 . Let W 2 = [ { u } ]. If dimW 1 = 5, dimV = 10, then find dim ( W 1 + W 2 ). [2] Q2(b). Let U and W be two distinct ( n - 1)-dimensional subspace of an n - dimensional vector space V . Then prove that dim ( U W ) = n - 2 . [3] Q3(a). Find all the value of z for which sin z = 3+ i 2 . [4] Q3(b). Find the principal argument Arg z when z = - 2 1+ 3 i . [2] Q3(c).
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Unformatted text preview: Find the principal value of h e 2 (-1-√ 3 i ) i 3 πi [2] Q4(a). State the Cauchy integral formula. [2] Q4(b). Show that if C is the boundary of the triangle with vertices at the point 0 , 3 i , and-4 oriented in the counterclockwise direction, the [3] | Z C ( e z-¯ z ; dz |≤ 60 Q4(c). Evaluate Z 1+ i ( x-y + ix 2 ) dz ; along the real axis from z = 0 to z = 1 and along the parallel to imaginary axis from z = 1 to z = 1 + i . [4] Q5. Use residues to evaluate the integrals ( i ) Z C tan z dz, where C is the positively oriented circle | z | = 2. ( ii ) Z ∞ x sin2 xdx x 2 + 3 ( iii ) Z 2 π dθ 1 + a sin θ (-1 < a < 1) [4+5+5]...
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