1 . R ad i u s of c o nve r g e n c e of ∞ n = 0 a n z n i s ( a) L i m n → ∞ | a n | +1 n (b) Lim n→∞ |an | -1 n (c) |an | -1 n (d) |an | 1 n 2 . R ad i u s of c o nve r g e n c e of M a c l au r i n s s e r i e s of 1 ( z 2 + 1 ) ( z + 2 ) i s ( a) - 1 ( b ) 0 ( c ) 2 ( d ) 1 3 . T h e L au r e nt S e r i e s e x p an s i on of 1 z - 1 a b ou t z = 0 va l i d i n | z | < 1 i s ( a) - ∞ n = 0 z n ( b ) ∞ n = 0 z n ( c ) ∞ n = 1 1 z n ( d ) ∞ n=0 1 z n 4 . T h e L au r e nt s e r i e s e x p a n s i on of z ( z - 1 ) ( 2 - z ) va l i d i n | z - 1 | > 1 i s ( a) - 2 ∞ n = 1 ( z - 1 ) - n ( b ) - ( z - 1 ) - 1 - 2 ∞ n = 2 ( z - 1 ) -n (c) -(z - 1)-1 - 2 ∞ n=2 (z - 1)n (d) (z - 1)-1 +2 ∞ n=2 (z - 1)-n 5 . z = 0 f o r s i n z z i s ( a) s i m p l e p o le ( b ) N o t a s i n g u l ar p o i nt ( c ) E s s e nti a l s i n gu l a r ity ( d ) R e m ovab l e s i n gu l a r i ty 6 . R e s i d u e of e 2 z ( z ) 2 ( z 2 + 2 z + 2 ) a t z = 0 i s . ( a) 1 ( b ) 1 / 2 ( c ) -1/2 (d) ∞ 7 . T h e r e s i d u e o f 1 - e 2 z z 4 a t z = 0 i s . . ( a) - 4 / 3 ( b ) - 2 ( c ) - 3 / 4 (d) -2/3 8 . ∞ 0 d x ( 1 + x 2 ) ( x 2 + 4 ) 2 = ( a) 5 π 2 8 8 ( b ) 5 π 1 8 8 ( c ) 3 π 2 8 8 (d) π 288 9. 2π
0 c o s 2 θ 5 + 4 c o s θ d θ = ( a) 3 π 2 ( b ) π 6 ( c ) π 2 ( d ) π 1 2
1 0. ∞ 0 S i n x x d x i s f ou n d u s i n g C e iz z dz whereCisthe b o u n d a ry of . . ( a) | z | < R → ∞ & I m( z ) ≥ 0 ( b ) 0 < ε < | z | < R → ∞ (c) 0 ← ε< |z |
|z |=π f
( z ) f ( z ) d z = ( a) 1 2 π i (
1 2. N u mb e r o f z e r os of e z - 4 z 6 + 1 i n s i d e | z | = 1 ( a) O n e ( b ) s i x ( c ) e i g ht ( d ) z e r o 1 3. T h e M a xi mu m val u e o f | C o s 3 z | i n | z | ≤ 1 i s ( a) C a n ‘ t b e d e t e r mined (b) e3 +e-3 2 (c) Cos3 (d) 1 1 4. Fu n d a m e nt al T h e o re m o f a l ge b r a i s ( a) A n nt h d e gr e e p ol y n o m i al e q u at i o n h a s a t l e a s t o n e r o ot w h e r e n ≥ 1 ( b ) T h e nu mb e r of i m a gi n a ry ro o t s o f a n nt h d e g r e e p ol y n o m i al e q u at i o n i s n ( c ) T h e nu mb e
r of R e al r o o t s o f a n nt h d e gr e e Re a l p ol y n o m i al e qu a t i on i s n ( d ) T h e nu mb e r of d i s t i n c t r o ot s of a n nt h d e g r e e p o l y n om i a l e qu a t i on i s n 1 5. I f f i s a n al y t i c a n d b o u n d e d by M i n | z - a | = R t h e n ( a) | f (a)|≤ R M (b) |f (a)|≤ M R (c) |f (a)|≥ M R (d) |f (a) |≥ R M 1 6. T h e m ag n i t u d e o f a n g l e o f r o ta t i on i nvo l ve d i n f ( z ) = ( 1 - i ) z - ( 4 +2 i ) ( a) 5 π 4 ( b ) 3 π 4 ( c ) 7 π 4 ( d ) π 4 1 7. I f u +i v = w = f ( z )= l n z [ z = r e i θ ] t h e n u , v r e s p e c t i ve l y ar e . . . . . . . . . . . . . . . . . . ( a) l n r ; θ ( b ) θ ; l n r ( c ) 1 /r ; θ ( d ) l n ( 1/ r ) ; θ 1 8. w = C o s z m a p s L i n e s p ar a l l e l t o X a x i s t o ( a) hy p e r b o l as ( b ) e l l i p s e s ( c ) L i n e s p ar a l l e l t o X a x i s ( d ) L i n e s p ar a l l e l t o Y a x i s 1 9. u + i v= w = 1 z m a p s t h e x an d y ax e s i n z p l an e t o . . . . . . . . . . . . . . . . i n w - p l an e ( a) C i r c l e s t ou ch i n g u an d v ax e s r e p e c t i ve l y ( b ) v a n d u ax e s re s p e c t i ve l y ( c ) u a n d v ax e s re s p e c t i ve l y ( d ) C i r c l e s t ou ch i n g v a n d u ax e s 2 0. A B i l i n e a r t ra n s f o r m at i o n m ap p i n g u p p e r h al f p l a n e t o u n i t d i s c s u ch t h a t z = i i s m a p p e d t o w = 0 an d z = ∞ t o w = - 1 ( a) w = i + z i - z (b) w = i-z i+z (c) w = z +1 z -1 (d) w = i z -1 z +1 ANS:B,D,A,B,D,B,A,A,,B,C,B,B,B,A,B,C,A,B,C,B.