a) the sum of their amplitudes b) the difference of their amplitudes c) the product of their amplitudes d) the quotient of their amplitudes ? K?

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1 (! ) 1. The amplitude of a complex number a) 0 b) 8/: c) 8 d) :8 2.The amplitude of the quotient of two complex numbers a) the sum of their amplitudes b) the difference of their amplitudes c) the product of their amplitudes d) the quotient of their amplitudes 3. The =+B then C = a) : +B : b) : +B : c) : B : d) imaginary 4. If & : are any two complex numbers, then K :. : a) K : : b) K :. c) K : : d) : 5.MNO : Q:+B : R : = a) S b) S c) T d) U 6.The equation +U + U =W represents a) Circle b) ellipse c) parabola d) hyperbola 7.The equation +Z = Z represents a) Circle b) ellipse c) parabola d) Real ax 8.The equation + = : represents a) Circle b) ellipse c) parabola d) hyperbola 9.The equation +: + : =U represents a) Circle b) a st. line c) parabola d) hyperbola 10.The equation : = : represents a) Circle b) a st. line c) parabola d) hyperbola 11.The equation : + : : =U represents a) Circle b) a st. line c) parabola d) hyperbola 12.The equation!_` b=c represents a) Circle b) a st. line c) parabola d) hyperbola 13.The equation!_` b=8/d represents K 1

2 j a) Circle b) a st. line c) parabola d) hyperbola h, W 14.. Let f()=g j and W, =W _()=k W a) both f(z) & g(z) are continues at the origin d (K)aB d (a) b) both f(z) & g(z) are not continues at the origin : KB :, W,=W c) f(z) continues at the origin & g(z) not continues at the origin d) f(z) not continues at the origin & g(z) continues at the t origin 15.The multiplicative identity if C a)(0,0) b) (1,1) c) (0,1) d) (1,0) 16. n K8 S K8 S K 8 S a8 S S o = a) 1+i b) 1-i 1 i c) 1 d) If q a complex number such that q : +q+=w then q d = a) q b) q : c) 0 d) If p a complex number & r = then s ar s tra a) i b) -i i c) 1 d) MNO. :`: au b = : a) : b) U c) d) If & : are the image of the complex plane of two diametrically opposite point on the Rieman s sphere then CCCCCCC : a) 0 b) 1 c) -1 d) vw! cb

3 : ( vb f ) 1.. The function w= : K : a analytic a) for all z b) for all z except 1,-1 1 c) for all z except, d) for all z except +: 2.. Which of the following not an entire function a) b) a c) : d)mz{ 3.. Which of the following called C-R C equation for the function f()=+ to be analytic at z a) = B & B = b) = B & B = c) = B & B = d) = & B = B 4.. Which of the following called C-R C equation for polar coordinates a)! = & =! b)! =! &! =! c)! =! &! =! d)!!= &! =! 5.Define f()= z 0 then a) MNO W f() does not exts b) f(z) continues at z=0 b) f(z) does not continues at z=0 d) MNO W f()= 6.For the function f()=+ to be differentiable a) necessary & sufficient b) necessary & not sufficient b) not necessary & sufficient d) none 12.For the function f()=+ to be differentiable,which of the following true a), B, B, exts b) = B & B = holds b), B, B, exts & continuous d) all, W 7.. Let f()=g j then, =W f(z) at z=0 a) f(z) continues b) f(z) not continues c) f(z) well d) none 8.For the function f(z).a point W said to be singular point if a) f(z) analytic at W b) f(z) not analytic at W c) f(z) continues at W n) none 9.For the function f()= (K:)d (:)() : which of the following true a) z = -2 a zero of order 3 b) a) z = 1 a pole of order 2 c) a) z = 2 a pole of order d) all 10.For the function f()=c 3

4 a) entire function b) differentiable at z =2 c) a) z = 2 a pole of order d) all 11. If f()=q : B : R+ B+ differentiable at every point,the constants a) 2b=a b) 4b=a c) 2a=b d) a=b 12.The function f()= : a) differentiable at z = 0 b) no where differentiable c) not harmonic d) differentiable at z 0 13.If f()=+ analytic & =:( B) then f()= a) :+ : b) : : c) + : /: d) : /: 14.If f()=+ analytic & =:B then f()= a) U + b) : + c) + d) : /: 15.. If f()=q : +B : :BR+Q : B : +:BR analytic,then the value v of a, b are respectually a) -1,1 b) -1,2 c) 1,-1 c) 2, If f()=q : B : R+ B+ differentiable at every point,the constants a) 2b=a b) 4b=a c) 2a=b d) a=b 17.. If =:+ : B : harmonic, then the value of m a) 3 b) 2 c) 0 d) If the function f()=,+= K a) B : KB : b) : KB : K : KB : (K) : KB :+ B c) : KB : (K) : KB : (K) : KB :+ B d) (K) : KB : 19.Any two harmonic conjugates of a given harmonic function u(x,y) differ by a) x b) y c) xy d) constant 20.. The image of the strip 0<x<1 under the transformation w=iz a) W<ƒ<1 b) W< <1 c) <ƒ <2 d) < <2 d) Q: +ZR+B vw! cb

5 d ( h f!_ ) 1.. The radius of convergence of ` b! a) b) c) / d) 2.. The radius of convergence of ` K b a) b) c) / d) K 3.. The radius of convergence of ` (K:)(Kd) b a) b) c) / d) 4.. The radius of convergence of `!: :! b a) b) c) / d) U 5.. The radius of convergence of `(a) b( :) a) b) c) / d) 6.. The radius of convergence of (d+u) a) b) c) /Z d) U 7.. The radius of convergence of `+ b: a) b) c) / d) W 8.The invariants points of w= Z a) 3,-5 b) 3,-3 3 c) 3,5 d) -3, 3,-5 9.The invariants points of w= du a) :, : b) W, c) ± d) ±d 10.The invariants points s of w= :a a) :, : b) W, c) ± d) ±d K 11.The invariants points of w= a) :, : b) W, c) ± d) ±d 12.The invariants points of w= : 5

6 a) :, : b) W, c) ± d) ±d K 13.The invariants points of w= dz a) :, : b) W, c) ±: d) ±d 14.The invariants points of w= K :K a) ± : b) ± d c) ± : d) ± d 15.The transformation w=+ said to be a) Translation b) Magnification c) Rotation d) Inversion 16.The transformation w= said to be a) Translation b) Magnification & Rotation c) ) Inversion d) none 17.The transformation w=/ said to be a) Translation b) Magnification c) Rotation d) Inversion & Reflection 18.The curve & their images under w=/ are given below, state which of the following true 1) Circle not thro 0 into circle not thro 0 2) St. line not thro 0 into circle thro 0 3) Circle thro 0 into St.line not thro 0 4) St.line thro 0 into St.line thro 0 a) 1,2,3 b) 2,3,4 c) 1,2,3,4 d) 3,4 19.The angle of rotation on at z = 1+i under the map w= : a) 45 b) 0 c) 90 d) The image of the circle d =under the map d w=/ a) circle b) st.line c) square d) rectangle vw! cb

7 U (!f! & Œ Ž ) 1.. The image of the strip 0<x<1 under the transformation w=iz a) W<ƒ<1 b) W< <1 c) <ƒ <2 d) < <2 2.. The image of the strip 0 < x <1 under the transformation w= a) >1 b) = c) W< <1 d) ) < <0 3. The image of the strip 0 < x <1 under the transformation : + : =š a) : + : =/š b) =/š c) =/š d) none 4.. The image of the strip 0 < y <1 under the transformation w=/ a) : + : +>0 b) : : :>0 c) : + : >0 d) none 5.. The image of the strip 0 < y <1/2c under the transformation w = / a) : + : +:>0 b) : : :>0 c) : + : >0 d) none 6.. The transformation w = / transforms a) circle not through the origin in the z- z plane into the circle through the origin in the w-plane w b) circle through the origin in the z- z plane into the st.line passing through the origin in the w-plane w c) st.line not through the origin in the z- z plane into the circle through the origin in the w-plane w d) st.line through the origin in the z- z plane into the st.line passing through the origin in the w-plane w 7.The transformation w= transforms the imaginary ax in the Z ax into a) a unit circle with center at origin in the W plane b) a circle of radius 2 with center at origin in the W plane c) a straight line in the W plane d) an ellipse in the W plane 8.The bilinear transformation that map the points =W, : =, d = w =,w : = :,w d = a)w= :K K: b) w= :K : c) w= a:k K: d) w= a:k : 9.The bilinear transformation that map the points =, : =, d =W w =W,w : =,w d = a)w=/ b) w= / c) w= d) w= 10.The bilinear transformation that map the points =W, : =, d = w =,w : =,w d =W 7

8 a)w=() K b) w=( ) K c) w=() K d) w=( ) K 11. If f() an entire function & if œ f() =W then a) z=a lies inside C b) z=a lies on the boundary of C c) z=a lies outside of C d) none 12.. If f() analytic with in & on C & z=a lies inside C, then œ f() a) 0 b) :8f() c) 8f() d) :8 13.If f(z) analytic in a simple closed curve C,then œf() a) 0 b) :8f() c) 8f() d) :8 14.If C the positively oriented circl =:,then œ : a a) 0 b) :8 c) 8f() d) :8 15. If C the unit circle =, then œ (C) a a) 0 b) :8 c) + d) 16.. If C the unit circle =, then œc a) 0 b) 8 c) :8 d) d If C a strigh line from z=0 to z=4+2i, then œc a) 0 b) W+ S d c) W S d d) S+ W d 18.. If C the unit circle =, then œ : a) 0 b) :8 c) 8 d) d If C the unit circle =, then œ a) 0 b) :8 c) 8 d) d If C the unit circle =, then œ a a) 0 b) :8 c) 8 d) d8 vw! cb

9 Z ( ž_!) 1. If C the unit circle =, then œ a) 0 b) :8 c) 8 d) d8 2.. If C the unit circle =, then œ : au a) 0 b) :8 c) 8 d) d8 3.. If C the unit circle =, then œ : K:K: a) 0 b) :8 c) 8 d) d8 4.. If C the unit circle =, then œ : d a) 0 b) :8 c) 8 d) d8 5. If C the unit circle =, then œ : d a) 0 b) :8 c) 8 d) d8 6. If C the circle : =d, then œ a) 0 b) :8 c) :8 : d) :8 a: 7.. If C the circle : =d, then œ a) 0 b) :8 c) :8 : d) :8 a: 8.. If C the circle =:, then œ : (:) a) 0 b) :8 c) :8 : d) :8 a 9.. If C the circle =:, then œ Ÿ ` 8 : b: a) 0 b) :8 c) :8 : d) :8 a: 10.. If C the circle =:, then œ Ÿ : (:) a) 0 b) 8/: c) :8 : d) :8 a 11.. If C the circle =, then œ : KZ d a) 0 b) :8 c) :8 d) :S8 9

10 12.. If C the circle + =, then œ d: KK K a) 0 b) :8 c) d8 d) T If C the circle =:, then œ d: KdZ a) 0 b) :8 c) d8 d) T8 26. If C the circle =, then œ : KZKT : a) 0 b) :W8 c) :8 d) UW If C =h<1 then œ : K : a a) 0 b) :8 c) 8 d) : If C the circle =, then œ Z a) :8/Z b) 8/: c) :8 : d) :8 a: 16.. If C the circle =d, then œ Ÿ : (:) a) :8/Z b) 8/: c) :8 : d) :8 a: 17.. If C the circle =, then œ d a) :8/Z b) 8/: c) :8 : d) If C the circle =:, then œ U (:) a) :8/Z b) 8/d c) :8 : d) 8 19.If C a closed curve enclosing the origin, then the value of œ a) 0 b) :8 c) :8 d) U If C = then œ : a) 0 b) :8 c) 8 d) :8 (a)! vw! cb

11 T ( h ) 1.. If C the positively oriented circle =:, then œ : KU a) 8 : b) 8 : Q: : R c) 8 : Q: R d) :8 : 2.The poles of z cot z are a) :8 b) 8 c) (:+)8 d) (: )8 3.The residue of f()= K :(:) at z=2 a) 1/5 b) 1/4 c) 1/3 d) 3/2 4.The residue of f()= U d (:) at a simple pole a) 1/5 b) 1/4 c) 1/3 d) 1/2 5.The residue of f()= at z=0 a) 1/5 b) 1/4 c) 1/3 d) -1/2 6.The residue of f()= at =8/: a) 1/5 b) 1/4 c) 1/3 d) 7.The residue of f()= K are : a: a) 1/2 b) /: c) d/: d) d/: 8.The residue of f()= d a at z=0 a) 1/5 b) 1/4 c) 1/3 d) 3 9.The residue of f()= K : (:) at z=0 a) 1/5 b) 1/4 c) 1/3 d) -3/4 10.The residue of f()= K: at : : at =: a) 1/5 b) 1/4 c) 5/3 5 d) 11.The residue of f()= N at () : at = a) cos 1 b) sin 1 c) 1/3 d) 12.The residue of f()= / at =W 11

12 a) 1/5 b) 1/4 c) 1/3 d) /: 13.. If C the circle =d/:, then œ 8: K8 : ()(:) a) 0 b) :8 c) :8 : d) :8 a 14.. If C the circle =:, then œ d U (K) a) 0 b) 8/: c) 8/ d d) :8 a 15.. If C the circle =:, then œ Kdd : a a) 0 b) U8 c) :8 : d) :8 a 16. If C the circle =:, then œ d K a) 0 b) U8 c) :8( ) d) :8 a 17.The singular point of f()= K are :( : K) a) 0,i, -i i b),n, N c) W,N d), N 18. For f()= : () d ; Z=1 a a) Removable singularity b) Pole of order 2 c)simple pole 19.. For f()=( U) (Kd) d) Essential singularity ; Z= -3 a a) Removable singularity b) Pole of order 1 c)simple pole d) Essential singularity 20.. For f()= ( d ; Z= 0 a a) Removable singularity b) Pole of order 3 c)simple pole d) Essential singularity 21. The function f()=,=w a) not a singular point b) a removable singularity c) simple pole d) essential singularity 22.The Taylor s expansion of n N about the origin a) + : + U :! U! + b) : :! +U U! c) ++: + d :! d! + d) d d! +Z Z! 12

13 23. Maclaurin s expansion d + Z d! Z! a) b) c) Mz{(+) d) N 24.In,the series ` :b a) convergent but not uniform b) divergent c) uniformly convergent d) neither convergent nor divergent 25.. in the open dc <1, then the series ++ : + d + a) uniformly convergent b) convergent c) divergent d) absolutely convergent 26.The Taylor s expansion of n ªz about the origin a) + : + U :! U! + b) : :! +U U! c) ++: + d :! d! 27.. The Taylor s expansion of n Mz{ (+) about the origin + d) d d! +Z Z! a) + : : +d d + b) : : +d d c) : : d d d) d d! +Z Z! 28.Tailar s series for 1/z 1 about z = 1 a) ++ : + d + b) +( )+( ) : + c) ( )+( ) : d) +! +: :! + 29.The expansion ()(:) = `+ + : + b : n+: +`: b: + o a) not a valued function b) valued in < <2 c) valued in <1 d) valued in >2 30.The only singularity of a single value function f(z) are poles of order 1,2 & at z = -1 & z = 2 with residue at these poles 1 & 2 resp. f(w)= & U «()=Z : a) + + : + d b) K : (:) : b) U + + : K : c) the function f(z) + d K d) (:) : d) + : + d : (:) : vw! cb

14 ( B!! ) 1. The amplitude of a complex number a) 0 b) 8/: c) 8 d) :8 2.The amplitude of the quotient of two complex numbers a) the sum of their amplitudes b) the difference of their amplitudes c) the product of their amplitudes d) the quotient of their amplitudes 3. The =+B then C = a) : +B : b) : +B : c) : B : d) imaginary 4. If & : are any two complex numbers, then K :. : a) K : : b) K :. c) K : : d) : 5. The function w= : K : a analytic a) for all z b) for all z except 1,-1 1 c) for all z except, d) for all z except +: 6. Which of the following not an entire function a) b) a c) : d)mz{ 7. If C =h<1 then œ : K : a a) 0 b) :8 c) 8 d) :8 8. If f() an entire function & if œ f() =W then a) z=a lies inside C b) z=a lies on the boundary of C c) z=a lies outside of C d) none 9. If f() analytic with in & on C & z=a lies inside C, then œ f() a) 0 b) :8f() c) 8f() d) :8 10. Which of the following called C-R C equation for the function f()=+ to be analytic at z a) = B & B = b) = B & B = c) = B & B = d) = & B = B 11.The Taylor s expansion of n ªz about the origin 14

15 a) + : + U :! U! + b) : :! +U U! c) ++: + d :! d! 12. The Taylor s expansion of n Mz{ (+) about the origin + d) d d! +Z Z! a) + : : +d d + b) : : +d d c) : : d d d) d d! +Z Z! 13. If & : are the image of the complex plane of two diametrically opposite point on the Rieman s sphere then CCCCCCC : a) 0 b) 1 c) -1 d) 14. in the open dc <1, then the series ++ : + d + a) uniformly convergent b) convergent c) divergent d) absolutely convergent 15. If C the unit circle =, then œ : d a) 0 b) :8 c) 8 d) d8 16. If C the circle =d, then œ : a) 0 b) :8 c) :8 : d) :8 a: 17.MNO : Q:+B : R : = a) S b) S c) T d) U 18. The radius of convergence of ` b! a) b) c) / d) 19. The image of the strip 0<x<1 under the transformation w=iz a) W<ƒ<1 b) W< <1 c) <ƒ <2 d) < <2 20. If C the circle =, then œ Z a) :8/Z b) 8/: c) :8 : d) :8 a: 21. If C the circle + =, then œ d: KK K a) 0 b) :8 c) d8 d) T8 22.The equation +U + U =W represents ents a) Circle b) ellipse c) parabola d) hyperbola 23. Maclaurin s expansion d + Z d! Z! 15

16 j a) b) c) Mz{(+) d) N 24.The invariants s points of w= Z a) 3,-5 b) 3,-3 3 c) 3,5 d) -3, 3,-5 25.The bilinear transformation that map the points =, : =, d =W w =W,w : =,w d = a)w=/ b) w= / c) w= d) w= 26. If C the circle =, then œ : KZKT : a) 0 b) :W8 c) :8 d) UW8 27.The Taylor s expansion of n N about the origin a) + : + U :! U! + b) : :! +U U! 28.In,the series ` :b c) ++: + d :! d! + d) d a) convergent but not uniform b) divergent c) uniformly convergent d) neither convergent nor divergent h, W 29. Let f()=g j and W, =W _()=k W d (K)aB d (a) : KB :, W,=W a) both f(z) & g(z) are continues at the origin b) both f(z) & g(z) are not continues at the origin c) f(z) continues at the origin & g(z) not continues at the origin d) f(z) not continues at the origin & g(z) continues at the origin 30.If f()=+ analytic & =:( B) then f()= a) :+ : b) : : c) + : /: d) : /: 31.The multiplicative identity if C a)(0,0) b) (1,1) c) (0,1) d) (1,0) d! +Z Z! 32. n K8 S K8 S K 8 S a8 S S o = a) 1+i b) 1-i 1 i c) 1 d) If q a complex number such that q : +q+=w then q d = a) q b) q : c) 0 d) 1 16

17 34. If p a complex number & r = then s ar s tra a) i b) -i c) 1 d) If C a closed curve enclosing the origin, then the value of œ a) 0 b) :8 c) :8 d) U8 36.If f()=+ analytic & =:B then f()= a) U + b) : + c) + d) : /: 37. If f()=q : +B : :BR+Q : B : +:BR analytic,then the value of a, b are respectually a) -1,1 b) -1,2 c) 1,-1 1 c) 2,-1 : 38.The residue of f()= U d (:) at a simple pole a) 1/5 b) 1/4 c) 1/3 d) 1/2 39. If C the circle =, then œ : KZ d a) 0 b) :8 c) :8 d) :S8 40.The transformation w= transforms the imaginary ax in the Z ax into a) a unit circle with center at origin in the W plane b) a circle cle of radius 2 with center at origin in the W plane c) a straight line in the W plane d) an ellipse in the W plane 41.The bilinear transformation that map the points =W, : =, d = w =,w : = :,w d = a)w= :K K: b) w= :K : c) w= a:k K: d) w= a:k : 42.The invariants points of w= K :K a) ± : b) ± d c) ± : d) ± d 43.The poles of z cot z are a) :8 b) 8 c) (:+)8 d) (: )8 44. If =:+ : B : harmonic, then the value of m a) 3 b) 2 c) 0 d) 1 45.The residue of f()= K :(:) at z=2 17

18 a) 1/5 b) 1/4 c) 1/3 d) 3/2 46..MNO. :`: au b = : a) : b) U c) d) If f()=q : B : R+ B+ differentiable at every point,the constants a) 2b=a b) 4b=a c) 2a=b d) a=b 47.The angle of rotation at z = 1+i under the map w= : a) 45 b) 0 c) 90 d) Tailar s series for 1/z about z = 1 a) ++ : + d + b) +( )+( ) : + c) ( )+( ) : d) +! +: :! + 49.The function f()= : a) differentiable at z = 0 b) no where differentiable c) not harmonic d) differentiable d at z If the function f()=,+= K a) B : KB : b) : KB : K : KB : (K) : KB :+ B c) : KB : (K) : KB : (K) : KB :+ B d) (K) : KB : 51.The image of the circle d =under the map d w=/ a) circle b) st.line c) square d) rectangle 52. If C = then œ d) Q: +ZR+B a) 0 b) :8 c) 8 d) :8 (a)! 53.The singular point of f()= K are :( : K) a) 0,i, -i i b),n, N c) W,N d), N 54.Any two harmonic conjugates of a given harmonic function u(x,y) differ by a) x b) y c) xy d) constant 55. The function f()=,=w a) not a singular point b) a removable singularity c) simple pole d) essential singularity 53.The residue of f()= K are : a: 18

19 a) 1/2 b) /: c) d/: d) d/: 54. The radius of convergence of `+ b: a) b) c) / d) W 55.The bilinear transformation that map the points =W, : =, d = w =,w : =,w d =W a)w=() K b) w=( ) K c) w=() K d) w=( ) K 56. If C the positively oriented circle =:, then œ : KU a) 8 : b) 8 : Q: : R c) 8 : Q: R d) :8 : 57. The transformation w = / transforms a) circle not through the origin in the z- z plane into the circle through the origin in the w-plane w b) circle through the origin in the z- z plane into the st.line passing through the origin in the w-plane w c) st.line not through the origin in the z- z plane into the circle through the origin in the w-plane w d) st.line through the origin in the z- z plane into the st.line passing through the origin in the w-plane w 58.The only singularity of a single value function f(z) are poles of order 1,2 & at z = -1 & z = 2 with residue at these poles 1 & 2 resp. r f(w)= & U «()=Z : a) + + : + d b) K : (:) : b) U + + : K : c) the function f(z) + d K d) (:) : d) + : + d : (:) : 59.The expansion ()(:) = `+ + : + b : n+: +`: b: + o a) not a valued function b) valued in < <2 c) valued in <1 d) valued in >2 vw! cb

Multivariable Calculus

Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation