# Real And Complex Analysis

20 Questions | Attempts: 58
Share  Settings  • 1.
Which is compact in Rn?
• A.

{(x1,x2,x3,……….xn )   ∶ |xi | ≤ 1,   1 ≤ i ≤n}.

• B.

{(x1,x2,x3,……….xn )   ∶ x1+x2+x3+......+xn=0}.

• C.

{(x1,x2,x3,……….xn )   ∶  xi > 0,   1 ≤ i ≤ n}.

• D.

{(x1,x2,x3,……….xn )   ∶ 1≤ |xi | < 2,   1 ≤ i ≤ n}.

• 2.
If G is an open set and   f : G → C is differentiable function then
• A.

F is analytic on G

• B.

F is non-analytic on G

• C.

F is constant on C

• D.

None of these

• 3.
Listed below are four subsets of C2. Which is bounded in C2.      ( R(z) denotes  real part of z )
• A.

{ (z,w) ∈ C2  :   z 2+w 2 = 1}

• B.

{ (z,w) ∈ C 2   :   | R(z)| 2+ |R(w)| 2=1}

• C.

{ (z,w) ∈ C 2  :  |z| 2+  |w| 2  = 1}

• D.

{ (z,w) ∈ C 2  :  |z| 2 - |w| 2  =1}

• 4.
The image of the region {z ∈ C:  Re (z )>0, Im (z) >0 } under the mapping z → e z^2 is
• A.

{w ∈ C:  Re (w )>0, Im (w) >0 }

• B.

{w ∈ C:  Re (w )>0, Im (w) >0, |w| >1 }

• C.

{w ∈ C:   |w|>1 }

• D.

{w ∈ C:   Im (w) >0,  |w| >1 }

• 5.
Consider the function f(z) = z(1-cosz),   z C (Choose all possible options)
• A.

The function f has zero of order 2 at 0

• B.

The function f has zero of order 1 at  2nπ, n = ±1, ±2, ….

• C.

The function f has zero of order 4 at 0

• D.

The function f has zero of order 2 at  2nπ,  n = ±1, ±2, ….

• 6.
Which is closed in C ?
• A.

{ z ∈ C:   Re z= Im z}

• B.

{z∈ C:   Re z ≠ 0}

• C.

{z∈ C:   |z| ≠0}

• D.

{z∈ C:  Im z ≠ 0}

• 7.
F : D→ D be holomorphic with f(0)= 0,  f(1/2)=0 where D = {z : |z| < 1}. Then   (Choose all possible options)
• A.

| f'(1/2) |  ≤ 4/3

• B.

|f(1/2)| ≤ 1

• C.

| f'(1/2) |  ≤ 4/3  and   |f'(0)|  ≤1

• D.

F(z) = z,   z∈ D

• 8.
Let f(z) = (z - 1) / (e2πi/z -1). Then   (Choose all possible options)
• A.

F has isolated singularity at z = 0.

• B.

F has removable singularity at z=1.

• C.

F has infinitely many poles.

• D.

Each pole of f is of order 1.

• 9.
Suppose f : C C is a holomorphic function such that the real part of f''(z) is strictly positive for all z∈ C. What is the possible number of solutions of the equation f(z)= az + b as a and b vary over C?
• A.

1

• B.

0

• C.

• D.

2

• 10.
Let f : C → C be a non-constant entire function and let  Im (f) = {w∈ C :  z ∈ C such that f(z) = w}. Then
• A.

Then interior of Im (f) is empty

• B.

Im (f) intersects every line passing through the origin

• C.

There exists a disc in complex plane, which is disjoint from Im (f)

• D.

Im(f) contains all its limit points

• 11.
For z ∈ C  of the form z = x+ iy, define  H+ = {z∈ C :  y > 0 } , H - = {z∈ C :  y < 0 },   L+ = {z∈ C :  x> 0 },  L- = {z∈ C :  x < 0 }. The function f(z)= (2z+1)/(5z+3)
• A.

Maps H+ onto H and H- onto H

• B.

Maps H+ onto H- and H- onto H +

• C.

Maps H+ onto L and H- onto L-

• D.

Maps H+ onto L and H- onto L+

• 12.
Let A be a closed subset of  R,  A ≠ ∅ ,  A ≠  R. Then A is
• A.

The closure of the interior of A

• B.

A countable set

• C.

A compact set

• D.

Not open

• 13.
Let I = { 1 } ∪ { 2 } for x ∈ R,  ϕ (x )= dist (x , I) = Inf { |x - y| : y I },  then ϕ is
• A.

Discontinuous somewhere.

• B.

Continuous on R  but not differentiable only at x=1.

• C.

Continuous on R  but  not differentiable only at x=1, 2.

• D.

Continuous on R  but not differentiable only at   x=1, 3 /2 ,2.

• 14.
Let X ⊂  R  be an infinite countable bounded subset of R  then which of the statements is true
• A.

X cannot be compact.

• B.

X contains an interior point.

• C.

X may be closed.

• D.

Closure of X is countable.

• 15.
Let A be a connected open subset of R2.The number of continuous surjective functions from the closure of A in R2  to Q is
• A.

1

• B.

0

• C.

2

• D.

Not finite

• 16.
Let f : X→X  such that f [ f x ) ] = x,  for all  x ∈ X   then
• A.

F is one-to-one and onto.

• B.

F is one-to-one but not onto

• C.

F is onto but not one-to-one

• D.

F need not be either one-to-one or onto

• 17.
Suppose f :  R → R  is a function that satisfies  | f(x) - f(y) | ≤  | x - y | β ,  β > 0    (Choose all possible options)
• A.

If β =1  then f is differentiable.

• B.

If β >0  then f is uniformly continuous .

• C.

If β >1 then f is constant function.

• D.

F must be a polynomial.

• 18.
Which of the following subsets of Ris /are convex?    (Choose all possible options)
• A.

{(x,y) :  x ≤ 5 ,  y ≤ 10 }.

• B.

{(x , y) :  x 2 + y 2 =1}.

• C.

{(x , y) :  y ≥  x 2 }.

• D.

{(x , y) :  y ≤  x 2 }.

• 19.
Let  f(x) : R→ R  be defined by  fn(x) = x / (1+n x2) , n∈ N. Then (Choose all possible options)
• A.

The sequence {fn(x)} converges uniformly on R.

• B.

The sequence {fn(x)}  converges uniformly  on [1, b]  for any b > 1.

• C.

The sequence {fn'(x)}  converges uniformly on R.

• D.

The sequence {fn'(x)}  converges uniformly on  [1,b] for any b > 1.

• 20.
Let  G1 and  G2  be two subsets of R 2 and  f: R 2→ R 2 be a function, then  (Choose all possible options)
• A.

f -1( G1 ∪ G )  = f -1(G1) ∪  f -1( G2 ) .

• B.

f -1(G1 )c = ( f -1( G1 ) ) c .

• C.

f (G 1 ∩ G 2) = f (G 1 ) ∩  f ( G ).

• D.

If G1 is open and G2 is closed then  G1 + G 2 = { x+y  :  x∈ G 1 , y∈ G 2 }  is neither open nor closed.

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