Real And Complex Analysis

20 Questions | By Jothimani.k | Updated: Mar 20, 2022 | Attempts: 58

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1.

Which is compact in Rⁿ?
- A.
  
  {(x₁,x₂,x₃,……….x_n ) ∶ |x_i | ≤ 1, 1 ≤ i ≤n}.
- B.
  
  {(x₁,x₂,x₃,……….x_n ) ∶ x₁+x₂+x₃+......+x_n=0}.
- C.
  
  {(x₁,x₂,x₃,……….x_n ) ∶ x_i > 0, 1 ≤ i ≤ n}.
- D.
  
  {(x₁,x₂,x₃,……….x_n ) ∶ 1≤ |x_i | < 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) = z²(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) / (e^2πi/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 R²to 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 R²is /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 ≥ x² }.
- D.
  
  {(x , y) : y ≤ x ² }.
19.

Let f_n(x) : R→ R be defined by f_n(x) = x / (1+n x²) , n∈ N. Then (Choose all possible options)
- A.
  
  The sequence {f_n(x)} converges uniformly on R.
- B.
  
  The sequence {f_n(x)} converges uniformly on [1, b] for any b > 1.
- C.
  
  The sequence {f_n^'(x)} converges uniformly on R.
- D.
  
  The sequence {f_n^'(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}( G_₁ ∪ G ₂) = f ^{^-¹}(G_₁) ∪ f ^{^-¹}( G_₂ ) .
- B.
  
  f ^{^-¹}(G_₁ )^{^c} = ( f ^{^-¹}( G_₁) ) ^{^c .}
- C.
  
  f (G ₁ ∩ G 2) = f (G ₁ ) ∩ f ( G ₂).
- D.
  
  If G_₁ is open and G_₂ is closed then G_₁+ G _₂ = { x+y : x∈ G _₁ , y∈ G _₂ } is neither open nor closed.