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Let u be a real valued harmonic function on ℂ. Let g:ℝ2 → ℝ be defined byWhich of the following statements is TRUE
g is neither harmonic nor a polynomial
g is a harmonic polynomial
g is a polynomial but not harmonic
g is harmonic but not a polynomial
Which of the following statements are TRUE?P: If f∈L1(ℝ), then F iscontinuous.
Q: If f∈L1(ℝ) and lim|x|→∞f(x) exists, then the limit iszero.
R: If f∈L1(ℝ), then f isbounded.
S: If f∈L1(ℝ) is uniformly continuous, then lim|x|→∞f(x) exists and equals zero
P and Q only
Q and S only
R and S only
P and R only
The image of the region {z ∈ℂ∶ Re(z)>Im(z)>0} under the mapping z↦
(ez)2 is
{w∈ℂ∶ Im(w)>0,|w|>1}
{w∈ℂ∶ |w|>1}
{w∈ℂ∶ Re(w)>0, Im(w)>0}
{w∈ℂ∶ Re(w)>0, Im(w)>0,|w|>1}
Let M be the real vector space of 2*3 matrices with realentries.
Let T:M → M be defined byThe determinant of T is ______
-2
-1
1
Let x0 = 0. Define xn+1 = cos xn for every n≥0. Then
{xn} is not convergent
{xn} is decreasing and convergent
{xn} is convergent and x2n < limm-->∞ xm < x2n+1 for every n ∈ ℕ
{xn} is increasing and convergent
Which of the following is a field
ℤ[x]/〈x2+2〉
ℝ[x]/〈x2-2〉
ℂ[x]/〈x2+2〉
ℚ[x]/〈x2-2〉
The value of the integralis ______
8
4
2
6
Let f:ℝ →ℝ be a continuous function with f
(1)=5 and f
(3)=11. If g(x) = then g′(0) is equal to ______
3
4
5
6
The coefficient of (z − π)2 in the Taylor series expansion ofaround π is
1/2
-1/6
1/6
-1/2
Consider the Fredholm integral equation .The solution of this integral equation is
x2/(1 - λ2)
x/(1 - λ)
x/(1 + λ2)
(x + 1)/(1 - λ)
Consider the system ofequations.
Using Jacobi’s method with the initial guess [x(0) y(0) z(0)]T = [2.0 3.0 0.0]T, the approximate solution [x
(2) y
(2) z
(2)]T after two iterations, is
[2.64 -1.70 1.12]T
[2.64 1.70 -1.12]T
[2.64 1.70 1.12]T
[2.64 -1.70 -1.12]T
The number of 5-Sylow subgroup(s) in a group of order 45 is
3
4
1
2
The function Ø(x) satisfying the integral equation is
1 + x2/2
x - x2/2
x + x2/2
x2/2
If a transformation y = uv transforms the given differential equation f(x)y'' - 4f'(x)y' + g(x)y = 0 into the equation of the form v'' + h(x)v = 0, then u must be
xf
F2
1/f2
1/2f
The function u(r, θ) satisfying the Laplace equationsubject to the conditions u
(e, θ) = 1, u
(e2 ,θ) = 0 is
ln(e/r2)
.
ln(e2/r)
ln(e/r)
Let R = ℤ*ℤ*ℤ and I = ℤ*ℤ*{0}. Then which of the following statement is correct
I is a maximal ideal but not a prime ideal of R .
I is a prime ideal but not a maximal ideal of R .
I is both maximal ideal as well as a prime ideal of R .
I is neither a maximal ideal nor a prime ideal of R .
Let α = e2πi/5 and the matrixThen the trace of the matrix I + M + M2 is
-5
3
5
Let f(x) and xf(x) be the particular solutions of a differential equation y'' + R(x)y' + S(x)y = 0. Then the solution of the differential equation y'' + R(x)y' + S(x)y = f(x) is
y = [x3 + αx + β] f(x)
y = [x2/2 + αx + β] f(x)
y = [-x2 + αx + β] f(x)
y = [-x2/2 + αx + β] f(x)
Suppose that R is a unique factorization domain and that a,b ∈R are distinct irreducibleelements.
Which of the following statements is TRUE
The ideal 〈a+b〉 is a prime ideal
The ideal 〈1+a〉 is a prime ideal
The ideal 〈1+ab〉 is a prime ideal
The ideal 〈a〉 is not necessarily a maximal ideal
The order of the smallest possible non trivial group containing elements x and y such that x7 = y2 = e and yx = x4y is
7
1
14
2
1
2
3
5
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