20 questions
If T1z = (z+2)/(z+3) and T2z = z/(z+1) then T1T2z
(3z+1)/(4z+3)
(z+2)/(4z+3)
(3z+2)/(z+3)
(3z+2)/(4z+3)
Cross ratio is
real
invariant under linear transformations
four distinct points
linear transformation
If 2π is the smallest positive period of eiz then eiπ is
e0
1
-1
½
If ∑anzn and ∑bnzn have radii of convergence R1 and R2, then radius of convergence ∑anbnzn is
R1R2
at most R1R2
at least R1R2
R12R22
Radius of convergence of ∑npzn
modulus of eiπ
1
e0
e-1
Radius of convergence of ∑ (n!/nn) zn and ∑ zn/n!
0 and 1/e
∞ and 1
∞ and 1/e
1/0 and e-1
Any linear transformation which transforms the real axis into itself can be written with real coefficient
True
False
Jordan arc is
z(t1) = z(t2) for t1 = t2
z(t1) = z(t2)
simple
z(t1) = z(t2) only for t1 = t2
The reflection z → complement of z is a linear transformation
True
False
If the cross ratio (z1, z2, z3, z4) is real then
four points lie on a circle
four points lie on a straight line
(z1, z2, z3, z4) = complement of (z1, z2, z3, z4)
All the above
Let {an} be a sequence of real numbers.
1) {an} is convergent
2) {an} is a Cauchy sequence
1 implies 2
2 implies 1
1 implies 2 is always true
2 implies 1 need not always be true
Another name(s) for linear transformation
Mobius transformation
bilinear transformation
linear fractional transformation
singular transformation
Every linear transformation is not a conformal map
True
False
Symmetric point of a(center of C) with respect to C is
a
0
∞
R2 + a
If S(w) = (az+b)/(cz+d) and T(w) = (-dw+b)/(cw-a) then S-1 =
ST
TS
T
T-1
Elementary linear transformations are
Dilation
Contraction
Expansion
Homothetic
An arc z = z(t) is rectifiable if and only if
real part of z(t) is of bounded variation
z(t) is a piecewise differentiable arc
all the above
none of the above
The arc length of a circle equals
its perimeter
its circumference
2πr
2πi
True
False
If S is a linear transformation, then S(-d/c) is
a/c
0
∞
(-ad+bc)/0