Math Answers

) Let T be a normal operator on a finite dimensional Hilbert space H

with spectrum𝜆1, 𝜆2, … … , 𝜆𝑚 , . Then prove that

i) T is self - adjoint⟺ each 𝜆𝑖,,is real

ii) T is positive ⟺ each 𝜆𝑖 ≥ 0

iii) T is unitary ⟺ 𝜆𝑖 =1 for each .

Prove that 𝑇1 and 𝑇2 are self adjoint operators on a Hilbert space

H, prove that 𝑇1 𝑇2 +𝑇2 𝑇1 is self adjoint

1.Show that an operator T on a Hilbert Space H is unitary iff

T(𝑒𝑖) complete orthonormal set whenever 𝑒𝑖

is.

Let X be a normed linear space and Y a closed subspace of X 

with Y ≠ X ,if 0 < 𝑟 < 1 prove that there exist 

𝑋r element of X such that ||X|| = 1And 𝑟 < 𝑑 𝑋𝑟

, 𝑌 ≤ 1

Find the norm of the linear functional f defined by

𝑓 𝑥 =integral -1 to 0𝑥 𝑡 𝑑𝑡 − integral 0 to 1𝑥 𝑡 𝑑𝑡

where𝑥 ∈ [−1 , 1]

Nalini wants to open a small restaurant stand selling specialty burgers. The fixed

cost component on a daily basis is Rs 2500. For each burger made, the cost is Rs

50. In addition the daily special cost is 0.25x2 where x is the number of burgers she

will make for the day.

a) Write down the expression of the Total Cost function for a day (1 mark)

b) Write down the expression for the Average Cost function for a day

c) How many burgers should she make a day to minimize the cost per burger per

day? ( 4 marks)

d) What is the total cost per day associated with the minimized cost per burger per

day?

e) The price per burger is 250. What is the maximum daily revenue Nalini can earn

provided she decides to make at the minimum unit cost per burger?

f) What is the daily profit she would earn in e) ?

g) How many burgers would she have to sell to break even on a daily basis?

Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an edge in common.

(A).If a population is given as (23 6 8 11), after selecting all possible samples of size 2 with replacement, find out and standard error ofx ̅

(B). Discuss the relationship between x ̅ and μ

b) Persons who visit the restroom of a certain fast-food outlet were asked to state their opinion of the quality of the restroom facilities, The following tables show the responses from a sample of 100 persons. Gender of Respondent Totals Male Female Quality of Facilities Above Average 8 7 15 Average 26 24 50 Below Average 7 28 35 Totals 41 59 100 A 𝜒 2 test is carried out to determine whether there is an association between the gender of persons and their opinion. i) State appropriate null and alternative hypotheses [2] ii) Determine the critical region of the test at the 1% level of significance [1] iii) Calculate the expected value for Male and below average [2] iv) For a test statistic of 9.825, explain with reason, the conclusion of your test. [3] 

Write down 10 uses of Surface Integration.