Express    as a sum of partial fractions.

If z-1+2i|= 4, show that the point z+ i describes a circle. Also draw this circle.

The set R of real numbers with the usual addition (+) and usual multiplication (.) is given. Define (*) on R as:

Is (*) associative in R? Is (.) distributive (*) in R? Check.

Find the domain of the function f given by f(x) = 

Which of the following statements are true or false? Give reasons for your answer in the form of a short proof or a counter-example, whichever is appropriate.

a) The set  {S ∈R :x² - 3x + 2  = 0}is an infinite set.

b) The greatest interger function is continuous on R.

c)  

d) Every integrable function is monotonic.

e) a⊕b = a + b defines a binary operation on Q, the set of rational numbers.

A set which has no limit point

A compact set.

A set which is neither open nor closed.

A sequence which is divergent.

Evaluate  using Riemann integration.

10. a) Find the value/s of x for which the series 

is convergent.

Show that the sequence {f_n}of functions, where 

is uniformly convergent in ,0[ k],where k > .0 Show further that } { n f is not uniformly convergent in ,0[ ∞[.

Show that 

Check whether f is uniformly continuous on [− ]1,1 or not.

Find the radius of convergence of the series .

For x∈ ]2,0[ and n∈N, define f_n (x) = 3 x² + 2x/n. Find the limit function ' f ' of the sequence (f_n ) _n∈N , Is f continuous? Check if  and  are equal or not.

Apply the Cauchy’s integral test to evaluate:

Let f : R → R be a function defined by:

Check whether f ′ is continuous on R.

Prove that between any two real roots of ,2 e cos2x = x there is at least one real root of e^x sin 2x = 1.

Let x and y be two real numbers such that x < y.Show that there exists an irrational number λ such that

x < λ < y.

Show that [a,∞)is closed set.

b) Write the following statement, and its negation, using logical quantifiers. Also interpret its negation in words.

∃ x∈R such that x-1/3 > 0.