What do you mean by active power, reactive power and power factor in electrical systems?

Explain resistance, inductance and capacitance. Name various electrical appliances and mention the resistive, inductive and capacitive type of load these belong.

Give one example each for the following. Justify your choice of examples.

i) A sequence which is divergent.

ii) A set which is neither open nor closed.

iii) A compact set.

iv) A set which has no limit point

Find the value/s of x for which the series 

is convergent.

Evaluate  using Riemann integration

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

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.

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

∃ x∈R such that x-1/3 > 0.

 Show that [a,∞)is closed set.

Find the supremum and infimum of the set:

Disprove the statement:

by providing a suitable counter-example.

 Prove that a function f : S → S (where S is a finite non-empty set) is injective if it is surjective.

Test the following series for convergence: