Find the length of the cycloid  and show that the line  divides it in the ratio 1: 3. 

Trace the curve y²(x+1)=x²(3-x), clearly stating all the properties used for tracing it.

 If the revenue function is given by dR /dx 15+2x-x², x being the input, find the maximum revenue. Also find the revenue function R, if the initial revenue is 0.

Find the area between the curve y²(4-x) = x³ and its asymptote parallel to y-axis.

Find all the roots of f(x) = 0.

 Find the value of a for which f is continuous at x = -3.

 Find the value of a for which f is continuous at x = -3.

Let f and g be two functions defined on R by:

f(x) = x³-x²-8x+12

and

 For any two sets S and T, show that:

Depict this situation in the Venn diagram.

 Find the least value of a² sec² x + b² cosec²x, where a>0,b>0.

b) Evaluate:

 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 

  defines a binary operation on Q, the set of rational numbers.

Every integrable function is monotonic.

The greatest interger function is continuous on R.

c) 

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

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.

ई- नीलामी का महत्त्व स्पष्ट करें।