Evaluate:

Solve the equation:

x² +15x³ +70x²+120x+64=0

given that its roots are in G.P.

Find the largest subset of R on which the function f: R→R defined as:

is continuous.

If y = e™ sin-¹ x, then show that (1-x²)y₂-xy, -m²y = 0. Hence using Leibnitz's formula, find the value of (1-x²)y_n+2-(2n+1)xy _n+1

Find the condition for the curves, ax² + by² =1 and a'x² + b'y² = 1 intersecting orthogonally.

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.

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

f(x) = x³-x²-8x+12

and

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

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

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

b) Evaluate:

c) For any two sets S and T, show that:

Depict this situation in the Venn diagram.

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 

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

Explain the organization of a typical microprocessor with the help of block diagram.

Explain with the help of circuit diagram working of 4-bit R-2R ladder network DAC using op-amp. 

Draw the circuit diagram of monostable multivibrator using an op-amp and explain its operation.

Explain the principle of switch mode power supply with the help of schematic diagram. State its advantages over linear power supply.